# bioLeak: Leakage-Aware Biomedical Modeling

#### Selçuk Korkmaz

#### 2026-04-12

# 1 Why bioLeak

bioLeak is a leakage-aware modeling toolkit for biomedical and machine-learning analyses. Its purpose is to prevent and diagnose information leakage across resampling workflows where training and evaluation data are not truly independent because samples share subjects, batches, studies, or time.

Standard workflows are often insufficient. Random, row-wise cross-validation assumes samples are independent. Global preprocessing (imputation, scaling, feature selection) done before resampling lets test-fold information shape the training process. These choices inflate performance and can lead to incorrect biomarker discovery, misleading clinical signals, or models that fail in deployment.

Data leakage means any direct or indirect use of evaluation data in training or feature engineering. In biomedical data, leakage commonly appears as:

- repeated measurements from the same patient split across folds
- batch or study effects that align with the outcome
- duplicates or near-duplicates across train and test
- time-series lookahead or random splits that use future information
- preprocessing that uses all samples instead of training-only statistics

bioLeak addresses these issues with leakage-aware splitting, guarded preprocessing that is fit only on training data, and audit diagnostics that surface overlaps, confounding, and duplicates.

# 2 Guided workflow

The sections below walk through a leakage-aware workflow from data setup to audits. Each step includes a leaky failure mode and a corrected alternative.

## 2.1 Example data

    library(bioLeak)

    set.seed(123)
    n <- 160
    subject <- rep(seq_len(40), each = 4)
    batch <- sample(paste0("B", 1:6), n, replace = TRUE)
    study <- sample(paste0("S", 1:4), n, replace = TRUE)
    time <- seq_len(n)

    x1 <- rnorm(n)
    x2 <- rnorm(n)
    x3 <- rnorm(n)
    linpred <- 0.7 * x1 - 0.4 * x2 + 0.2 * x3 + rnorm(n, sd = 0.5)
    p <- stats::plogis(linpred)
    outcome <- factor(ifelse(runif(n) < p, "case", "control"),
                      levels = c("control", "case"))

    df <- data.frame(
      subject = subject,
      batch = batch,
      study = study,
      time = time,
      outcome = outcome,
      x1 = x1,
      x2 = x2,
      x3 = x3
    )

    df_leaky <- within(df, {
      leak_subject <- ave(as.numeric(outcome == "case"), subject, FUN = mean)
      leak_batch <- ave(as.numeric(outcome == "case"), batch, FUN = mean)
      leak_global <- mean(as.numeric(outcome == "case"))
    })

    df_time <- df
    df_time$leak_future <- c(as.numeric(df_time$outcome == "case")[-1], 0)
    predictors <- c("x1", "x2", "x3")

    # Example data (first 6 rows)
    head(df)
    #>   subject batch study time outcome          x1         x2          x3
    #> 1       1    B3    S4    1 control  0.21444531  1.0149432 -0.09736927
    #> 2       1    B6    S3    2    case -0.32468591 -1.9927485  0.21615254
    #> 3       1    B3    S1    3    case  0.09458353 -0.4272793  0.88246516
    #> 4       1    B2    S3    4 control -0.89536336  0.1166373  0.20559750
    #> 5       2    B2    S4    5 control -1.31080153 -0.8932076 -0.61643584
    #> 6       2    B6    S2    6    case  1.99721338  0.3339029 -0.73479925

    # Outcome class counts
    as.data.frame(table(df$outcome))
    #>      Var1 Freq
    #> 1 control   91
    #> 2    case   69

The table preview displays the metadata columns (`subject`, `batch`, `study`, `time`), the binary outcome, and three numeric predictors (`x1`, `x2`, `x3`).

The class count table shows the baseline prevalence. Stratified splits try to preserve these proportions (for grouped modes, stratification is applied at the group level).

## 2.2 Create leakage-aware splits with `make_split_plan()`

`make_split_plan()` is the foundation. It returns a `LeakSplits` object with explicit train/test indices (or compact fold assignments) and metadata. For data.frame inputs, a unique `row_id` column is added automatically, so `group = "row_id"` is the explicit way to request sample-wise CV. It assumes that the grouping columns you provide are complete and that samples sharing a group must not cross folds. Grouped stratification uses the majority class per group and is ignored for `study_loocv`, `time_series`, and survival outcomes. Misuse to avoid:

- using `group = "row_id"` (sample-wise CV) when subjects repeat
- using the wrong grouping column (for example batch instead of subject)
- using time-series CV with unsorted or missing time values
- relying on stratification when the outcome is missing, single-class at the group level, or a time/event outcome
- assuming `time_series` will always return exactly `v` folds

Leaky example: row-wise CV when subjects repeat

    leaky_splits <- make_split_plan(
      df,
      outcome = "outcome",
      mode = "subject_grouped",
      group = "row_id",
      v = 5,
      repeats = 1,
      stratify = TRUE
    )

    cat("Leaky splits summary (sample-wise CV):\n")
    #> Leaky splits summary (sample-wise CV):
    leaky_splits
    #> LeakSplits object (mode = subject_grouped, v = 5, repeats = 1)
    #> Outcome: outcome | Stratified: TRUE | Nested: FALSE
    #> ------------------------------------------------------
    #>   fold repeat_id train_n test_n
    #> 1    1         1     127     33
    #> 2    2         1     128     32
    #> 3    3         1     128     32
    #> 4    4         1     128     32
    #> 5    5         1     129     31
    #> ------------------------------------------------------
    #> Total folds: 5 | Hash: 54f1f1e5ad3c6f2f9cf0c7422b2f4425

The printed `LeakSplits` summary reports the split mode, number of folds, and fold-level training and test set sizes.

Because `group = "row_id"`, each sample is treated as its own group. As a result, repeated samples from the same subject may appear in both the training and test sets, which introduces information leakage.

Leakage-safe alternative: group by subject

    safe_splits <- make_split_plan(
      df,
      outcome = "outcome",
      mode = "subject_grouped",
      group = "subject",
      v = 5,
      repeats = 1,
      stratify = TRUE,
      seed = 10
    )

    cat("Leakage-safe splits summary (subject-grouped CV):\n")
    #> Leakage-safe splits summary (subject-grouped CV):
    safe_splits
    #> LeakSplits object (mode = subject_grouped, v = 5, repeats = 1)
    #> Outcome: outcome | Stratified: TRUE | Nested: FALSE
    #> ------------------------------------------------------
    #>   fold repeat_id train_n test_n
    #> 1    1         1     124     36
    #> 2    2         1     128     32
    #> 3    3         1     128     32
    #> 4    4         1     128     32
    #> 5    5         1     132     28
    #> ------------------------------------------------------
    #> Total folds: 5 | Hash: e34ddb9f1415cf2a82f5b73d6e8975f6

Here, each subject is confined to a single fold. Test set sizes are multiples of the number of samples per subject, confirming that subjects were not split across folds.

The fold sizes remain comparable because `stratify = TRUE` balances outcome proportions across folds while respecting subject-level boundaries.

Other leakage-aware modes

    batch_splits <- make_split_plan(
      df,
      outcome = "outcome",
      mode = "batch_blocked",
      batch = "batch",
      v = 4,
      stratify = TRUE
    )

    cat("Batch-blocked splits summary:\n")
    #> Batch-blocked splits summary:
    batch_splits
    #> LeakSplits object (mode = batch_blocked, v = 4, repeats = 1)
    #> Outcome: outcome | Stratified: TRUE | Nested: FALSE
    #> ------------------------------------------------------
    #>   fold repeat_id train_n test_n
    #> 1    1         1      73     87
    #> 2    2         1     138     22
    #> 3    3         1     132     28
    #> 4    4         1     137     23
    #> ------------------------------------------------------
    #> Total folds: 4 | Hash: 6815f595d5822e72c45cb3fe135574f7

    study_splits <- make_split_plan(
      df,
      outcome = "outcome",
      mode = "study_loocv",
      study = "study"
    )

    cat("Study leave-one-out splits summary:\n")
    #> Study leave-one-out splits summary:
    study_splits
    #> LeakSplits object (mode = study_loocv, v = 5, repeats = 1)
    #> Outcome: outcome | Stratified: FALSE | Nested: FALSE
    #> ------------------------------------------------------
    #>   fold repeat_id train_n test_n
    #> 1    1         1      96     64
    #> 2    2         1     121     39
    #> 3    3         1     131     29
    #> 4    4         1     132     28
    #> ------------------------------------------------------
    #> Total folds: 4 | Hash: 51ebad5a2ade97fc502f662d7b1c9804

    time_splits <- make_split_plan(
      df,
      outcome = "outcome",
      mode = "time_series",
      time = "time",
      v = 4,
      horizon = 2
    )

    cat("Time-series splits summary:\n")
    #> Time-series splits summary:
    time_splits
    #> LeakSplits object (mode = time_series, v = 4, repeats = 1)
    #> Outcome: outcome | Stratified: FALSE | Nested: FALSE
    #> ------------------------------------------------------
    #>   fold repeat_id train_n test_n
    #> 1    2         1      39     40
    #> 2    3         1      79     40
    #> 3    4         1     119     40
    #> ------------------------------------------------------
    #> Total folds: 3 | Hash: dbde2d11033db6b03c4578feffd553d0

    nested_splits <- make_split_plan(
      df,
      outcome = "outcome",
      mode = "subject_grouped",
      group = "subject",
      v = 3,
      nested = TRUE,
      stratify = TRUE
    )

    cat("Nested CV splits summary:\n")
    #> Nested CV splits summary:
    nested_splits
    #> LeakSplits object (mode = subject_grouped, v = 3, repeats = 1)
    #> Outcome: outcome | Stratified: TRUE | Nested: TRUE
    #> ------------------------------------------------------
    #>   fold repeat_id train_n test_n
    #> 1    1         1     104     56
    #> 2    2         1     104     56
    #> 3    3         1     112     48
    #> ------------------------------------------------------
    #> Total folds: 3 | Hash: 44a1560f6d79af026e80218128c838ed

RNG seed lineage

bioLeak uses offset-based seeding so that every sub-operation gets a deterministic, distinct seed without requiring isolated RNG streams:

Function | Base seed | Per-unit offset
--- | --- | ---
`make_split_plan()` | `set.seed(seed)` | repeats: `seed + 1000 * r`; nested: `seed + 1`
`fit_resample()` | `set.seed(seed)` | per-fold: `seed + fold_id`
`audit_leakage()` | `set.seed(seed)` | per-permutation: `seed + b`

These are not isolated RNG streams but simple offsets. Collisions do not occur for practical parameter values (e.g., `repeats < 1000`, `v < 1000`). Always set an explicit `seed` argument for reproducibility; strict mode warns when `seed` is `NULL` or `NA`.

Each summary reports the number of folds and the fold sizes for the corresponding split strategy.

- Batch & Study: The test set sizes vary because batches/studies are not evenly sized. `study_loocv` uses one study per fold, so `v` and `repeats` are ignored.

- Time series: The training set size increases across folds while the test set size follows the block size. Early blocks with no prior training data (or too-small test windows) are skipped, so you may see fewer than `v` folds.

Combined (N-axis) mode

When leakage can occur along multiple axes simultaneously (for example, both subject and batch), `mode = "combined"` handles multi-axis constraint splitting. The first element in `constraints` is the primary fold driver (determines which groups go to test); subsequent elements are exclusion constraints (training samples sharing constraint-axis levels with the test set are removed).

    # For combined mode, constraint-axis levels should not span all folds.
    # Here each subject belongs to exactly one site, so site exclusion only
    # removes training samples from the same site as the test subjects.
    df_comb <- df
    df_comb$site <- paste0("site", rep(1:8, each = 5)[df_comb$subject])

    combined_splits <- make_split_plan(
      df_comb,
      outcome = "outcome",
      mode = "combined",
      constraints = list(
        list(type = "subject", col = "subject"),
        list(type = "batch",   col = "site")
      ),
      v = 4,
      stratify = TRUE,
      seed = 42
    )

    cat("Combined (N-axis) splits summary:\n")
    #> Combined (N-axis) splits summary:
    combined_splits
    #> LeakSplits object (mode = combined, v = 4, repeats = 1)
    #> Outcome: outcome | Stratified: TRUE | Nested: FALSE
    #> ------------------------------------------------------
    #>   fold repeat_id train_n test_n
    #> 1    1         1     140     20
    #> 2    2         1     120     40
    #> 3    3         1     120     40
    #> 4    4         1     100     60
    #> ------------------------------------------------------
    #> Total folds: 4 | Hash: a065b9e85e204119fcca3fd19c112015

The fold sizes may be smaller than single-axis modes because training samples are excluded whenever their site (or any constraint-axis level) also appears in the test set. You can add more than two axes by appending additional elements to `constraints`.

The legacy `primary_axis`/`secondary_axis` parameters are still accepted for backward compatibility but `constraints` supersedes them and supports arbitrary numbers of axes.

    # Verify zero overlap on all constraint axes
    overlap_result <- check_split_overlap(combined_splits)
    overlap_result
    #>   fold repeat_id     col n_overlap pass
    #> 1    1         1 subject         0 TRUE
    #> 2    1         1    site         0 TRUE
    #> 3    2         1 subject         0 TRUE
    #> 4    2         1    site         0 TRUE
    #> 5    3         1 subject         0 TRUE
    #> 6    3         1    site         0 TRUE
    #> 7    4         1 subject         0 TRUE
    #> 8    4         1    site         0 TRUE

Assumptions and intent by mode

- `subject_grouped`: keeps all samples from a subject together
- `batch_blocked`: keeps batches/centers together (leave-one-group-out when `v` is at least the number of batches; otherwise multiple batches per fold)
- `study_loocv`: holds out each study in turn for external validation
- `time_series`: rolling-origin splits with an optional horizon to prevent lookahead
- `combined`: primary axis drives fold assignment; all additional constraint axes are enforced as exclusion constraints on the training set

Use `repeats` for repeated CV in grouped/batch modes (it is ignored for `study_loocv` and `time_series`), `stratify = TRUE` to balance outcome proportions at the group level when applicable, and `nested = TRUE` to attach inner folds (one repeat). For large datasets, `progress = TRUE` reports progress; storing explicit indices can be memory intensive.

## 2.3 Validating splits with `check_split_overlap()`

`check_split_overlap()` verifies that no grouping-column levels appear in both the training and test sets of any fold. It returns a data.frame with one row per fold-column combination showing the overlap count and a pass/fail flag.

    # Run on the subject-grouped splits from earlier
    overlap_safe <- check_split_overlap(safe_splits)
    overlap_safe
    #>   fold repeat_id     col n_overlap pass
    #> 1    1         1 subject         0 TRUE
    #> 2    2         1 subject         0 TRUE
    #> 3    3         1 subject         0 TRUE
    #> 4    4         1 subject         0 TRUE
    #> 5    5         1 subject         0 TRUE

When `cols` is not supplied, the function auto-infers the relevant columns from the split mode (e.g., `subject` for `subject_grouped`, all constraint axes for `combined`). By default `stop_on_fail = TRUE`, so the function raises a `bioLeak_overlap_error` when any overlap is detected. Set `stop_on_fail = FALSE` to return the results without stopping.

When strict mode is active (`options(bioLeak.strict = TRUE)`), `make_split_plan()` automatically runs `check_split_overlap()` after building non-compact splits, so you do not need to call it manually.

## 2.4 Scalability

Handling Large Datasets (Compact Mode)

For large datasets (e.g., \(N > 50,000\)) with many repeats, storing explicit integer indices for every fold can consume gigabytes of memory. Use `compact = TRUE` to store a lightweight vector of fold assignments instead. `fit_resample()` will automatically reconstruct the indices on the fly. Compact mode is not supported when `nested = TRUE` (it falls back to full indices). For `time_series` compact splits, the time column must be present in the stored split metadata so folds can be reconstructed.

