Sample Size Estimation for Machine Learning

Introduction

Determining the minimum sample size needed to achieve a target performance in Machine Learning models is a fundamental challenge in applied research. Unlike traditional statistical analysis where tools like G*Power provide closed-form solutions, machine learning lacks analytical formulas for sample-size planning.

The ml_sample_size() function in PsyML implements a Monte Carlo simulation approach that:

Generates learning curves via repeated model evaluation
Fits a power-law model for extrapolation
Identifies the sample size meeting explicit decision criteria

This vignette demonstrates the complete workflow for sample size estimation.

Quick Start

library(PsyML)

# Basic usage with simulated data
result <- ml_sample_size(
  task = "classification",
  metric = "auc",
  target = 0.80,
  seed = 2026,
  reps = 100
)

# View recommendation
result$recommend_n
# [1] 600

Core Concepts

The Learning Curve

A learning curve plots model performance as a function of training sample size. It typically shows:

Rapid improvement at small sample sizes
Progressively slower gains as data increases
Asymptotic plateau determined by problem difficulty

Power-Law Model

The learning curve is modeled as:

\[\text{Performance}(n) = a - b \cdot n^{-c}\]

Where:

\(a\) = asymptotic performance (maximum achievable)
\(b\) = improvement magnitude
\(c\) = convergence rate

Decision Criteria

The recommended sample size \(n^*\) is the smallest \(n\) satisfying:

Mean performance ≥ target
Probability of success \(p_k\) ≥ 0.80 (at least 80% of runs meet target)
Stability (optional): SD ≤ threshold

Example 1: Basic Classification

Let’s estimate the sample size needed to achieve AUC ≥ 0.80 with Random Forest.

library(PsyML)

# Simulate a classification problem
result <- ml_sample_size(
  task = "classification",
  metric = "auc",
  target = 0.80,
  n_grid = c(100, 200, 300, 400, 600, 800, 1000),
  reps = 100,  # 100 Monte Carlo repetitions
  sim_params = list(
    p = 20,           # 20 predictors
    prevalence = 0.20, # 20% positive class
    signal = 2.5       # Moderate signal (AUC ~0.85-0.90 achievable)
  ),
  seed = 123
)

Interpreting the Output

# View full results
print(result)

# Recommended sample size
result$recommend_n
# [1] 600

# Summary table with SE of success proportion
result$summary[, c("n", "mean", "sd", "p_meet", "se_p_meet")]
#     n  mean    sd p_meet se_p_meet
# 1 100 0.760 0.025   0.12     0.032
# 2 200 0.795 0.018   0.45     0.050
# 3 300 0.815 0.015   0.78     0.041
# 4 400 0.825 0.012   0.92     0.027
# 5 600 0.840 0.010   0.98     0.014
# ...

Key Output Components

Component	Description
`mean`	Average metric across R repetitions
`sd`	Standard deviation (variability)
`p_meet`	Proportion of runs meeting target (\(p_k\))
`se_p_meet`	Standard error of \(p_k\)

Visualizing the Learning Curve

# Plot the learning curve
plot_learning_curve(result)

The plot shows:

Points: observed mean performance
Shaded band: 95% confidence interval
Horizontal line: target performance
Vertical line: recommended \(n^*\)
Curve: fitted power law

Example 2: Choosing the Number of Repetitions (R)

The precision of \(p_k\) depends on R. The standard error follows:

\[SE(p_k) \approx \sqrt{\frac{p_k(1-p_k)}{R}}\]

R	SE at p=0.80	Use Case
50	0.057	Exploratory
100	0.040	Standard
200	0.028	Publication

# For exploratory analysis
result_quick <- ml_sample_size(
  task = "classification",
  target = 0.80,
  seed = 2026,
  reps = 50  # Quick exploration
)

# For publication-quality results
result_precise <- ml_sample_size(
  task = "classification",
  target = 0.80,
  seed = 2026,
  reps = 200  # High precision
)

The function automatically displays SE guidance:

Monte Carlo settings:
  - Repetitions (R): 100
  - Approx SE(p_k) at p=0.80: 0.04

Example 3: Small Datasets (N < 1000)

When your dataset is small, a single test partition may yield unstable estimates. The robustness protocol uses multiple independent partitions.

# Load a small dataset
data(PimaIndiansDiabetes, package = "mlbench")

# Use 5 independent test partitions
result <- ml_sample_size(
  data = PimaIndiansDiabetes,
  formula = diabetes ~ .,
  task = "classification",
  metric = "auc",
  target = 0.75,
  positive = "pos",
  seed = 2026,
  reps = 100,
  n_outer_splits = 5,  # 5 independent partitions
  verbose = TRUE
)

Interpreting Multiple Partition Results

# View results across partitions
result$outer_splits_results
# $median: 50       # Median n* (use this as recommendation)
# $range: [50, 150] # Range of estimates
# $cv: 0.559        # Coefficient of variation

# If CV > 50%, there's high uncertainty
# Report: "n* = 50 (range: 50-150, CV = 56%)"

Guideline: If the range exceeds 50% of the median, report the full range and acknowledge high uncertainty.

Example 4: Bootstrap Confidence Intervals

For formal inference about \(n^*\), use bootstrap CIs:

result <- ml_sample_size(
  task = "classification",
  target = 0.80,
  seed = 2026,
  reps = 100,
  bootstrap_ci = TRUE,  # Enable bootstrap
  n_boot = 1000         # 