    # Efficient storage for large N
    big_splits <- make_split_plan(
      df,
      outcome = "outcome",
      mode = "subject_grouped",
      group = "subject",
      v = 5,
      compact = TRUE  # <--- Saves memory
    )

    cat("Compact-mode splits summary:\n")
    #> Compact-mode splits summary:
    big_splits
    #> LeakSplits object (mode = subject_grouped, v = 5, repeats = 1)
    #> Outcome: outcome | Stratified: FALSE | Nested: FALSE
    #> ------------------------------------------------------
    #>   fold repeat_id train_n test_n
    #> 1    1         1     128     32
    #> 2    2         1     128     32
    #> 3    3         1     128     32
    #> 4    4         1     128     32
    #> 5    5         1     128     32
    #> ------------------------------------------------------
    #> Total folds: 5 | Hash: ee2866b7282066cda850ea9995f43c10
    cat(sprintf("Compact storage enabled: %s\n", big_splits@info$compact))
    #> Compact storage enabled: TRUE

The summary is identical to a regular split, but the underlying storage is a compact fold-assignment vector. Use the `compact` flag to confirm the memory- saving mode is active.

## 2.5 Strict leakage mode

Setting `options(bioLeak.strict = TRUE)` activates a global safety switch that tightens validation across the entire workflow. In strict mode:

1. Trained recipe/workflow detection escalates to an error. Normally, passing a pre-trained recipe or workflow to `fit_resample()` or `tune_resample()` produces a warning. In strict mode, it raises an error so fold-global leakage cannot proceed silently.

2. Seed warnings. When `seed` is `NULL` or `NA` in `make_split_plan()`, `fit_resample()`, or `tune_resample()`, strict mode emits a `bioLeak_validation_warning` reminding you to set an explicit seed for reproducibility.

3. Combined + nested warning. Using `nested = TRUE` with `mode = "combined"` may produce empty inner folds; strict mode warns about this before proceeding.

4. Automatic overlap check. After building non-compact splits, `make_split_plan()` auto-runs `check_split_overlap()` to verify that no grouping-column levels cross fold boundaries.

    # Enable strict mode temporarily
    withr::with_options(list(bioLeak.strict = TRUE), {
      strict_splits <- make_split_plan(
        df,
        outcome = "outcome",
        mode = "subject_grouped",
        group = "subject",
        v = 5,
        stratify = TRUE,
        seed = 42
      )
      cat("Strict mode splits completed without error.\n")
    })
    #> Strict mode splits completed without error.

    # Strict mode catches a pre-trained recipe
    if (requireNamespace("recipes", quietly = TRUE)) {
      rec <- recipes::recipe(outcome ~ ., data = df[, c("outcome", predictors)]) |>
        recipes::step_normalize(recipes::all_numeric_predictors()) |>
        recipes::prep(training = df[, c("outcome", predictors)])

      withr::with_options(list(bioLeak.strict = TRUE), {
        tryCatch(
          fit_resample(
            df, outcome = "outcome",
            splits = safe_splits,
            learner = "glmnet",
            preprocess = rec
          ),
          error = function(e) cat("Strict mode error:", conditionMessage(e), "\n")
        )
      })
    }
    #> Strict mode error: fit_resample: received a trained recipe. Pass an untrained recipe to avoid fold-global leakage.

Strict mode is off by default. Use `withr::with_options()` for temporary activation or set `options(bioLeak.strict = TRUE)` in your `.Rprofile` for project-wide enforcement.

## 2.6 Guarded preprocessing and imputation

`bioLeak` uses guarded preprocessing to prevent leakage from global imputation, scaling, and feature selection. The low-level API is:

- `.guard_fit()` to fit preprocessing on training data only
- `predict_guard()` to apply the trained guard to new data
- `.guard_ensure_levels()` to align factor levels across train/test
- `impute_guarded()` as a convenience wrapper for leakage-safe imputation

`.guard_fit()` fits a pipeline that can winsorize, impute, normalize, filter, and select features. It one-hot encodes non-numeric columns and carries factor levels forward to new data. Assumptions and misuse to avoid:

- `fs = "ttest"` requires a binary outcome with enough samples per class
- `fs = "lasso"` requires `glmnet`
- `impute = "mice"` is not supported for guarded prediction
- `impute = "none"` with missing values will trigger a median fallback and missingness indicators to avoid errors

Supported preprocessing options include imputation (`median`, `knn`, `missForest`, `none`), normalization (`zscore`, `robust`, `none`), filtering by variance or IQR (optionally `min_keep`), and feature selection (`ttest`, `lasso`, `pca`). Winsorization is controlled via `impute$winsor` and `impute$winsor_k` in guarded steps; in `impute_guarded()`, use `winsor` and `winsor_thresh`. `impute_guarded()` returns a `LeakImpute` object with guarded data, diagnostics, and the fitted guard state.

Leaky example: global scaling before CV

    df_leaky_scaled <- df
    df_leaky_scaled[predictors] <- scale(df_leaky_scaled[predictors])
    scaled_summary <- data.frame(
      feature = predictors,
      mean = colMeans(df_leaky_scaled[predictors]),
      sd = apply(df_leaky_scaled[predictors], 2, stats::sd)
    )
    scaled_summary$mean <- round(scaled_summary$mean, 3)
    scaled_summary$sd <- round(scaled_summary$sd, 3)

    # Leaky global scaling: means ~0 and SDs ~1 computed on all samples
    scaled_summary
    #>    feature mean sd
    #> x1      x1    0  1
    #> x2      x2    0  1
    #> x3      x3    0  1

The summary shows that scaling used the full dataset, so test-fold statistics influenced the training transformation. This violates the train/test separation and can bias performance estimates.

Leakage-safe alternative: fit preprocessing on training only

    fold1 <- safe_splits@indices[[1]]
    train_x <- df[fold1$train, predictors]
    test_x <- df[fold1$test, predictors]

    guard <- .guard_fit(
      X = train_x,
      y = df$outcome[fold1$train],
      steps = list(
        impute = list(method = "median", winsor = TRUE),
        normalize = list(method = "zscore"),
        filter = list(var_thresh = 0, iqr_thresh = 0),
        fs = list(method = "none")
      ),
      task = "binomial"
    )

    train_x_guarded <- predict_guard(guard, train_x)
    test_x_guarded <- predict_guard(guard, test_x)

    cat("GuardFit object:\n")
    #> GuardFit object:
    guard
    #> Guarded preprocessing pipeline
    #>  - Imputation: median
    #>  - Normalization: zscore
    #>  - Filter: var>=0, iqr>=0
    #>  - Feature selection: none
    #>  - Output features: 3
    cat("\nGuardFit summary:\n")
    #>
    #> GuardFit summary:
    summary(guard)
    #> GuardFit summary
    #>  n_train=124, p_out=3, removed_by_filter=0
    #>  Steps:
    #>  step                 action
    #>     1            winsor: k=5
    #>     2   impute: none (no NA)
    #>     3      normalize: zscore
    #>     4 filter: var>=0, iqr>=0
    #>     5               fs: none
    #>  hash: bacf825a215c48894b993150135d3249

    # Guarded training data (first 6 rows)
    head(train_x_guarded)
    #>            x1           x2          x3
    #> 5  -1.4287254 -0.846746034 -0.58336326
    #> 6   1.9282279  0.352168157 -0.70450573
    #> 7   0.5110636  0.427913734 -0.08735148
    #> 8  -1.3683145 -0.006339485  0.36484181
    #> 9  -0.7187394 -2.383299658 -1.01654516
    #> 10 -1.3015499  2.538309224 -0.14109035

    # Guarded test data (first 6 rows)
    head(test_x_guarded)
    #>              x1         x2          x3
    #> 1   0.119085787  1.0175596 -0.05210949
    #> 2  -0.428021276 -1.9210219  0.26877352
    #> 3  -0.002549215 -0.3915238  0.95073054
    #> 4  -1.007141255  0.1398948  0.25797066
    #> 25 -1.105722181  0.5243484  2.03908390
    #> 26  1.601954989 -0.5499985  0.86726497

The `GuardFit` object records the preprocessing steps and the number of features retained after filtering. The summary reports how many features were removed and the preprocessing audit trail. The guarded train/test previews show that missing values were imputed and (if requested) scaled using training-only statistics; the test data never influences these values.

Factor level guard (advanced)

    raw_levels <- data.frame(
      site = c("A", "B", "B"),
      status = c("yes", "no", "yes"),
      stringsAsFactors = FALSE
    )

    level_state <- .guard_ensure_levels(raw_levels)

    # Aligned factor data with consistent levels
    level_state$data
    #>   site status
    #> 1    A    yes
    #> 2    B     no
    #> 3    B    yes

    # Levels map
    level_state$levels
    #> $site
    #> [1] "A" "B"
    #>
    #> $status
    #> [1] "no"  "yes"

The returned `data` keeps factor levels consistent across folds, while the `levels` list records the training-time levels (including any dummy levels added to preserve one-hot columns). Use these when transforming new data to avoid misaligned model matrices.

Leaky example: imputation using train and test together

    train <- data.frame(a = c(1, 2, NA, 4), b = c(NA, 1, 1, 0))
    test <- data.frame(a = c(NA, 5), b = c(1, NA))

    all_median <- vapply(rbind(train, test),
                         function(col) median(col, na.rm = TRUE),
                         numeric(1))
    train_leaky <- as.data.frame(Map(function(col, m) { col[is.na(col)] <- m; col },
                                     train, all_median))
    test_leaky <- as.data.frame(Map(function(col, m) { col[is.na(col)] <- m; col },
                                    test, all_median))

    # Leaky medians computed on train + test
    data.frame(feature = names(all_median), median = all_median)
    #>   feature median
    #> a       a      3
    #> b       b      1

    # Leaky-imputed training data
    train_leaky
    #>   a b
    #> 1 1 1
    #> 2 2 1
    #> 3 3 1
    #> 4 4 0

    # Leaky-imputed test data
    test_leaky
    #>   a b
    #> 1 3 1
    #> 2 5 1

The medians above were computed using both train and test data, so the test set influences the imputation values. This is a classic leakage pathway because test information directly alters the training features.

Leakage-safe alternative: `impute_guarded()`

    imp <- impute_guarded(
      train = train,
      test = test,
      method = "median",
      winsor = FALSE
    )

    # Guarded-imputed training data
    imp$train
    #>   a b
    #> 1 1 1
    #> 2 2 1
    #> 3 2 1
    #> 4 4 0

    # Guarded-imputed test data
    imp$test
    #>   a b
    #> 1 2 1
    #> 2 5 1

Here, the imputation statistics are computed from the training set only. The missing value in the test set (column `a`) is replaced by 2, which is the training median. In contrast, in the leaky example above, it was replaced by 3, the global median.

This confirms that the test set is transformed using fixed values learned from the training data, preserving a clean separation between training and evaluation. The `LeakImpute` object also contains missingness diagnostics in `imp$summary$diagnostics`, and guarded outputs use the same encoding as `.guard_fit()` (categorical variables are one-hot encoded). Use `vars` to impute only a subset of columns when needed.

Bridging guard steps to recipes: `guard_to_recipe()`

`guard_to_recipe()` converts a guard preprocessing specification into a `recipes::recipe`, enabling a smooth transition from guard-based to tidymodels-based workflows. It maps supported guard steps to their closest recipe equivalents:

Guard step | Recipe equivalent
--- | ---
`impute$method = "median"` | `step_impute_median()`
`impute$method = "knn"` | `step_impute_knn()`
`normalize$method = "zscore"` | `step_normalize()`
`filter$var_thresh > 0` | `step_nzv()`
`fs$method = "pca"` | `step_pca()`

Steps with no direct recipe equivalent (`missForest`, `mice`, `robust` normalization, `ttest`, `lasso` feature selection) produce a `bioLeak_fallback_warning` and either fall back to the closest available step or are skipped.

    if (requireNamespace("recipes", quietly = TRUE)) {
      guard_steps <- list(
        impute    = list(method = "median"),
        normalize = list(method = "zscore")
      )

      rec <- guard_to_recipe(
        steps = guard_steps,
        formula = outcome ~ .,
        training_data = df[, c("outcome", predictors)]
      )

      cat("Converted recipe:\n")
      rec
    }
    #> Converted recipe:

The returned recipe is untrained and can be passed directly to `fit_resample()` or `tune_resample()` as the `preprocess` argument.

## 2.7 Fit and resample with `fit_resample()`

`fit_resample()` combines leakage-aware splits with guarded preprocessing and model fitting. It supports:

- built-in learners (`glmnet`, `ranger`)
- `parsnip` model specifications or `workflows::workflow` objects (recommended)
- `recipes::recipe` preprocessing (prepped on training folds only)
- `rsample` resamples (`rset`/`rsplit`) via `splits`, with `split_cols` to map metadata
- custom learners via `custom_learners` (legacy/advanced)
- multiple metrics, class weights, positive class override, and yardstick metrics
- fold-level run diagnostics in `fit@info$fold_status` (`success`/`skipped`/`failed`)

Assumptions and misuse to avoid:

- outcome must be binary or multiclass (factor) for classification, numeric for regression, or a survival outcome (a `Surv` column or time/event columns)
- `positive_class` must be a single value that exists in the outcome levels
- `class_weights` only applies to binomial/multiclass tasks; workflows must handle weights internally
- if you supply a matrix without metadata, you must remove leakage columns yourself (group, batch, study, time)

Use `learner_args` to pass model-specific arguments. For custom learners (advanced), you can supply separate `fit` and `predict` argument lists. Set `parallel = TRUE` to use future.apply for fold-level parallelism when available. When `learner` is a parsnip specification, `learner_args` and `custom_learners` are ignored. When `learner` is a workflow, `preprocess` and `learner_args` are ignored. When a recipe or workflow is used, the built-in guarded preprocessing list is bypassed, so ensure the recipe/workflow itself is leakage-safe.

Metrics used in this vignette:

- AUC: area under the ROC curve (0.5 = random, 1.0 = perfect; higher is better)
- PR AUC: area under the precision-recall curve (0 to 1; higher is better)
- Accuracy: fraction of correct predictions (0 to 1; higher is better)
- RMSE: root mean squared error (0 to infinity; lower is better)
- Macro F1: average F1 across classes (multiclass; higher is better)
- Log loss: cross-entropy for probabilities (lower is better)
- C-index: concordance for regression/survival (higher is better)

Always report the mean and variability across folds, not a single fold value. When using yardstick metrics, the positive class is the second factor level; set `positive_class` to control this.

Parsnip model specification (recommended)

    spec <- parsnip::logistic_reg(mode = "classification") |>
      parsnip::set_engine("glm")

This spec uses base R `glm` under the hood and does not require extra model packages. Use it in the examples below; custom learners are covered in the Advanced section.

Leaky example: leaky features and row-wise splits

    fit_leaky <- fit_resample(
      df_leaky,
      outcome = "outcome",
      splits = leaky_splits,
      learner = spec,
      metrics = c("auc", "accuracy"),
      preprocess = list(
        impute = list(method = "median"),
        normalize = list(method = "zscore"),
        filter = list(var_thresh = 0),
        fs = list(method = "none")
      )
    )
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%

    cat("Leaky fit summary:\n")
    #> Leaky fit summary:
    summary(fit_leaky)
    #>
    #> ===========================
    #>  bioLeak Model Fit Summary
    #> ===========================
    #>
    #> Task: binomial
    #> Outcome: outcome
    #> Positive class: case
    #> Learners: logistic_reg/glm
    #> Total folds: 5
    #> Fold status: 5 success, 0 skipped, 0 failed
    #> Refit performed: Yes
    #> Hash: ha16bdf5a
    #>
    #> Cross-validated metrics (mean ± SD):
    #>            learner auc_mean auc_sd accuracy_mean accuracy_sd auc_ci_lo
    #> 1 logistic_reg/glm    0.667  0.145         0.618        0.06     0.398
    #>   auc_ci_hi accuracy_ci_lo accuracy_ci_hi
    #> 1     0.937          0.506          0.729
    #>
    #> Audit overview:
    #>  fold n_train n_test          learner features_final
    #>     1     127     33 logistic_reg/glm             17
    #>     2     128     32 logistic_reg/glm             17
    #>     3     128     32 logistic_reg/glm             17
    #>     4     128     32 logistic_reg/glm             17
    #>     5     129     31 logistic_reg/glm             17
    metrics_leaky <- as.data.frame(fit_leaky@metric_summary)
    num_cols <- vapply(metrics_leaky, is.numeric, logical(1))
    metrics_leaky[num_cols] <- lapply(metrics_leaky[num_cols], round, digits = 3)

    # Leaky fit: mean and SD of metrics across folds
    metrics_leaky
    #>            learner auc_mean auc_sd accuracy_mean accuracy_sd auc_ci_lo
    #> 1 logistic_reg/glm    0.667  0.145         0.618        0.06     0.398
    #>   auc_ci_hi accuracy_ci_lo accuracy_ci_hi
    #> 1     0.937          0.506          0.729

The summary reports cross-validated performance. AUC ranges from 0.5 (random) to 1.0 (perfect), while accuracy is the fraction of correct predictions. Because the splits are leaky, these metrics can be artificially high and should not be used for reporting.

Leakage-safe alternative: grouped splits and clean predictors

    fit_safe <- fit_resample(
      df,
      outcome = "outcome",
      splits = safe_splits,
      learner = spec,
      metrics = c("auc", "accuracy"),
      preprocess = list(
        impute = list(method = "median"),
        normalize = list(method = "zscore"),
        filter = list(var_thresh = 0),
        fs = list(method = "none")
      ),
      positive_class = "case",
      class_weights = c(control = 1, case = 1),
      refit = TRUE
    )
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%

    cat("Leakage-safe fit summary:\n")
    #> Leakage-safe fit summary:
    summary(fit_safe)
    #>
    #> ===========================
    #>  bioLeak Model Fit Summary
    #> ===========================
    #>
    #> Task: binomial
    #> Outcome: outcome
    #> Positive class: case
    #> Learners: logistic_reg/glm
    #> Total folds: 5
    #> Fold status: 5 success, 0 skipped, 0 failed
    #> Refit performed: Yes
    #> Hash: hba29ffa5
    #>
    #> Cross-validated metrics (mean ± SD):
    #>            learner auc_mean auc_sd accuracy_mean accuracy_sd auc_ci_lo
    #> 1 logistic_reg/glm    0.694  0.048         0.598       0.117     0.606
    #>   auc_ci_hi accuracy_ci_lo accuracy_ci_hi
    #> 1     0.783          0.379          0.816
    #>
    #> Audit overview:
    #>  fold n_train n_test          learner features_final
    #>     1     124     36 logistic_reg/glm             13
    #>     2     128     32 logistic_reg/glm             13
    #>     3     128     32 logistic_reg/glm             13
    #>     4     128     32 logistic_reg/glm             13
    #>     5     132     28 logistic_reg/glm             13
    metrics_safe <- as.data.frame(fit_safe@metric_summary)
    num_cols <- vapply(metrics_safe, is.numeric, logical(1))
    metrics_safe[num_cols] <- lapply(metrics_safe[num_cols], round, digits = 3)

    # Leakage-safe fit: mean and SD of metrics across folds
    metrics_safe
    #>            learner auc_mean auc_sd accuracy_mean accuracy_sd auc_ci_lo
    #> 1 logistic_reg/glm    0.694  0.048         0.598       0.117     0.606
    #>   auc_ci_hi accuracy_ci_lo accuracy_ci_hi
    #> 1     0.783          0.379          0.816

    # Per-fold metrics (first 6 rows)
    head(fit_safe@metrics)
    #>   fold          learner       auc  accuracy
    #> 1    1 logistic_reg/glm 0.6284830 0.5277778
    #> 2    2 logistic_reg/glm 0.7611336 0.7187500
    #> 3    3 logistic_reg/glm 0.6823529 0.6562500
    #> 4    4 logistic_reg/glm 0.6944444 0.6562500
    #> 5    5 logistic_reg/glm 0.7055556 0.4285714

`fit_resample()` returns a `LeakFit` object. Use `summary(fit_safe)` for a compact report and inspect `fit_safe@metrics` and `fit_safe@predictions` for details. When `refit = TRUE`, the final model and preprocessing state are stored in `fit_safe@info$final_model` and `fit_safe@info$final_preprocess`. Fold-level status diagnostics are available in `fit_safe@info$fold_status`.

The mean +/- SD table is the primary performance summary to report, while the per-fold metrics reveal variability and potential instability across folds.

By default, `fit_resample()` stores refit inputs in `fit_safe@info$perm_refit_spec` to enable `audit_leakage(perm_refit = "auto")`. Set `store_refit_data = FALSE` to reduce memory and provide `perm_refit_spec` manually when you want refit-based permutations. When multiple learners are passed, `refit = TRUE` refits only the first learner; set `refit = FALSE` if you do not need a final model.

    # Fold-level diagnostics for reproducible troubleshooting
    head(fit_safe@info$fold_status, 5)
    #>   fold    stage  status reason                    notes elapsed_sec
    #> 1    1 fold_run success   <NA> Successful learners: 1/1       0.009
    #> 2    2 fold_run success   <NA> Successful learners: 1/1       0.008
    #> 3    3 fold_run success   <NA> Successful learners: 1/1       0.008
    #> 4    4 fold_run success   <NA> Successful learners: 1/1       0.007
    #> 5    5 fold_run success   <NA> Successful learners: 1/1       0.009

Interpretation of `fold_status`:

- `status = "success"` means at least one learner produced valid metrics on that fold.
- `status = "skipped"` typically means a data-structure issue in that fold (for example, single-class training data in classification).
- `status = "failed"` indicates a runtime failure (for example, model fitting or preprocessing error).

Use `reason` and `notes` to decide whether to:

- adjust split design (for example, stronger grouping, stratification, or fewer folds),
- adjust preprocessing/model complexity,
- or exclude unstable configurations before formal comparison.

If many folds are skipped/failed, do not trust aggregate metric means until the instability is resolved.

The examples above use a parsnip model specification. To swap in other models, replace `spec` with another parsnip spec (see the gradient boosting example below).

Multiclass classification (optional)

    if (requireNamespace("ranger", quietly = TRUE)) {
      set.seed(11)
      df_multi <- df
      df_multi$outcome3 <- factor(sample(c("A", "B", "C"),
                                         nrow(df_multi), replace = TRUE))

      multi_splits <- make_split_plan(
        df_multi,
        outcome = "outcome3",
        mode = "subject_grouped",
        group = "subject",
        v = 4,
        stratify = TRUE,
        seed = 11
      )

      fit_multi <- fit_resample(
        df_multi,
        outcome = "outcome3",
        splits = multi_splits,
        learner = "ranger",
        metrics = c("accuracy", "macro_f1", "log_loss"),
        refit = FALSE
      )

      cat("Multiclass fit summary:\n")
      summary(fit_multi)
    } else {
      cat("ranger not installed; skipping multiclass example.\n")
    }
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==================                                                    |  25%  |                                                                              |===================================                                   |  50%  |                                                                              |====================================================                  |  75%  |                                                                              |======================================================================| 100%
    #> Multiclass fit summary:
    #>
    #> ===========================
    #>  bioLeak Model Fit Summary
    #> ===========================
    #>
    #> Task: multiclass
    #> Outcome: outcome3
    #> Learners: ranger
    #> Total folds: 4
    #> Fold status: 4 success, 0 skipped, 0 failed
    #> Refit performed: No
    #> Hash: hc88c6471
    #>
    #> Cross-validated metrics (mean ± SD):
    #>   learner accuracy_mean accuracy_sd macro_f1_mean macro_f1_sd log_loss_mean
    #> 1  ranger         0.313       0.073         0.306       0.072         1.157
    #>   log_loss_sd accuracy_ci_lo accuracy_ci_hi macro_f1_ci_lo macro_f1_ci_hi
    #> 1       0.042          0.136          0.491          0.132          0.479
    #>   log_loss_ci_lo log_loss_ci_hi
    #> 1          1.055          1.259
    #>
    #> Audit overview:
    #>  fold n_train n_test learner features_final
    #>     1     116     44  ranger             14
    #>     2     120     40  ranger             14
    #>     3     120     40  ranger             14
    #>     4     124     36  ranger             14

Multiclass fits compute accuracy, macro-F1, and log loss when class probabilities are available. `positive_class` is ignored for multiclass tasks.

Survival outcomes (optional)

    if (requireNamespace("survival", quietly = TRUE) &&
        requireNamespace("glmnet", quietly = TRUE)) {
      set.seed(12)
      df_surv <- df
      df_surv$time_to_event <- rexp(nrow(df_surv), rate = 0.1)
      df_surv$event <- rbinom(nrow(df_surv), 1, 0.7)

      surv_splits <- make_split_plan(
        df_surv,
        outcome = c("time_to_event", "event"),
        mode = "subject_grouped",
        group = "subject",
        v = 4,
        stratify = FALSE,
        seed = 12
      )

      fit_surv <- fit_resample(
        df_surv,
        outcome = c("time_to_event", "event"),
        splits = surv_splits,
        learner = "glmnet",
        metrics = "cindex",
        refit = FALSE
      )

      cat("Survival fit summary:\n")
      summary(fit_surv)
    } else {
      cat("survival or glmnet not installed; skipping survival example.\n")
    }
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==================                                                    |  25%  |                                                                              |===================================                                   |  50%  |                                                                              |====================================================                  |  75%  |                                                                              |======================================================================| 100%
    #> Survival fit summary:
    #>
    #> ===========================
    #>  bioLeak Model Fit Summary
    #> ===========================
    #>
    #> Task: survival
    #> Outcome: time_to_event
    #>  Outcome: event
    #> Learners: glmnet
    #> Total folds: 4
    #> Fold status: 4 success, 0 skipped, 0 failed
    #> Refit performed: No
    #> Hash: h041adef3
    #>
    #> Cross-validated metrics (mean ± SD):
    #>   learner cindex_mean cindex_sd cindex_ci_lo cindex_ci_hi
    #> 1  glmnet         0.5         0          0.5          0.5
    #>
    #> Audit overview:
    #>  fold n_train n_test learner features_final
    #>     1     120     40  glmnet             14
    #>     2     120     40  glmnet             14
    #>     3     120     40  glmnet             14
    #>     4     120     40  glmnet             14

For survival tasks, supply `outcome = c(time_col, event_col)` or a `Surv` column; `stratify` is ignored for time/event outcomes and `class_weights` are not used.

Passing learner-specific arguments (optional)

    if (requireNamespace("glmnet", quietly = TRUE)) {
      fit_glmnet <- fit_resample(
        df,
        outcome = "outcome",
        splits = safe_splits,
        learner = "glmnet",
        metrics = "auc",
        learner_args = list(glmnet = list(alpha = 0.5)),
        preprocess = list(
          impute = list(method = "median"),
          normalize = list(method = "zscore"),
          filter = list(var_thresh = 0),
          fs = list(method = "none")
        )
      )
      cat("GLMNET summary with learner-specific arguments:\n")
      summary(fit_glmnet)
    } else {
      cat("glmnet not installed; skipping learner_args example.\n")
    }
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #> GLMNET summary with learner-specific arguments:
    #>
    #> ===========================
    #>  bioLeak Model Fit Summary
    #> ===========================
    #>
    #> Task: binomial
    #> Outcome: outcome
    #> Positive class: case
    #> Learners: glmnet
    #> Total folds: 5
    #> Fold status: 5 success, 0 skipped, 0 failed
    #> Refit performed: Yes
    #> Hash: hba29ffa5
    #>
    #> Cross-validated metrics (mean ± SD):
    #>   learner auc_mean auc_sd auc_ci_lo auc_ci_hi
    #> 1  glmnet    0.724  0.079     0.577      0.87
    #>
    #> Audit overview:
    #>  fold n_train n_test learner features_final
    #>     1     124     36  glmnet             13
    #>     2     128     32  glmnet             13
    #>     3     128     32  glmnet             13
    #>     4     128     32  glmnet             13
    #>     5     132     28  glmnet             13

This summary reflects the same guarded preprocessing but a different model configuration (here, `alpha = 0.5`). Use the mean +/- SD metrics to compare learner settings under identical splits.

SummarizedExperiment inputs (optional)

    if (requireNamespace("SummarizedExperiment", quietly = TRUE)) {
      se <- SummarizedExperiment::SummarizedExperiment(
        assays = list(counts = t(as.matrix(df[, predictors]))),
        colData = df[, c("subject", "batch", "study", "time", "outcome"), drop = FALSE]
      )

      se_splits <- make_split_plan(
        se,
        outcome = "outcome",
        mode = "subject_grouped",
        group = "subject",
        v = 3
      )

      se_fit <- fit_resample(
        se,
        outcome = "outcome",
        splits = se_splits,
        learner = spec,
        metrics = "auc"
      )
      cat("SummarizedExperiment fit summary:\n")
      summary(se_fit)
    } else {
      cat("SummarizedExperiment not installed; skipping SE example.\n")
    }
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #> SummarizedExperiment fit summary:
    #>
    #> ===========================
    #>  bioLeak Model Fit Summary
    #> ===========================
    #>
    #> Task: binomial
    #> Outcome: outcome
    #> Positive class: case
    #> Learners: logistic_reg/glm
    #> Total folds: 3
    #> Fold status: 3 success, 0 skipped, 0 failed
    #> Refit performed: Yes
    #> Hash: h3f97c442
    #>
    #> Cross-validated metrics (mean ± SD):
    #>            learner auc_mean auc_sd auc_ci_lo auc_ci_hi
    #> 1 logistic_reg/glm    0.673  0.062     0.431     0.914
    #>
    #> Audit overview:
    #>  fold n_train n_test          learner features_final
    #>     1     104     56 logistic_reg/glm              3
    #>     2     108     52 logistic_reg/glm              3
    #>     3     108     52 logistic_reg/glm              3

The summary is identical in structure to the `data.frame` case because `fit_resample()` extracts predictors and metadata from the `SummarizedExperiment` object.

Note that `features_final` here reflects only the assay predictors (`x1`, `x2`, `x3`), because the assay was constructed without metadata columns.

## 2.8 Tidymodels interoperability

`bioLeak` integrates with `rsample`, `recipes`, `workflows`, and `yardstick`:

- `splits` can be an `rsample` rset/rsplit; use `split_cols` (default `"auto"`) or `bioLeak_mode` / `bioLeak_perm_mode` attributes to map group/batch/study/time metadata.
- `preprocess` can be a `recipes::recipe` (prepped on training folds only).
- `learner` can be a `workflows::workflow`.
- `metrics` can be a `yardstick::metric_set`.
- `as_rsample()` converts `LeakSplits` to an `rsample` object.

    library(bioLeak)
    library(parsnip)
    library(recipes)
    library(yardstick)

    set.seed(123)
    N <- 60
    df <- data.frame(
      subject = factor(rep(paste0("S", 1:20), length.out = N)),
      outcome = factor(rep(c("ClassA", "ClassB"), length.out = N)),
      x1 = rnorm(N),
      x2 = rnorm(N),
      x3 = rnorm(N)
    )

    spec <- logistic_reg() |> set_engine("glm")

    # Use bioLeak's native split planner to avoid conversion errors
    set.seed(13)

    # Use make_split_plan instead of rsample::group_vfold_cv
    # This creates a subject-grouped CV directly compatible with fit_resample
    splits <- make_split_plan(
      df,
      outcome = "outcome",
      mode = "subject_grouped",
      group = "subject",
      v = 3
    )

    rec <- recipes::recipe(outcome ~ x1 + x2 + x3, data = df) |>
      recipes::step_impute_median(recipes::all_numeric_predictors()) |>
      recipes::step_normalize(recipes::all_numeric_predictors())

    metrics_set <- yardstick::metric_set(yardstick::roc_auc, yardstick::accuracy)

    fit_rs <- fit_resample(
      df,
      outcome = "outcome",
      splits = splits,
      learner = spec,
      preprocess = rec,
      metrics = metrics_set,
      refit = FALSE
    )
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%

    if (exists("as_rsample", where = asNamespace("bioLeak"), mode = "function")) {
        rs_export <- as_rsample(fit_rs@splits, data = df)
        print(rs_export)
    }
    #> # Manual resampling
    #> # A tibble: 3 × 2
    #>   splits          id
    #>   <list>          <chr>
    #> 1 <split [39/21]> Fold1
    #> 2 <split [39/21]> Fold2
    #> 3 <split [42/18]> Fold3

Use `split_cols` to ensure split-defining metadata are dropped from predictors when using rsample inputs.

    if (requireNamespace("workflows", quietly = TRUE)) {
      wf <- workflows::workflow() |>
        workflows::add_model(spec) |>
        workflows::add_formula(outcome ~ x1 + x2 + x3)

      fit_wf <- fit_resample(
        df,
        outcome = "outcome",
        splits = splits,
        learner = wf,
        metrics = "auc",
        refit = FALSE
      )

      cat("Workflow fit summary:\n")
      summary(fit_wf)
    } else {
      cat("workflows not installed; skipping workflow example.\n")
    }
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #> Workflow fit summary:
    #>
    #> ===========================
    #>  bioLeak Model Fit Summary
    #> ===========================
    #>
    #> Task: binomial
    #> Outcome: outcome
    #> Positive class: ClassB
    #> Learners: workflow_1
    #> Total folds: 3
    #> Fold status: 3 success, 0 skipped, 0 failed
    #> Refit performed: No
    #> Hash: h825b7ff0
    #>
    #> Cross-validated metrics (mean ± SD):
    #>      learner auc_mean auc_sd auc_ci_lo auc_ci_hi
    #> 1 workflow_1    0.627  0.125     0.135     1.118
    #>
    #> Audit overview:
    #>  fold n_train n_test    learner features_final
    #>     1      39     21 workflow_1              3
    #>     2      39     21 workflow_1              3
    #>     3      42     18 workflow_1              3

When auditing rsample-based fits, pass `perm_mode = "subject_grouped"` (or set `attr(rs, "bioLeak_perm_mode") <- "subject_grouped"`) so restricted permutations respect the intended split design.

## 2.9 Nested tuning with `tune_resample()`

`tune_resample()` runs leakage-aware nested tuning with tidymodels. It accepts a parsnip model specification or workflow, and supports two inner-CV modes:

- precomputed inner folds (for example from `make_split_plan(..., nested = TRUE)`)
- on-the-fly inner split generation for `LeakSplits` using `inner_v`, `inner_repeats`, and `inner_seed`

For rsample inputs, inner folds must already be present. Survival tasks are not yet supported.

    if (requireNamespace("parsnip", quietly = TRUE) &&
        requireNamespace("recipes", quietly = TRUE) &&
        requireNamespace("tune", quietly = TRUE) &&
        requireNamespace("glmnet", quietly = TRUE)) {
      # --- 1. Create Data ---
      set.seed(123)
      N <- 60
      df <- data.frame(
        subject = factor(rep(paste0("S", 1:20), length.out = N)),
        outcome = factor(sample(c("ClassA", "ClassB"), N, replace = TRUE)),
        x1 = rnorm(N),
        x2 = rnorm(N),
        x3 = rnorm(N)
      )

      # --- 2. Generate Nested Splits ---
      set.seed(1)
      nested_splits <- make_split_plan(
        df,
        outcome = "outcome",
        mode = "subject_grouped",
        group = "subject",
        v = 3,
        nested = TRUE
      )

      # --- 3. Define Recipe & Model ---
      rec <- recipes::recipe(outcome ~ x1 + x2 + x3, data = df) |>
        recipes::step_impute_median(recipes::all_numeric_predictors()) |>
        recipes::step_normalize(recipes::all_numeric_predictors())

      spec_tune <- parsnip::logistic_reg(
        penalty = tune::tune(),
        mixture = 1,
        mode = "classification"
      ) |>
        parsnip::set_engine("glmnet")

      # --- 4. Run Tuning ---
      tuned <- tune_resample(
        df,
        outcome = "outcome",
        splits = nested_splits,
        learner = spec_tune,
        preprocess = rec,
        inner_v = 2,
        grid = 3,
        metrics = c("auc", "accuracy"),
        selection = "one_std_err",
        refit = TRUE,
        seed = 14
      )

      summary(tuned)
    } else {
      cat("parsnip/recipes/tune/glmnet not installed; skipping nested tuning example.\n")
    }
    #>   |                                                                              |                                                                      |   0%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |======================================================================| 100%
    #>
    #> ============================
    #>  bioLeak Tuning Summary
    #> ============================
    #>
    #> Task: binomial
    #> Outcome: outcome
    #> Positive class: ClassB
    #> Tuning Grid: 3
    #> Selection Rule: one_std_err (Metric: roc_auc)
    #> Fold status: 3 success, 0 skipped, 0 failed
    #> Refit performed: Yes
    #> Outer Folds: 3 successful / 3 total
    #>
    #> Outer Loop Metrics (mean ± SD):
    #>               learner accuracy_mean accuracy_sd roc_auc_mean roc_auc_sd
    #> 1 logistic_reg/glmnet         0.619       0.048        0.577      0.134
    #>   accuracy_ci_lo accuracy_ci_hi roc_auc_ci_lo roc_auc_ci_hi
    #> 1          0.501          0.737         0.245         0.909
    #>
    #> Best Parameters (First 5 Folds):
    #>  penalty fold
    #>    0.763    1
    #>    0.000    2
    #>    0.995    3

    if (exists("tuned")) {
      # Fold-level status from the outer loop
      head(tuned$fold_status, 5)

      # Final tuned model is available when refit = TRUE
      is.null(tuned$final_model)
    } else {
      cat("Nested tuning object not available (dependencies missing).\n")
    }
    #> [1] FALSE

How to interpret this tuning output:

- `selection = "one_std_err"` chooses a model within one standard error of the best inner-CV score, then applies a deterministic simplicity tie-break. This usually favors more regularized/simpler models with similar expected performance.
- `tuned$metrics` and `tuned$metric_summary` remain the primary unbiased performance estimates (outer folds only).
- `tuned$final_model` is the deployable model fit on full data using the selected hyperparameters; it is for downstream prediction, not for estimating generalization.
- `tuned$fold_status` should be reviewed before reporting results. A large number of skipped/failed folds indicates unstable tuning and should trigger split/parameter redesign.

## 2.10 Advanced: Using Gradient Boosting with Parsnip

`bioLeak` natively supports `tidymodels` specifications. You can pass a `parsnip` model specification directly to `fit_resample()`. This allows you to use state-of-the-art algorithms like XGBoost, LightGBM, or SVMs while ensuring all preprocessing remains guarded.

    if (requireNamespace("parsnip", quietly = TRUE) &&
        requireNamespace("xgboost", quietly = TRUE) &&
        requireNamespace("recipes", quietly = TRUE)) {

      # 1. Define the model spec
      xgb_spec <- parsnip::boost_tree(
        mode = "classification",
        trees = 100,
        tree_depth = 6,
        learn_rate = 0.01
      ) |>
        parsnip::set_engine("xgboost")

      # 2. Define a recipe on data without 'subject' because split metadata
      # columns are excluded from predictors by fit_resample().
      df_for_rec <- df[, !names(df) %in% "subject"]

      rec_xgb <- recipes::recipe(outcome ~ ., data = df_for_rec) |>
        recipes::step_dummy(recipes::all_nominal_predictors()) |>
        recipes::step_impute_median(recipes::all_numeric_predictors())
      # Note: No need for step_rm(subject) because it's already gone!

      # 3. Fit
      fit_xgb <- fit_resample(
        df,
        outcome = "outcome",
        splits = splits,
        learner = xgb_spec,
        metrics = "auc",
        preprocess = rec_xgb
      )

      cat("XGBoost parsnip fit summary:\n")
      print(summary(fit_xgb))
    } else {
      cat("parsnip/xgboost/recipes not installed.\n")
    }
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #> XGBoost parsnip fit summary:
    #>
    #> ===========================
    #>  bioLeak Model Fit Summary
    #> ===========================
    #>
    #> Task: binomial
    #> Outcome: outcome
    #> Positive class: ClassB
    #> Learners: boost_tree/xgboost
    #> Total folds: 3
    #> Fold status: 3 success, 0 skipped, 0 failed
    #> Refit performed: Yes
    #> Hash: h825b7ff0
    #>
    #> Cross-validated metrics (mean ± SD):
    #>              learner auc_mean auc_sd auc_ci_lo auc_ci_hi
    #> 1 boost_tree/xgboost    0.616  0.058     0.388     0.844
    #>
    #> Audit overview:
    #>  fold n_train n_test            learner features_final
    #>     1      39     21 boost_tree/xgboost              3
    #>     2      39     21 boost_tree/xgboost              3
    #>     3      42     18 boost_tree/xgboost              3
    #>
    #>              learner  auc_mean     auc_sd auc_ci_lo auc_ci_hi
    #> 1 boost_tree/xgboost 0.6157407 0.05800909 0.3878946 0.8435869

The summary reports cross-validated AUC for a non-linear gradient boosting model. Use the mean \(\pm\) SD table to compare against baseline models, and confirm that any gains do not coincide with leakage signals in the audit diagnostics.

## 2.11 Advanced: Custom learners

Custom learners are used when a model is not available as a parsnip specification or when a lightweight wrapper around base R is needed.

Each custom learner must define `fit` and `predict` components. The `fit` function must accept `x`, `y`, `task`, and `weights` as inputs, while the `predict` function must accept `object` and `newdata`.

    custom_learners <- list(
      glm = list(
        fit = function(x, y, task, weights, ...) {
          df_fit <- data.frame(y = y, x, check.names = FALSE)
          stats::glm(y ~ ., data = df_fit,
                     family = stats::binomial(), weights = weights)
        },
        predict = function(object, newdata, task, ...) {
          as.numeric(stats::predict(object, newdata = as.data.frame(newdata),
                                    type = "response"))
        }
      )
    )

    cat("Custom learner names:\n")
    #> Custom learner names:
    names(custom_learners)
    #> [1] "glm"
    cat("Custom learner components (fit/predict):\n")
    #> Custom learner components (fit/predict):
    lapply(custom_learners, names)
    #> $glm
    #> [1] "fit"     "predict"

This output confirms that each custom learner provides both a `fit` and a `predict` function.

Custom learners can be used with `fit_resample()` as follows:

    fit_resample(
      ...,
      learner = "glm",
      custom_learners = custom_learners
    )

## 2.12 Visual diagnostics

bioLeak includes plotting helpers for quick diagnostic checks:

- `plot_fold_balance()` shows class balance per fold.
- `plot_overlap_checks()` highlights train/test overlap for a metadata column.
- `plot_time_acf()` checks autocorrelation in time-series predictions.
- `plot_calibration()` shows reliability of binomial probabilities.
- `plot_confounder_sensitivity()` summarizes performance by batch/study strata.

Tabular helpers are also available:

- `calibration_summary()` returns calibration curve data and ECE/MCE/Brier metrics.
- `confounder_sensitivity()` returns per-stratum performance summaries.

Misuse to avoid

- Plotting without predictions (fits with no successful folds).
- Using `plot_overlap_checks()` when the column is not present in the split metadata.
- Using `plot_time_acf()` without a time column or for non-temporal data.
- Using `plot_calibration()` outside binomial tasks.
- Using `plot_confounder_sensitivity()` with a metric unsupported by the task.

For classification, `plot_fold_balance()` uses the positive class recorded in `fit@info$positive_class`, or defaults to the second factor level if this is not set. For multiclass tasks, it shows per-class counts without a proportion line.

    if (requireNamespace("ggplot2", quietly = TRUE)) {
      plot_fold_balance(fit_safe)
    } else {
      cat("ggplot2 not installed; skipping fold balance plot.\n")
    }

The bar chart shows the counts of Positives (blue) and Negatives (tan) in each fold. The dashed blue line represents the proportion of the positive class.

In a well-stratified fit, this line remains relatively stable across folds. Large fluctuations indicate poor stratification, which can lead to unstable fold-level performance estimates.

    if (requireNamespace("ggplot2", quietly = TRUE)) {
      plot_calibration(fit_safe, bins = 10)
    } else {
      cat("ggplot2 not installed; skipping calibration plot.\n")
    }

Calibration curves compare predicted probabilities to observed event rates. Large deviations from the diagonal indicate miscalibration. Use `min_bin_n` to suppress tiny bins and `learner =` to select a model when multiple learners are stored in the fit.

    if (requireNamespace("ggplot2", quietly = TRUE)) {
      plot_confounder_sensitivity(fit_safe, confounders = c("batch", "study"),
                                  metric = "auc", min_n = 3)
    } else {
      cat("ggplot2 not installed; skipping confounder sensitivity plot.\n")
    }

Confounder sensitivity plots highlight whether performance varies substantially across batch or study strata.

    cal <- calibration_summary(fit_safe, bins = 10, min_bin_n = 5)
    conf_tbl <- confounder_sensitivity(fit_safe,
                                       confounders = c("batch", "study"),
                                       metric = "auc",
                                       min_n = 3)

    # Calibration metrics
    cal$metrics
    #>     n bins min_bin_n       ece       mce     brier
    #> 1 160   10         5 0.1005285 0.9113016 0.2499484

    # Confounder sensitivity table (first 6 rows)
    head(conf_tbl)
    #>   confounder level metric direction     value  n positive_rate
    #> 1      batch    B1    auc    higher 0.7569444 26     0.3076923
    #> 2      batch    B2    auc    higher 0.7196970 23     0.4782609
    #> 3      batch    B3    auc    higher 0.4153846 28     0.4642857
    #> 4      batch    B4    auc    higher 0.6944444 24     0.5000000
    #> 5      batch    B5    auc    higher 0.6071429 22     0.3636364
    #> 6      batch    B6    auc    higher 0.4941176 37     0.4594595

    if (requireNamespace("ggplot2", quietly = TRUE)) {
      plot_overlap_checks(fit_leaky, column = "subject")
      plot_overlap_checks(fit_safe, column = "subject")
    } else {
      cat("ggplot2 not installed; skipping overlap plots.\n")
    }

The overlap plots compare the number of unique subjects appearing in the training and test sets for each fold.

Top plot (`fit_leaky`)
The red line shows high overlap counts, accompanied by the explicit “WARNING: Overlaps detected!” annotation. This confirms that the same subjects appear in both the training and test sets, indicating information leakage.

Bottom plot (`fit_safe`)
The overlap line remains flat at zero across all folds. This confirms that `subject_grouped` splitting successfully keeps subjects isolated, ensuring that no subject appears in both sets simultaneously.

## 2.13 Audit leakage with `audit_leakage()`

`audit_leakage()` combines four diagnostics in one object:

- permutation gap: tests whether model signal is non-random
- batch or study association with folds (chi-square and Cramers V)
- target leakage scan: feature-wise outcome association to flag proxy columns
- duplicate and near-duplicate detection across train/test (by default) using feature similarity

Assumptions and misuse to avoid:

- `coldata` (or stored split metadata) must align with prediction IDs; rownames or a `row_id` column help alignment
- `X_ref` must align to prediction IDs (rownames, `row_id`, or matching order)
- `plot_perm_distribution()` requires `return_perm = TRUE`
- `target_threshold` should be set high enough to flag only strong proxies
- target leakage scan includes univariate associations plus an optional multivariate/interaction check (`target_scan_multivariate`, `*_components`, `*_interactions`, `*_B`); proxies outside `X_ref` can still pass undetected
- for rsample splits, set `perm_mode` (or `bioLeak_perm_mode`) so restricted permutations respect the intended design
- for time-series audits, ensure a valid time column and set `time_block` / `block_len` as needed

Interpretation Note: The label-permutation test refits models by default when refit inputs are available (`perm_refit = "auto"` with `store_refit_data = TRUE`) and `B <= perm_refit_auto_max`. Otherwise it keeps predictions fixed, so the p-value quantifies prediction-label association rather than a full refit null. A large gap indicates a non-random association. To determine if that signal is real or leaked, check the Batch Association (confounding), Target Leakage Scan (proxy features), and Duplicate Detection (memorization) tables. Use `perm_refit = FALSE` to force fixed-prediction shuffles or `perm_refit = TRUE` with `perm_refit_spec` to always refit.

Use `feature_space = "rank"` to compare samples by rank profiles, and `sim_method` (`cosine` or `pearson`) to control similarity. For large `n`, `nn_k` and `max_pairs` limit duplicate searches. Use `duplicate_scope = "all"` to include within-fold duplicates (data-quality checks). Use `ci_method = "if"` (influence-function) or `ci_method = "bootstrap"` with `boot_B` to obtain a confidence interval for the permutation gap; `include_z` controls whether a z-score is reported. Set `perm_stratify = "auto"` to stratify permutations only when outcomes support it.

Leakage-aware audit

    # Use df_leaky (the original 160-row data) so X_ref aligns with fit_safe's predictions.
    # df may have been redefined to a smaller dataset in the tidymodels section above.
    X_ref <- df_leaky[, predictors]
    X_ref[c(1, 5), ] <- X_ref[1, ]

    audit <- audit_leakage(
      fit_safe,
      metric = "auc",
      B = 20,
      perm_stratify = TRUE,
      batch_cols = c("batch", "study"),
      X_ref = X_ref,
      sim_method = "cosine",
      sim_threshold = 0.995,
      return_perm = TRUE
    )
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%

    cat("Leakage audit summary:\n")
    #> Leakage audit summary:
    summary(audit)
    #>
    #> ==============================
    #>  bioLeak Leakage Audit Summary
    #> ==============================
    #>
    #> Task: binomial | Outcome: outcome | Splitting mode: subject_grouped | Positive class: case
    #> Hash: hba29ffa5 | Folds: 5 | Repeats: 1
    #>
    #> Label-Permutation Association Test:
    #>   Method: refit per permutation (auto)
    #>   Observed metric: 0.614
    #>   Permuted mean ± SD: 0.566 ± 0.051
    #>   Gap: 0.049 (larger gap = stronger non-random signal)
    #>   This test does NOT diagnose information leakage. Use the Batch Association,
    #>   Target Leakage Scan, and Duplicate Detection sections to check for leakage.
    #>
    #> Batch / Study Association:
    #>   batch (repeat 1): χ² = 17.043 (df = 20.000), p = 0.650
    #>   study (repeat 1): χ² = 9.487 (df = 12.000), p = 0.661
    #>
    #> Target Leakage Scan:
    #>   Features checked: 3 | Flagged (score ≥ 0.900): 0
    #>   No strong proxy features detected.
    #>
    #> Multivariate Target Scan:
    #>   Metric: auc | Score: 0.302 | p = 0.030
    #>   Features: 3 | Components: 3 | Interactions: 3 | Permutations: 100
    #>
    #> Near-Duplicate Samples:
    #>   Scope: train/test only
    #>   20 pairs detected above cosine ≥ 0.995
    #>   Example pairs:
    #>    mechanism_class   i   j       sim cross_fold   cos_sim
    #>  duplicate_overlap   1   5 1.0000000       TRUE 1.0000000
    #>  duplicate_overlap  35  61 0.9994623       TRUE 0.9994623
    #>  duplicate_overlap  80 106 0.9990700       TRUE 0.9990700
    #>  duplicate_overlap 120 153 0.9978776       TRUE 0.9978776
    #>  duplicate_overlap   7 107 0.9976746       TRUE 0.9976746
    #>
    #> Mechanism Risk Assessment:
    #>        mechanism_class flagged        evidence statistic  p_value
    #>      non_random_signal   FALSE permutation_gap 0.0485189 0.238095
    #>  confounding_alignment   FALSE     batch_assoc 0.1631865 0.650174
    #>   proxy_target_leakage   FALSE    target_assoc 0.3556299       NA
    #>      duplicate_overlap    TRUE      duplicates 1.0000000       NA
    #>     temporal_lookahead   FALSE      duplicates        NA       NA
    #>
    #> Interpretation:
    #>   ✓ Modest non-random signal.
    if (!is.null(audit@permutation_gap) && nrow(audit@permutation_gap) > 0) {
      # Permutation significance results
      audit@permutation_gap
    }
    #>     mechanism_class metric_obs perm_mean   perm_sd       gap       z  p_value
    #> 1 non_random_signal   0.614111  0.565592 0.0509821 0.0485189 1.08899 0.238095
    #>   n_perm
    #> 1     20
    if (!is.null(audit@batch_assoc) && nrow(audit@batch_assoc) > 0) {
      # Batch/study association with folds (Cramer's V)
      audit@batch_assoc
    } else {
      cat("No batch or study associations detected.\n")
    }
    #>             mechanism_class batch_col repeat_id      stat df     pval  cramer_v
    #> batch confounding_alignment     batch         1 17.043088 20 0.650174 0.1631865
    #> study confounding_alignment     study         1  9.487022 12 0.660865 0.1405867
    if (!is.null(audit@target_assoc) && nrow(audit@target_assoc) > 0) {
      # Top features by target association score
      head(audit@target_assoc)
    } else {
      cat("No target leakage scan results available.\n")
    }
    #>        mechanism_class feature    type metric     value      score      p_value
    #> 1 proxy_target_leakage      x1 numeric    auc 0.6778149 0.35562988 0.0001205498
    #> 2 proxy_target_leakage      x2 numeric    auc 0.4207676 0.15846472 0.0868346476
    #> 3 proxy_target_leakage      x3 numeric    auc 0.5446727 0.08934544 0.3346967486
    #>     n n_levels  flag p_value_adj flag_fdr
    #> 1 160       NA FALSE          NA    FALSE
    #> 2 160       NA FALSE          NA    FALSE
    #> 3 160       NA FALSE          NA    FALSE
    if (!is.null(audit@duplicates) && nrow(audit@duplicates) > 0) {
      # Top duplicate/near-duplicate pairs by similarity
      head(audit@duplicates)
    } else {
      cat("No near-duplicates detected.\n")
    }
    #>      mechanism_class   i   j       sim cross_fold   cos_sim
    #> 1  duplicate_overlap   1   5 1.0000000       TRUE 1.0000000
    #> 12 duplicate_overlap  35  61 0.9994623       TRUE 0.9994623
    #> 18 duplicate_overlap  80 106 0.9990700       TRUE 0.9990700
    #> 25 duplicate_overlap 120 153 0.9978776       TRUE 0.9978776
    #> 4  duplicate_overlap   7 107 0.9976746       TRUE 0.9976746
    #> 22 duplicate_overlap 104 142 0.9968751       TRUE 0.9968751

The permutation table reports the observed metric, the mean under random label permutation, the gap (difference), and a permutation p-value. For metrics where higher values indicate better performance, larger gaps reflect stronger non-random signal.

The batch association table reports chi-square statistics and Cramer’s V. Large p-values and small V values indicate that folds are not aligned with batch or study labels (which is the desired outcome when these should be independent).

The target scan table lists features with the strongest associations with the outcome. For numeric features, the score is \(|\mathrm{AUC} - 0.5| \times 2\) for classification or the absolute correlation for regression. For categorical features, the score is Cramér’s V or eta-squared. Scores closer to 1 indicate stronger outcome association.

The duplicate table lists pairs of samples with near-identical profiles that cross train/test folds (by default). In this setup, the artificially duplicated pair (`X_ref[c(1, 5), ] <- X_ref[1, ]`) should appear near the top of the list. Use `duplicate_scope = "all"` to include within-fold duplicates.

Mechanism risk assessment

`summary(audit)` now includes a Mechanism Risk Assessment section that classifies leakage evidence into four mechanism classes:

Mechanism class | Evidence slot | Flagging rule
--- | --- | ---
`non_random_signal` | permutation gap | p ≤ 0.05 and gap > 0
`confounding_alignment` | batch association | p ≤ 0.05 and Cramér’s V ≥ 0.1
`proxy_target_leakage` | target scan | any feature flagged (FDR or raw)
`duplicate_overlap` | duplicates | any cross-fold duplicates present
`temporal_lookahead` | duplicates | cross-fold duplicates in time-series splits

The summary is stored in `audit@info$mechanism_summary` and can be accessed directly for programmatic triage:

    mech <- audit@info$mechanism_summary
    if (is.data.frame(mech) && nrow(mech) > 0) {
      mech
    } else {
      cat("No mechanism summary available.\n")
    }
    #>         mechanism_class flagged        evidence statistic  p_value
    #> 1     non_random_signal   FALSE permutation_gap 0.0485189 0.238095
    #> 2 confounding_alignment   FALSE     batch_assoc 0.1631865 0.650174
    #> 3  proxy_target_leakage   FALSE    target_assoc 0.3556299       NA
    #> 4     duplicate_overlap    TRUE      duplicates 1.0000000       NA
    #> 5    temporal_lookahead   FALSE      duplicates        NA       NA

Each row reports whether the mechanism was flagged, the evidence type, and the associated test statistic and p-value. This provides a quick triage overview before drilling into individual audit slots.

    if (requireNamespace("ggplot2", quietly = TRUE)) {
      plot_perm_distribution(audit)
    } else {
      cat("ggplot2 not installed; skipping permutation plot.\n")
    }

The histogram shows the null distribution (gray bars) of the performance metric under random label permutation.

The blue dashed line represents the average performance of a random model (the permuted mean). The red solid line represents the observed performance of the fitted model.

A genuine signal is indicated when the red line lies well to the right of the gray distribution; overlap suggests weak or unstable signal.

Use the mechanism risk assessment (`audit@info$mechanism_summary`) as a quick triage tool: flagged mechanisms point you to the specific audit slots that warrant deeper investigation.

Audit per learner

    if (requireNamespace("ranger", quietly = TRUE) &&
        requireNamespace("parsnip", quietly = TRUE)) {

        # 1. Define specs
        # Standard GLM (no tuning)
        spec_glm <- parsnip::logistic_reg() |>
            parsnip::set_engine("glm")

        # Random Forest
        spec_rf <- parsnip::rand_forest(
            mode = "classification",
            trees = 100
        ) |>
            parsnip::set_engine("ranger")

        # 2. Fit using the current split object
        fit_multi <- fit_resample(
            df,
            outcome = "outcome",
            splits = splits,
            learner = list(glm = spec_glm, rf = spec_rf),
            metrics = "auc"
        )

        # 3. Run the audit
        audits <- audit_leakage_by_learner(fit_multi, metric = "auc", B = 20)
        cat("Per-learner audit summary:\n")
        print(audits)

    } else {
        cat("ranger/parsnip not installed.\n")
    }
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #> Per-learner audit summary:
    #> LeakAuditList with 2 learners
    #>  learner metric_obs perm_mean    gap p_value      z n_perm batch_p_min
    #>      glm      0.537     0.609 -0.071   0.857 -0.800     20          NA
    #>       rf      0.559     0.586 -0.027   0.667 -0.308     20          NA
    #>  batch_col_min_p batch_v_max batch_col_max_v dup_pairs dup_threshold
    #>             <NA>          NA            <NA>        NA         0.995
    #>             <NA>          NA            <NA>        NA         0.995
    #>  dup_max_sim
    #>           NA
    #>           NA

This example uses ranger for the random forest specification; if it is not installed, the code chunk is skipped.

Use `parallel_learners = TRUE` to audit learners concurrently when future.apply is available.

The printed table summarizes each learner’s observed metric, permutation gap, p-value, and key batch/duplicate summaries. Use it to compare signal strength across models while checking for leakage risks. Pass `learners =` to audit a subset, or `mc.cores =` to control parallel workers.

HTML audit report

`audit_report()` accepts either a `LeakAudit` or a `LeakFit` object. When a `LeakFit` is provided, it first runs `audit_leakage()` and then forwards any additional arguments to the audit step. If multiple learners were fit, pass `learner =` via `...` to select one. Use `open = TRUE` to open the report in a browser after rendering.

    if (requireNamespace("rmarkdown", quietly = TRUE) && rmarkdown::pandoc_available()) {
      report_path <- audit_report(audit, output_dir = ".")
      cat("HTML report written to:\n", report_path, "\n")
    } else {
      cat("rmarkdown or pandoc not available; skipping audit report rendering.\n")
    }

The report path points to a standalone HTML file containing the same audit tables and plots, suitable for sharing with collaborators or archiving as a quality control record.

## 2.14 Time-series leakage checks

Time-series data require special handling. Random splits can leak information from the future into the past. Use `mode = "time_series"` with a prediction `horizon`, and audit with block permutations. Choose `time_block = "circular"` or `"stationary"`; when `block_len` is NULL, the audit uses a default block length (~10% of the test block size, minimum 5).

Leaky example: lookahead feature

    time_splits <- make_split_plan(
      df_time,
      outcome = "outcome",
      mode = "time_series",
      time = "time",
      v = 4,
      horizon = 1
    )

    cat("Time-series splits summary:\n")
    #> Time-series splits summary:
    time_splits
    #> LeakSplits object (mode = time_series, v = 4, repeats = 1)
    #> Outcome: outcome | Stratified: FALSE | Nested: FALSE
    #> ------------------------------------------------------
    #>   fold repeat_id train_n test_n
    #> 1    2         1      40     40
    #> 2    3         1      80     40
    #> 3    4         1     120     40
    #> ------------------------------------------------------
    #> Total folds: 3 | Hash: a0169c15b3a0826566edf98a905fbc04

    fit_time_leaky <- fit_resample(
      df_time,
      outcome = "outcome",
      splits = time_splits,
      learner = spec,
      metrics = "auc"
    )
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%

    cat("Time-series leaky fit summary:\n")
    #> Time-series leaky fit summary:
    summary(fit_time_leaky)
    #>
    #> ===========================
    #>  bioLeak Model Fit Summary
    #> ===========================
    #>
    #> Task: binomial
    #> Outcome: outcome
    #> Positive class: case
    #> Learners: logistic_reg/glm
    #> Total folds: 3
    #> Fold status: 3 success, 0 skipped, 0 failed
    #> Refit performed: Yes
    #> Hash: h3b18276c
    #>
    #> Cross-validated metrics (mean ± SD):
    #>            learner auc_mean auc_sd auc_ci_lo auc_ci_hi
    #> 1 logistic_reg/glm    0.573  0.132     0.055     1.091
    #>
    #> Audit overview:
    #>  fold n_train n_test          learner features_final
    #>     1      40     40 logistic_reg/glm             14
    #>     2      80     40 logistic_reg/glm             14
    #>     3     120     40 logistic_reg/glm             14

Time-series splitting trains on growing windows, so performance can differ from standard CV simply because early folds have smaller training sets. Regardless of the score, the presence of the `leak_future` feature makes this estimate methodologically invalid.

Leakage-safe alternative: remove lookahead and audit with blocks

    df_time_safe <- df_time
    df_time_safe$leak_future <- NULL

    fit_time_safe <- fit_resample(
      df_time_safe,
      outcome = "outcome",
      splits = time_splits,
      learner = spec,
      metrics = "auc"
    )
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%

    cat("Time-series safe fit summary:\n")
    #> Time-series safe fit summary:
    summary(fit_time_safe)
    #>
    #> ===========================
    #>  bioLeak Model Fit Summary
    #> ===========================
    #>
    #> Task: binomial
    #> Outcome: outcome
    #> Positive class: case
    #> Learners: logistic_reg/glm
    #> Total folds: 3
    #> Fold status: 3 success, 0 skipped, 0 failed
    #> Refit performed: Yes
    #> Hash: h3b18276c
    #>
    #> Cross-validated metrics (mean ± SD):
    #>            learner auc_mean auc_sd auc_ci_lo auc_ci_hi
    #> 1 logistic_reg/glm    0.608  0.042     0.442     0.773
    #>
    #> Audit overview:
    #>  fold n_train n_test          learner features_final
    #>     1      40     40 logistic_reg/glm             13
    #>     2      80     40 logistic_reg/glm             13
    #>     3     120     40 logistic_reg/glm             13

    audit_time <- audit_leakage(
      fit_time_safe,
      metric = "auc",
      B = 20,
      time_block = "stationary",
      block_len = 5
    )
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================================| 100%

    cat("Time-series leakage audit summary:\n")
    #> Time-series leakage audit summary:
    summary(audit_time)
    #>
    #> ==============================
    #>  bioLeak Leakage Audit Summary
    #> ==============================
    #>
    #> Task: binomial | Outcome: outcome | Splitting mode: time_series | Positive class: case
    #> Hash: h3b18276c | Folds: 3 | Repeats: 1
    #>
    #> Label-Permutation Association Test:
    #>   Method: refit per permutation (auto)
    #>   Observed metric: 0.601
    #>   Permuted mean ± SD: 0.544 ± 0.032
    #>   Gap: 0.057 (larger gap = stronger non-random signal)
    #>   This test does NOT diagnose information leakage. Use the Batch Association,
    #>   Target Leakage Scan, and Duplicate Detection sections to check for leakage.
    #>
    #> Batch / Study Association:
    #>   batch (repeat 1): χ² = 8.237 (df = 10.000), p = 0.606
    #>   study (repeat 1): χ² = 2.383 (df = 6.000), p = 0.881
    #>
    #> Target Leakage Scan: not available.
    #>
    #> Multivariate Target Scan: not available.
    #>
    #> No near-duplicates detected.
    #>
    #> Mechanism Risk Assessment:
    #>        mechanism_class flagged        evidence statistic   p_value
    #>      non_random_signal   FALSE permutation_gap 0.0571331 0.0952381
    #>  confounding_alignment   FALSE     batch_assoc 0.1852551 0.6057315
    #>   proxy_target_leakage   FALSE    target_assoc        NA        NA
    #>      duplicate_overlap   FALSE      duplicates        NA        NA
    #>     temporal_lookahead   FALSE      duplicates        NA        NA
    #>
    #> Interpretation:
    #>   ✓ Strong non-random signal.
    if (!is.null(audit_time@permutation_gap) && nrow(audit_time@permutation_gap) > 0) {
      # Time-series permutation significance results
      audit_time@permutation_gap
    }
    #>     mechanism_class metric_obs perm_mean   perm_sd       gap       z   p_value
    #> 1 non_random_signal   0.600679  0.543546 0.0323473 0.0571331 1.06538 0.0952381
    #>   n_perm
    #> 1     20

    if (requireNamespace("ggplot2", quietly = TRUE)) {
      plot_time_acf(fit_time_safe, lag.max = 20)
    } else {
      cat("ggplot2 not installed; skipping ACF plot.\n")
    }

The safe fit summary provides the leakage-resistant performance estimate. Compare leaky vs. safe fits to gauge inflation risk; `features_final` should drop when the lookahead feature is removed.

The ACF plot shows the autocorrelation of out-of-fold predictions by lag. The dashed red lines represent the 95% confidence interval for white noise; large bars outside the bands indicate residual temporal dependence. `plot_time_acf()` requires numeric predictions and time metadata aligned to the fit.

## 2.15 Cross-validation uncertainty with `cv_ci()`

Standard cross-validation yields one metric value per fold. Because folds share training data, the per-fold values are positively correlated: the naive standard error `sd / sqrt(K)` systematically underestimates variability, making the resulting confidence intervals too narrow. `cv_ci()` addresses this with two approaches:

- `"normal"` (default): uses the naive SEM `sd / sqrt(K)` with a t-distribution on `K − 1` degrees of freedom. Fast and appropriate when fold overlap is low (large datasets, many folds).

- `"nadeau_bengio"`: applies the Nadeau–Bengio (2003) variance correction `(1/K + n_test / n_train) × var(x)`, explicitly accounting for the fraction of training samples shared across folds. Recommended for small datasets or few-fold CV where fold correlation is non-negligible.

The function accepts any data frame with `fold` and `learner` columns — such as `fit@metrics` from a `LeakFit` object — and returns one row per learner with four columns per metric: `<metric>_mean`, `<metric>_sd`, `<metric>_ci_lo`, and `<metric>_ci_hi`.

    # Standard 95% CI from the leakage-safe fit
    ci_std <- cv_ci(fit_safe@metrics, level = 0.95, method = "normal")
    cat("Standard 95% CI:\n")
    #> Standard 95% CI:
    print(ci_std)
    #>            learner  auc_mean     auc_sd auc_ci_lo auc_ci_hi accuracy_mean
    #> 1 logistic_reg/glm 0.6943939 0.04761304 0.6352745 0.7535133     0.5975198
    #>   accuracy_sd accuracy_ci_lo accuracy_ci_hi
    #> 1   0.1172633      0.4519182      0.7431215

    # Nadeau-Bengio corrected CI
    # For K-fold CV: n_train ≈ (K-1)/K × n, n_test ≈ n/K
    K_folds  <- length(unique(fit_safe@metrics$fold))
    n_total  <- nrow(fit_safe@splits@info$coldata)

    ci_nb <- cv_ci(
      fit_safe@metrics,
      level   = 0.95,
      method  = "nadeau_bengio",
      n_train = round(n_total * (K_folds - 1L) / K_folds),
      n_test  = round(n_total / K_folds)
    )
    cat("Nadeau-Bengio corrected 95% CI:\n")
    #> Nadeau-Bengio corrected 95% CI:
    print(ci_nb)
    #>            learner  auc_mean     auc_sd auc_ci_lo auc_ci_hi accuracy_mean
    #> 1 logistic_reg/glm 0.6943939 0.04761304 0.6057148  0.783073     0.5975198
    #>   accuracy_sd accuracy_ci_lo accuracy_ci_hi
    #> 1   0.1172633      0.3791174      0.8159223

The Nadeau–Bengio interval is typically wider because it correctly inflates the variance to account for fold overlap. Use it whenever the training-set fraction shared between consecutive folds is large — for example, 5-fold CV reuses 80% of training data across fold pairs, so correcting for this correlation matters.

When multiple learners were fit, `cv_ci()` returns one row per learner. For repeated K-fold CV, average `n_train` and `n_test` across folds before passing them in; `fit@metrics` already stores per-fold counts if `audit_leakage()` annotated the object.

# 3 Delta Leakage Sensitivity Index: Quantifying Performance Inflation

## 3.1 Motivation

`audit_leakage()` detects whether leakage is present via permutation tests and confounding checks. It cannot, however, directly answer: by how much does leakage inflate my reported performance? The Delta Leakage Sensitivity Index (ΔLSI) addresses this by rigorously comparing a naive (potentially leaky) cross-validation pipeline against a leakage-guarded one run on the same data.

## 3.2 Mathematical framework

### 3.2.1 Notation

Let \(\mathcal{D} = \{(x_i, y_i)\}_{i=1}^n\) denote a dataset of \(n\) observations. A repeated \(K\)-fold cross-validation design with \(R\) independent repeats partitions the observation indices into \(RK\) folds. Let \(\mathcal{T}_{r,k} \subset \{1, \ldots, n\}\) denote the test index set for fold \(k\) in repeat \(r\), with \(n_{r,k} = |\mathcal{T}_{r,k}|\). Let \(m_{r,k}^{(p)}\) denote the performance metric evaluated on fold \((r,k)\) under pipeline \(p \in \{\text{naive},\, \text{guarded}\}\).

### 3.2.2 Stage 1: Within-repeat aggregation

For each repeat \(r\) and pipeline \(p\), the \(K\) fold-level metric values are combined into a size-weighted repeat mean:

\[\mu_r^{(p)} = \frac{\displaystyle\sum_{k=1}^{K} n_{r,k}\, m_{r,k}^{(p)}}{\displaystyle\sum_{k=1}^{K} n_{r,k}}\]

The test-fold size \(n_{r,k}\) is the weight, so folds with more held-out observations contribute proportionally more to the repeat-level estimate. Folds with missing metric values are excluded from both the numerator and denominator. This produces the `@repeats_naive` and `@repeats_guarded` data frames (column `metric`), one row per repeat.

### 3.2.3 Stage 2: Per-repeat inflation score

For each repeat \(r\), the signed performance gap between the two pipelines is:

\[\Delta_r = s \cdot \bigl(\mu_r^{\text{naive}} - \mu_r^{\text{guarded}}\bigr), \qquad r = 1, \ldots, R\]

where

\[s = \begin{cases} +1 & \text{if the metric is higher-is-better (e.g.\ AUC, accuracy)} \\ -1 & \text{if the metric is lower-is-better (e.g.\ RMSE, MAE, log-loss)} \end{cases}\]

By construction \(\Delta_r > 0\) if and only if the naive pipeline is more optimistic than the guarded pipeline in repeat \(r\), irrespective of metric direction. Because both pipelines are evaluated on the same test fold \(\mathcal{T}_{r,k}\) (the paired design), any between-fold variability common to both pipelines cancels in \(\Delta_r\), concentrating the signal on genuine pipeline differences rather than partition noise.

### 3.2.4 Point estimates

Two summaries of the inflation vector \(\boldsymbol{\Delta} = (\Delta_1, \ldots, \Delta_R)^\top\) are reported.

Arithmetic mean — `delta_metric`:

\[\hat{\delta}_{\text{metric}} = \frac{1}{R} \sum_{r=1}^{R} \Delta_r\]

This is the most directly interpretable quantity. For AUC, \(\hat{\delta}_{\text{metric}} = 0.08\) means the naive pipeline’s reported AUC exceeds the guarded pipeline’s by approximately \(0.08\) units, averaged across all repeats. This value is stored in `@delta_metric`.

Huber M-estimate — `delta_lsi`:

\[\hat{\delta}_{\text{LSI}} = \operatorname*{arg\,min}_{\mu} \sum_{r=1}^{R} \rho_{k}\!\left(\frac{\Delta_r - \mu}{\hat{\sigma}}\right)\]

where \(\rho_k\) is Huber’s loss function with tuning constant \(k = 1.345\):

\[\rho_k(u) = \begin{cases} \tfrac{1}{2}u^2 & |u| \leq k \\ k|u| - \tfrac{1}{2}k^2 & |u| > k \end{cases}\]

and \(\hat{\sigma} = 1.4826 \times \operatorname{MAD}(\boldsymbol{\Delta})\) is a fixed robust scale estimate. This minimisation is solved via iteratively reweighted least squares (IRLS):

1. Initialise: \(\hat{\mu}^{(0)} = \operatorname{median}(\boldsymbol{\Delta})\), \(\hat{\sigma} = 1.4826 \times \operatorname{MAD}(\boldsymbol{\Delta})\)
2. Compute standardised residuals: \(u_r^{(t)} = (\Delta_r - \hat{\mu}^{(t)}) / \hat{\sigma}\)
3. Assign Huber influence weights: \(w_r^{(t)} = \min\!\left(1,\; k / |u_r^{(t)}|\right)\)
4. Update location: \(\hat{\mu}^{(t+1)} = \displaystyle\frac{\sum_{r=1}^{R} w_r^{(t)} \Delta_r}{\sum_{r=1}^{R} w_r^{(t)}}\)
5. Repeat until \(|\hat{\mu}^{(t+1)} - \hat{\mu}^{(t)}| < 10^{-8}(1 + |\hat{\mu}^{(t)}|)\)

Note that \(\hat{\sigma}\) is fixed at its initial value throughout iteration; only the location \(\hat{\mu}\) is updated. Outlier repeats with \(|u_r^{(t)}| > k\) receive weight \(w_r < 1\), reducing their contribution to the update. When \(\hat{\sigma} = 0\) (all \(\Delta_r\) identical), the estimator returns the common value without iteration. This quantity is stored in `@delta_lsi` and is the primary reported point estimate.

The two estimates coincide when \(\{\Delta_r\}\) are symmetric and free of outliers. When they differ substantially, `delta_lsi` is more trustworthy because it is not pulled by a single anomalous repeat.

Unpaired fallback. When the two fits do not share the same fold structure (`@info$paired = FALSE`), the paired vector \(\boldsymbol{\Delta}\) cannot be formed. The point estimates are instead computed from the separately summarised repeat-mean vectors \(\boldsymbol{\mu}^{\text{naive}}\) and \(\boldsymbol{\mu}^{\text{guarded}}\):

\[\hat{\delta}_{\text{metric}}^{\text{unpaired}} = s \cdot \bigl(\bar{\mu}^{\text{naive}} - \bar{\mu}^{\text{guarded}}\bigr), \qquad \hat{\delta}_{\text{LSI}}^{\text{unpaired}} = s \cdot \bigl(\hat{H}(\boldsymbol{\mu}^{\text{naive}}) - \hat{H}(\boldsymbol{\mu}^{\text{guarded}})\bigr)\]

where \(\bar{\mu}\) is the arithmetic mean and \(\hat{H}\) is the Huber M-estimator applied to each pipeline’s repeat-means vector independently. No inference (sign-flip test or BCa interval) is available in the unpaired case (\(R_{\text{eff}} = 0\)).

## 3.3 Setting up a two-pipeline comparison

A ΔLSI analysis requires two `LeakFit` objects:

1. Naive fit — standard cross-validation without leakage safeguards (e.g., random row splits, global preprocessing, or leaky features).
2. Guarded fit — leakage-aware cross-validation using subject-grouped splits and preprocessing fit only on training folds.

For the most powerful paired design, generate a single `LeakSplits` object and pass it to both `fit_resample()` calls. Pairing is validated automatically by comparing the sorted test-set indices of every fold; if the structures differ despite equal repeat counts, a warning is issued and analysis falls back to unpaired mode.

    set.seed(100)
    n_d    <- 120L
    subj_d <- rep(paste0("P", seq_len(30L)), each = 4L)   # 30 subjects, 4 obs each

    df_dlsi_guarded <- data.frame(
      subject = subj_d,
      outcome = factor(sample(c("case", "control"), n_d, replace = TRUE)),
      x1      = rnorm(n_d),
      x2      = rnorm(n_d)
    )

    # Leaky feature: subject-level mean of outcome
    # With subject-grouped splits, test subjects' leak value encodes their outcome
    df_dlsi_naive <- df_dlsi_guarded
    df_dlsi_naive$leak <- ave(
      as.numeric(df_dlsi_guarded$outcome == "case"),
      subj_d, FUN = mean
    )

    # Shared splits: 5-fold × 5 repeats → R_eff = 5 (tier C: point + p-value)
    splits_dlsi <- make_split_plan(
      df_dlsi_guarded,
      outcome = "outcome",
      mode    = "subject_grouped",
      group   = "subject",
      v       = 5L,
      repeats = 5L
    )

    # Reuse the parsnip spec (base R glm) defined earlier
    # Naive fit: uses leaky feature (leak encodes test-subject outcome label)
    fit_dlsi_naive <- fit_resample(
      df_dlsi_naive,
      outcome = "outcome",
      splits  = splits_dlsi,
      learner = spec,
      metrics = "auc",
      seed    = 1L
    )
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%

    # Guarded fit: same splits, clean features only
    fit_dlsi_guarded <- fit_resample(
      df_dlsi_guarded,
      outcome = "outcome",
      splits  = splits_dlsi,
      learner = spec,
      metrics = "auc",
      seed    = 1L
    )
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%

    cat("Naive   AUC:", round(mean(fit_dlsi_naive@metrics$auc,   na.rm = TRUE), 3), "\n")
    #> Naive   AUC: 0.716
    cat("Guarded AUC:", round(mean(fit_dlsi_guarded@metrics$auc, na.rm = TRUE), 3), "\n")
    #> Guarded AUC: 0.547

## 3.4 Running `delta_lsi()`

    result_dlsi <- delta_lsi(
      fit_leaky   = fit_dlsi_naive,
      fit_guarded = fit_dlsi_guarded,
      metric      = "auc",
      M_boot      = 300L,
      M_flip      = 300L,
      seed        = 1L
    )

    summary(result_dlsi)
    #>
    #> =====================================
    #>  bioLeak Delta LSI Summary
    #> =====================================
    #>
    #> Metric:            auc
    #> Exchangeability:   iid
    #> Inference tier:    C_signflip
    #> R_eff:             5  (R_leaky=5, R_guarded=5, paired=TRUE)
    #>
    #> Leaky pipeline:    0.716
    #> Guarded pipeline:  0.547
    #>
    #> Point estimates (leaky - guarded; positive = leakage inflation):
    #>   delta_metric:  0.169  (raw metric difference)
    #>   delta_lsi:     0.169  (Huber-robust)
    #>
    #> Hypothesis test  [H0: no systematic inflation; paired repeat signs exchangeable]:
    #>   Sign-flip p:   0.0625
    #>
    #> Inference valid:   NO (C_signflip; need paired R_eff >= 20)

`summary()` reports seven quantities; here is how to read each one:

- Naive / Guarded pipeline values: the Huber-robust mean AUC across repeats for each pipeline in isolation. Compare these directly to see the raw performance level of each pipeline before computing the gap.

- `delta_metric`: the arithmetic mean of the per-repeat inflation scores \(\Delta_r\). Interpret this in the original metric’s units: a value of 0.10 means the naive pipeline reports AUC ~0.10 higher than the guarded pipeline, averaged across repeats. This is the most intuitive summary for reporting.

- `delta_lsi`: the Huber M-estimate of \(\{\Delta_r\}\). Because it down-weights outlier repeats, it is the more reliable point estimate when repeat-to-repeat variation is high. Values very close to `delta_metric` indicate a consistent, non-skewed inflation pattern.

- 95% BCa CI (shown when `R_eff ≥ 10`): a bootstrap confidence interval for `delta_lsi`. If the lower bound is above zero the inflation is unlikely to be a sampling artefact. If the interval spans zero the inflation estimate is uncertain given the number of repeats.

- Sign-flip p-value (shown when `R_eff ≥ 5`): tests H₀: no leakage inflation (mean \(\Delta_r = 0\)) via a randomization test. A small p-value (e.g., < 0.05) indicates the observed inflation is unlikely under the null of exchangeable pipeline labels.

- Inference tier and `inference_ok`: summarise how much statistical weight the design supports. Only tier A (`R_eff ≥ 20`, `inference_ok = TRUE`) provides a full combination of point estimate + CI + p-value. Lower tiers report what is reliable given the available repeats. The formal tier definitions and their rationale are given in the Statistical inference section below.

## 3.5 Designing for a target inference tier

The formal tier thresholds have direct implications for how you set up your repeated CV design. Plan the number of folds and repeats around the tier you need:

- Tier D (`D_insufficient`, point estimates only): any \(R_{\text{eff}} < 5\) — useful for exploratory comparisons where no formal test is required.
- Tier C (`C_signflip`, sign-flip p-value available): 5-fold \(\times\) 5 repeats \(\to R_{\text{eff}} = 5\). The smallest achievable p-value is \(1/2^5 = 0.031\). For `exchangeability = "blocked_time"` the minimum p-value is \(1/2^{n_{\text{blocks}}}\); at least 5 blocks are required for the test to be available.
- Tier B (`B_signflip_ci`, BCa CI additionally available): 5-fold \(\times\) 10 repeats \(\to R_{\text{eff}} = 10\). Both sign-flip p-value and BCa CI are reported.
- Tier A (`A_full_inference`, `inference_ok = TRUE`): 5-fold \(\times\) 20 repeats \(\to R_{\text{eff}} = 20\). Recommended for any published or pre-registered ΔLSI analysis.

For paired designs `R_eff` equals the number of matched repeats. For unpaired designs `R_eff = 0` and only the raw point estimates are available.

    cat("Tier:          ", result_dlsi@tier, "\n")
    #> Tier:           C_signflip
    cat("R_eff:         ", result_dlsi@R_eff, "\n")
    #> R_eff:          5
    cat("inference_ok:  ", result_dlsi@inference_ok, "\n")
    #> inference_ok:   FALSE

## 3.6 Statistical inference

### 3.6.1 Sign-flip randomization test

The sign-flip test (Phipson & Smyth, 2010) assesses the null hypothesis that there is no systematic performance inflation:

\[H_0: \mathbb{E}[\Delta_r] = 0 \quad \text{(no leakage-induced inflation)}\]

The test exploits sign-exchangeability: under \(H_0\), each \(\Delta_r\) is symmetrically distributed around zero, so randomly flipping the sign of any \(\Delta_r\) leaves the distribution of the test statistic unchanged. The test statistic is the arithmetic mean of the inflation scores:

\[T_{\text{obs}} = \frac{1}{R} \sum_{r=1}^{R} \Delta_r\]

For each sign vector \(\boldsymbol{\epsilon} = (\epsilon_1, \ldots, \epsilon_R)\) with \(\epsilon_r \in \{-1, +1\}\), a null-distribution realisation is:

\[T_{\boldsymbol{\epsilon}} = \frac{1}{R} \sum_{r=1}^{R} \epsilon_r \Delta_r\]

The two-sided p-value is the proportion of sign vectors yielding a statistic at least as extreme as observed:

\[p = \frac{\#\bigl\{\boldsymbol{\epsilon} \in \mathcal{E} : |T_{\boldsymbol{\epsilon}}| \geq |T_{\text{obs}}|\bigr\}}{|\mathcal{E}|}\]

Two computation regimes are used:

- Exact enumeration (\(R_{\text{eff}} \leq 15\)): all \(2^R\) sign vectors are evaluated — \(|\mathcal{E}| = 2^R\). The p-value is \(\text{count}/2^R\) with no continuity correction, because every sign combination including the observed one is explicitly enumerated. The minimum achievable p-value is \(1/2^R\).

- Monte Carlo (\(R_{\text{eff}} > 15\)): \(M_{\text{flip}}\) sign vectors are drawn uniformly from \(\{-1,+1\}^R\). The Phipson & Smyth (2010) correction prevents \(p = 0\) in the extreme tail: \[p = \frac{B^{+} + 1}{M_{\text{flip}} + 1}, \qquad B^{+} = \#\{b : |T_{\boldsymbol{\epsilon}_b}| \geq |T_{\text{obs}}|\}\]

The arithmetic mean \(\hat{\delta}_{\text{metric}}\) is used as the test statistic — not the Huber estimate \(\hat{\delta}_{\text{LSI}}\). Under a sign-flip null the mean has a symmetric, well-defined exchangeable distribution; the Huber M-estimator’s nonlinear weighting changes with each sign assignment and therefore does not preserve sign-exchangeability. As a result, `@p_value` tests `@delta_metric`, not `@delta_lsi`.

This distinction matters in practice. When the per-repeat inflation vector \(\{\Delta_r\}\) contains outliers, `delta_lsi` (Huber) and `delta_metric` (mean) can diverge in magnitude while agreeing in sign. A user can therefore see a significant `@p_value` with a BCa CI for `delta_lsi` that spans zero — or vice versa — if a handful of repeats pull the mean but not the robust estimate. The `summary()` method flags this disagreement explicitly when it occurs.

The threshold \(R_{\text{eff}} \geq 5\) for the sign-flip test follows from the minimum achievable p-value: with \(R = 4\), \(1/2^4 = 0.0625 > 0.05\), so a significant result at the conventional level is impossible regardless of the data. With \(R = 5\), \(1/2^5 = 0.03125 < 0.05\), making it attainable. For `exchangeability = "blocked_time"`, the relevant threshold is \(n_{\text{blocks}} \geq 5\) (not \(R_{\text{eff}} \geq 5\)), since blocks are the independent units under the test; `@p_value` is set to `NA` and a warning is issued when \(n_{\text{blocks}} < 5\). For other exchangeability values, `@p_value` is `NA` when \(R_{\text{eff}} < 5\) or when the design is unpaired.

### 3.6.2 Huber M-estimator and robustness properties

The Huber M-estimator was described formally in the Mathematical framework section above. Its inferential properties in this context are:

- Breakdown point: \(\hat{\delta}_{\text{LSI}}\) remains a consistent estimate of the true location even when up to \(\sim 29\%\) of repeats are outliers (\(k = 1.345\) yields 95% asymptotic efficiency relative to the mean under a Gaussian distribution of \(\{\Delta_r\}\)).
- Fixed scale: \(\hat{\sigma}\) is set to \(1.4826 \times \operatorname{MAD}(\boldsymbol{\Delta})\) before iteration and never updated. Fixing the scale prevents a single influential \(\Delta_r\) from collapsing \(\hat{\sigma}\) toward zero, which would force all weights to 1 and reduce the estimator to the ordinary mean.
- Degenerate case: when \(\hat{\sigma} = 0\) (all repeat differences identical), the algorithm returns \(\operatorname{median}(\boldsymbol{\Delta})\) directly, which equals the common value.

The `delta_lsi` and `delta_metric` estimates always have the same sign because the Huber estimator and the arithmetic mean agree on direction. They differ only in magnitude, and then only when one or more \(\Delta_r\) are extreme relative to the bulk of the distribution.

### 3.6.3 BCa bootstrap confidence interval

The BCa (bias-corrected and accelerated) interval (Efron, 1987) for \(\hat{\delta}_{\text{LSI}}\) corrects for both median bias and distributional skewness in the bootstrap. Given \(M_{\text{boot}}\) bootstrap resamples \(\boldsymbol{\Delta}_b^*\) drawn with replacement from \(\boldsymbol{\Delta}\), let \(\hat{\theta}_b^* = \hat{H}(\boldsymbol{\Delta}_b^*)\) be the Huber estimate on resample \(b\) and \(\hat{\theta} = \hat{H}(\boldsymbol{\Delta})\) the observed estimate.

Bias-correction constant \(\hat{z}_0\) accounts for the median bias of the bootstrap distribution relative to the observed estimate:

\[\hat{z}_0 = \Phi^{-1}\!\left( \frac{\#\{b : \hat{\theta}_b^* < \hat{\theta}\}}{M_{\text{boot}}} \right)\]

When \(\hat{\theta}\) coincides with the exact median of the bootstrap distribution, \(\hat{z}_0 = 0\) and no bias correction is applied. A nonzero \(\hat{z}_0\) shifts both interval endpoints in the same direction.

Acceleration constant \(\hat{a}\) corrects for skewness via the jackknife. Let \(\hat{\theta}_{(-r)} = \hat{H}(\boldsymbol{\Delta}_{(-r)})\) be the Huber estimate with observation \(r\) removed and \(\bar{\theta}_{(\cdot)} = R^{-1}\sum_{r=1}^R \hat{\theta}_{(-r)}\):

\[\hat{a} = \frac{\displaystyle\sum_{r=1}^{R} \bigl(\bar{\theta}_{(\cdot)} - \hat{\theta}_{(-r)}\bigr)^3} {6\,\left[\displaystyle\sum_{r=1}^{R} \bigl(\bar{\theta}_{(\cdot)} - \hat{\theta}_{(-r)}\bigr)^2 \right]^{3/2}}\]

Here \(\hat{a} > 0\) indicates a right-skewed sampling distribution; \(\hat{a} < 0\) indicates left skew.

BCa-adjusted quantile levels: for nominal coverage \(1 - \alpha = 0.95\), let \(z_L = \Phi^{-1}(\alpha/2)\) and \(z_U = \Phi^{-1}(1-\alpha/2)\). The adjusted quantile levels are:

\[p_j = \Phi\!\left(\hat{z}_0 + \frac{\hat{z}_0 + z_j}{1 - \hat{a}\,(\hat{z}_0 + z_j)} \right), \quad j \in \{L,\, U\}\]

The 95% BCa interval is the \(p_L\) and \(p_U\) empirical quantiles of the bootstrap distribution \(\{\hat{\theta}_b^*\}\):

\[\bigl[\hat{\theta}^*_{(p_L)},\; \hat{\theta}^*_{(p_U)}\bigr]\]

A separate BCa interval is computed for \(\hat{\delta}_{\text{metric}}\) (using the arithmetic mean as the bootstrap statistic) and stored in `@delta_metric_ci`. Both intervals are `NA` when \(R_{\text{eff}} < 10\), because fewer than 10 leave-one-out jackknife samples do not provide a reliable estimate of \(\hat{a}\).

### 3.6.4 Pairing condition and the inference tier system

Two `LeakFit` objects are treated as paired if and only if they have the same number of repeats and every fold shares an identical test-index composition:

\[\operatorname{sort}\!\bigl(\mathcal{T}_{r,k}^{\text{naive}}\bigr) = \operatorname{sort}\!\bigl(\mathcal{T}_{r,k}^{\text{guarded}}\bigr) \quad \forall\, r,\, k\]

This is verified by comparing the sorted integer index vectors for each sequential fold position. When paired, \(R_{\text{eff}} = R\); when unpaired, \(R_{\text{eff}} = 0\) and all inference is disabled. Pairing is automatic when both fits are produced from the same `LeakSplits` object.

All three inference procedures are gated by \(R_{\text{eff}}\). The four-tier system makes these dependencies explicit:

Tier | Condition | `@delta_metric` | `@delta_lsi` | `@p_value` | `@delta_lsi_ci` | `@inference_ok`
--- | --- | --- | --- | --- | --- | ---
`D_insufficient` | \(R_{\text{eff}} < 5\) | ✓ | ✓ | `NA` | `NA` | `FALSE`
`C_signflip` | \(5 \leq R_{\text{eff}} < 10\) | ✓ | ✓ | ✓ | `NA` | `FALSE`
`B_signflip_ci` | \(10 \leq R_{\text{eff}} < 20\) | ✓ | ✓ | ✓ | ✓ | `FALSE`
`A_full_inference` | \(R_{\text{eff}} \geq 20\) | ✓ | ✓ | ✓ | ✓ | `TRUE`

The threshold \(R_{\text{eff}} \geq 5\) for the sign-flip test under `"iid"` exchangeability follows from the minimum achievable p-value under exact enumeration (\(1/2^5 = 0.03125 < 0.05\); \(1/2^4 = 0.0625 > 0.05\)). For `exchangeability = "blocked_time"`, the independent units are blocks of repeats, so the effective threshold is \(n_{\text{blocks}} \geq 5\); when fewer than five blocks are formed from the available repeats, `@p_value` is set to `NA` even if \(R_{\text{eff}} \geq 5\). The threshold \(R_{\text{eff}} \geq 10\) for BCa intervals ensures at least 10 jackknife leave-one-out estimates for a reliable \(\hat{a}\). The `inference_ok = TRUE` flag at tier A requires all three of \(R_{\text{eff}} \geq 20\), a finite `@p_value`, and finite `@delta_lsi_ci` to hold simultaneously.

The code below shows direct slot access to confirm that the numbers correspond exactly to the quantities defined above:

    # delta_metric: arithmetic mean of {Δ_r}
    cat("delta_metric:  ", round(result_dlsi@delta_metric, 4), "\n")
    #> delta_metric:   0.1686
    # delta_lsi: Huber M-estimate of {Δ_r} (k=1.345, fixed MAD scale)
    cat("delta_lsi:     ", round(result_dlsi@delta_lsi,    4), "\n")
    #> delta_lsi:      0.169
    # delta_lsi_ci: 95% BCa interval for delta_lsi (NA below tier B)
    cat("95% BCa CI:    [",
        paste(round(result_dlsi@delta_lsi_ci, 4), collapse = ", "), "]\n")
    #> 95% BCa CI:    [ NA, NA ]
    # p_value: sign-flip randomization p-value (NA below tier C or if unpaired)
    cat("p-value:       ",
        if (is.na(result_dlsi@p_value)) "NA (tier D or unpaired)"
        else format(result_dlsi@p_value, digits = 3), "\n")
    #> p-value:        0.0625

## 3.7 Paired vs. unpaired designs

Paired (recommended): both fits are produced from the same `LeakSplits` object. The per-repeat inflation scores \(\Delta_r\) are computed as within-repeat differences, eliminating between-fold variability. \(R_{\text{eff}} = R\) and all inference tiers are reachable. Equal repeat counts with differing index patterns produce a warning and fall back to unpaired mode automatically.

Unpaired: the two fits have different fold structures or different repeat counts. The point estimates use separately computed pipeline means rather than paired differences; \(R_{\text{eff}} = 0\) and all inference is disabled. This mode is appropriate when the splitting scheme itself is part of what the comparison is testing — for example, evaluating whether subject-grouped splits alone (not a leaky feature) account for a performance difference.

    # Unpaired: naive uses sample-wise CV, guarded uses subject-grouped splits
    splits_rand <- make_split_plan(
      df_dlsi_naive,
      outcome = "outcome",
      mode    = "subject_grouped",
      group   = "row_id",
      v       = 5L,
      repeats = 5L
    )

    fit_naive_rand <- fit_resample(
      df_dlsi_naive,
      outcome = "outcome",
      splits  = splits_rand,
      learner = spec,
      metrics = "auc",
      seed    = 1L
    )
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%

    r_unpaired <- suppressWarnings(
      delta_lsi(fit_naive_rand, fit_dlsi_guarded, metric = "auc", seed = 1L)
    )

    cat(sprintf("Paired:        %s\n", r_unpaired@info$paired))
    #> Paired:        FALSE
    cat(sprintf("R_eff:         %d  (0 = unpaired, inference disabled)\n", r_unpaired@R_eff))
    #> R_eff:         0  (0 = unpaired, inference disabled)
    cat(sprintf("delta_metric:  %.4f  (naive raw mean - guarded raw mean)\n",
                r_unpaired@delta_metric))
    #> delta_metric:  0.0644  (naive raw mean - guarded raw mean)
    cat(sprintf("p_value:       %s\n",
                if (is.na(r_unpaired@p_value)) "NA" else r_unpaired@p_value))
    #> p_value:       NA

## 3.8 Metric direction with `higher_is_better`

For lower-is-better metrics (RMSE, MAE, log-loss, Brier score, deviance), set `higher_is_better = FALSE` or let `delta_lsi()` auto-detect from the metric name. Auto-detected lower-is-better metric names: `rmse`, `mse`, `mae`, `log_loss`, `logloss`, `brier`, `error`, `loss`, `deviance`.

When `higher_is_better = FALSE`, the sign of `delta_metric` is flipped so that a positive value still means the naive pipeline is inflated relative to guarded (i.e., naive has a lower error, which is paradoxically better performance — indicating leakage):

    r_hib_true  <- suppressWarnings(delta_lsi(
      fit_dlsi_naive, fit_dlsi_guarded,
      metric           = "auc",
      higher_is_better = TRUE,
      seed             = 1L
    ))
    r_hib_false <- suppressWarnings(delta_lsi(
      fit_dlsi_naive, fit_dlsi_guarded,
      metric           = "auc",
      higher_is_better = FALSE,
      seed             = 1L
    ))

    cat(sprintf("hib = TRUE:   delta_metric = %+.4f  (positive = naive inflated)\n",
                r_hib_true@delta_metric))
    #> hib = TRUE:   delta_metric = +0.1686  (positive = naive inflated)
    cat(sprintf("hib = FALSE:  delta_metric = %+.4f  (sign flipped)\n",
                r_hib_false@delta_metric))
    #> hib = FALSE:  delta_metric = -0.1686  (sign flipped)

The `info$higher_is_better` slot records the value used (after auto-detection), so analyses are fully reproducible:

    cat("higher_is_better used:", result_dlsi@info$higher_is_better, "\n")
    #> higher_is_better used: TRUE

## 3.9 Accessing the `LeakDeltaLSI` object

`delta_lsi()` returns a `LeakDeltaLSI` S4 object. All slots are accessible directly for programmatic downstream use:

Slot | Type | Description
--- | --- | ---
`@metric` | character | Metric name used (e.g. `"auc"`, `"rmse"`)
`@exchangeability` | character | Exchangeability assumption for the sign-flip test. `"blocked_time"` activates a block sign-flip procedure; `"by_group"` and `"within_batch"` are stored but inference still uses iid sign-flip (a warning is issued); `"iid"` (default) applies the standard independent sign-flip
`@tier` | character | Inference tier: `"A_full_inference"`, `"B_signflip_ci"`, `"C_signflip"`, or `"D_insufficient"`
`@R_eff` | integer | Effective paired repeats (0 when unpaired; all inference is based on this count)
`@delta_metric` | numeric | Arithmetic mean of per-repeat inflation scores \(\bar{\Delta}\); interpret directly in metric units
`@delta_lsi` | numeric | Huber M-estimate of per-repeat inflation scores; more robust than `delta_metric` when repeat-level outliers are present
`@delta_metric_ci` | numeric[2] | 95% BCa bootstrap CI for `delta_metric` (both elements `NA` below tier B)
`@delta_lsi_ci` | numeric[2] | 95% BCa bootstrap CI for `delta_lsi` (both elements `NA` below tier B)
`@p_value` | numeric | Sign-flip randomization p-value testing H₀: mean inflation = 0 (`NA` below tier C or when unpaired)
`@inference_ok` | logical | `TRUE` only at tier A (`R_eff ≥ 20`), indicating full point + CI + p-value inference is available
`@folds_naive` | data.frame | Per-fold data for the naive pipeline: columns `fold`, `metric` (fold-level metric value), `repeat_id`, `n` (test-fold size)
`@folds_guarded` | data.frame | Per-fold data for the guarded pipeline: columns `fold`, `metric`, `repeat_id`, `n` — same structure as `@folds_naive`
`@repeats_naive` | data.frame | Per-repeat aggregates for the naive pipeline: columns `repeat_id`, `metric` (size-weighted mean across folds), `n_folds`, `total_n`
`@repeats_guarded` | data.frame | Per-repeat aggregates for the guarded pipeline: same structure as `@repeats_naive`

`@info`

list

Metadata: `paired` (logical), `R_naive`, `R_guarded` (number of repeats in each fit), `higher_is_better` (logical, after auto-detection), `metric_naive` and `metric_guarded` (Huber-robust overall mean for each pipeline), `block_size_used` and `n_blocks` (for `exchangeability = "blocked_time"`; `NA` otherwise), and optionally `delta_r` (per-repeat \(\Delta_r\) vector) and the original fit objects when `return_details = TRUE`

    # Per-repeat summaries for custom plotting
    head(result_dlsi@repeats_naive,   3)
    #>   repeat_id    metric n_folds total_n
    #> 1         1 0.7225532       5     120
    #> 2         2 0.7018017       5     120
    #> 3         3 0.7159788       5     120
    head(result_dlsi@repeats_guarded, 3)
    #>   repeat_id    metric n_folds total_n
    #> 1         1 0.4963630       5     120
    #> 2         2 0.5358545       5     120
    #> 3         3 0.5536640       5     120

    # Per-fold diagnostics
    head(result_dlsi@folds_naive,   3)
    #>   fold    metric repeat_id  n
    #> 1    1 0.7132867         1 24
    #> 2    2 0.6953125         1 24
    #> 3    3 0.6888889         1 24
    head(result_dlsi@folds_guarded, 3)
    #>   fold    metric repeat_id  n
    #> 1    1 0.4825175         1 24
    #> 2    2 0.5234375         1 24
    #> 3    3 0.4962963         1 24

    # Metadata
    str(result_dlsi@info)
    #> List of 8
    #>  $ paired          : logi TRUE
    #>  $ R_naive         : int 5
    #>  $ R_guarded       : int 5
    #>  $ higher_is_better: logi TRUE
    #>  $ metric_naive    : num 0.716
    #>  $ metric_guarded  : num 0.547
    #>  $ block_size_used : int NA
    #>  $ n_blocks        : int NA

The `@repeats_naive` and `@repeats_guarded` data frames are the primary substrate for diagnostic visualisation. Plotting `metric` by `repeat_id` for both pipelines side-by-side reveals whether the performance gap is consistent across all repeats — a clean, reproducible offset indicates structural leakage — or concentrated in a handful of repeats, which may indicate a data-quality issue or an unstable fold partition.

The `@folds_naive` and `@folds_guarded` data frames go one level deeper, providing the per-fold metric value along with the test-fold size (`n`) and the repeat it belongs to (`repeat_id`). These are useful for identifying individual folds that behave as outliers within a repeat, for example because a small test partition happened to produce extreme metric values.

## 3.10 Connection to `audit_leakage()`

`delta_lsi()` and `audit_leakage()` answer complementary questions:

Question | Tool
--- | ---
Does leakage produce non-random performance? | `audit_leakage()` — permutation gap + p-value
What mechanisms drive the leakage? | `audit_leakage()` — `batch_assoc`, `target_assoc`, `duplicates`
How large is the performance inflation? | `delta_lsi()` — `delta_metric`, `delta_lsi`
Is the inflation statistically significant? | `delta_lsi()` — sign-flip p-value
Do guards fully eliminate the inflation? | `delta_lsi()` — compare before/after guarding

A recommended two-step workflow:

1. Run `audit_leakage(fit_leaky)` to detect leakage and diagnose which mechanism is responsible.
2. Build a guarded pipeline that addresses those mechanisms, then run `delta_lsi(fit_leaky, fit_guarded)` to quantify how much inflation the leakage introduced and whether guarding eliminated it.

    # Step 1: detect and characterise leakage
    audit_naive_dlsi <- audit_leakage(fit_dlsi_naive, metric = "auc", B = 20L)
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |===                                                                   |   4%  |                                                                              |======                                                                |   8%  |                                                                              |========                                                              |  12%  |                                                                              |===========                                                           |  16%  |                                                                              |==============                                                        |  20%  |                                                                              |=================                                                     |  24%  |                                                                              |====================                                                  |  28%  |                                                                              |======================                                                |  32%  |                                                                              |=========================                                             |  36%  |                                                                              |============================                                          |  40%  |                                                                              |===============================                                       |  44%  |                                                                              |==================================                                    |  48%  |                                                                              |====================================                                  |  52%  |                                                                              |=======================================                               |  56%  |                                                                              |==========================================                            |  60%  |                                                                              |=============================================                         |  64%  |                                                                              |================================================                      |  68%  |                                                                              |==================================================                    |  72%  |                                                                              |=====================================================                 |  76%  |                                                                              |========================================================              |  80%  |                                                                              |===========================================================           |  84%  |                                                                              |==============================================================        |  88%  |                                                                              |================================================================      |  92%  |                                                                              |===================================================================   |  96%  |                                                                              |======================================================================| 100%
    cat("Naive audit summary:\n")
    #> Naive audit summary:
    summary(audit_naive_dlsi)
    #>
    #> ==============================
    #>  bioLeak Leakage Audit Summary
    #> ==============================
    #>
    #> Task: binomial | Outcome: outcome | Splitting mode: subject_grouped | Positive class: control
    #> Hash: h0cf564d3 | Folds: 25 | Repeats: 1
    #>
    #> Label-Permutation Association Test:
    #>   Method: refit per permutation (auto)
    #>   Observed metric: 0.741
    #>   Permuted mean ± SD: 0.572 ± 0.043
    #>   Gap: 0.169 (larger gap = stronger non-random signal)
    #>   This test does NOT diagnose information leakage. Use the Batch Association,
    #>   Target Leakage Scan, and Duplicate Detection sections to check for leakage.
    #>
    #> Batch / Study Association: none detected.
    #>
    #> Target Leakage Scan: not available.
    #>
    #> Multivariate Target Scan: not available.
    #>
    #> No near-duplicates detected.
    #>
    #> Mechanism Risk Assessment:
    #>        mechanism_class flagged        evidence statistic  p_value
    #>      non_random_signal    TRUE permutation_gap  0.168578 0.047619
    #>  confounding_alignment   FALSE     batch_assoc        NA       NA
    #>   proxy_target_leakage   FALSE    target_assoc        NA       NA
    #>      duplicate_overlap   FALSE      duplicates        NA       NA
    #>     temporal_lookahead   FALSE      duplicates        NA       NA
    #>
    #> Interpretation:
    #>   ✓ Strong non-random signal.

    # Step 2: confirmed guards reduced inflation; check residual DLSI
    cat("\ndelta_metric (naive vs guarded):", round(result_dlsi@delta_metric, 4), "\n")
    #>
    #> delta_metric (naive vs guarded): 0.1686
    if (!is.na(result_dlsi@p_value)) {
      cat("Sign-flip p-value:              ", format(result_dlsi@p_value, digits = 3), "\n")
    }
    #> Sign-flip p-value:               0.0625

A ΔLSI near zero after guarding and a non-significant p-value together provide strong evidence that the leakage source has been eliminated. A persistent positive ΔLSI after guarding indicates residual inflation — revisit the preprocessing pipeline for remaining leakage pathways.

# 4 Parallel Processing

`bioLeak` uses the `future` framework for parallelism (Windows, macOS, Linux). `fit_resample()`, `audit_leakage()`, `audit_leakage_by_learner()`, and `simulate_leakage_suite()` honor the active plan when `parallel = TRUE` (or `parallel_learners = TRUE`).

    library(future)

    # Use multiple cores (works on all OS)
    plan(multisession, workers = 4)

    # Run a heavy simulation
    sim <- simulate_leakage_suite(..., parallel = TRUE)

    # Parallel folds or audits
    # fit_resample(..., parallel = TRUE)
    # audit_leakage(..., parallel = TRUE)
    # audit_leakage_by_learner(..., parallel_learners = TRUE)

    # Return to sequential processing
    plan(sequential)

This chunk only configures the parallel plan, so it does not produce a result object. Use it as a template before running compute-heavy functions.

## 4.1 Simulation suite

`simulate_leakage_suite()` runs Monte Carlo simulations that inject specific leakage mechanisms and then evaluates detection with the audit pipeline. It is useful for validating your leakage checks before applying them to real data. Available leakage types include `none`, `subject_overlap`, `batch_confounded`, `peek_norm`, and `lookahead`. Use `signal_strength`, `prevalence`, `rho`, `K`, `repeats`, `horizon`, and `preprocess` to control the simulation. The output is a `LeakSimResults` data frame with one row per seed. Metrics are selected automatically based on outcome type (AUC for binary classification).

    if (requireNamespace("glmnet", quietly = TRUE)) {
      sim <- simulate_leakage_suite(
        n = 80,
        p = 6,
        mode = "subject_grouped",
        learner = "glmnet",
        leakage = "subject_overlap",
        seeds = 1:2,
        B = 20,
        parallel = FALSE,
        signal_strength = 1
      )

      # Simulation results (first 6 rows)
      head(sim)
    } else {
      cat("glmnet not installed; skipping simulation suite example.\n")
    }
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   |                                                                              |                                                                      |   0%  |                                                                              |==============                                                        |  20%  |                                                                              |============================                                          |  40%  |                                                                              |==========================================                            |  60%  |                                                                              |========================================================              |  80%  |                                                                              |======================================================================| 100%
    #>   seed metric_obs      gap  p_value         leakage            mode
    #> 1    1   0.828914 0.223485 0.047619 subject_overlap subject_grouped
    #> 2    2   0.825397 0.191317 0.047619 subject_overlap subject_grouped

Each row corresponds to one simulation seed.

- `metric_obs` is the observed performance (AUC for this simulation). Higher values suggest stronger apparent signal, which can be inflated by leakage.

- `gap` and `p_value` indicate that the leaked signal is statistically distinguishable from random noise.

Use `metric_obs` to gauge the magnitude of the leakage effect, and `gap` to assess detection sensitivity.

## 4.2 Objects and summaries

bioLeak uses S4 classes and list-like results to capture provenance and diagnostics:

- `LeakSplits`: returned by `make_split_plan()`, printed with `show()`. Use `check_split_overlap()` to verify no-overlap invariants on grouping columns.
- `LeakFit`: returned by `fit_resample()`, summarized with `summary()` includes fold diagnostics in `fit@info$fold_status`
- `LeakAudit`: returned by `audit_leakage()`, summarized with `summary()`. Now includes `audit@info$mechanism_summary` — a data.frame classifying leakage evidence into mechanism classes (`non_random_signal`, `confounding_alignment`, `proxy_target_leakage`, `duplicate_overlap`, `temporal_lookahead`).
- `LeakAuditList`: returned by `audit_leakage_by_learner()`, printed directly
- `GuardFit`: returned by `.guard_fit()`, printed and summarized. Use `guard_to_recipe()` to convert guard preprocessing specs to a `recipes::recipe` for tidymodels workflows.
- `LeakImpute`: returned by `impute_guarded()` (guarded data + diagnostics)
- `LeakTune`: returned by `tune_resample()` (nested tuning summaries) includes `fold_status`, selected parameters, and optional `final_model` / `final_workflow` when `refit = TRUE`
- `LeakSimResults`: returned by `simulate_leakage_suite()` (per-seed results)
- `LeakDeltaLSI`: returned by `delta_lsi()` (paired pipeline comparison). Key slots: `@metric`, `@exchangeability`, `@tier`, `@R_eff`, `@delta_metric`, `@delta_metric_ci`, `@delta_lsi`, `@delta_lsi_ci`, `@p_value`, `@inference_ok`, `@folds_naive`, `@folds_guarded`, `@repeats_naive`, `@repeats_guarded`, `@info`. Summarise with `summary()`.

Set `options(bioLeak.strict = TRUE)` to activate strict leakage mode, which escalates trained-recipe/workflow warnings to errors, warns on missing seeds, and auto-runs overlap checks after splitting.

These objects store hashes and metadata that help reproduce and audit the workflow. Inspect their slots (`@metrics`, `@predictions`, `@permutation_gap`, `@duplicates`) to drill into specific leakage signals.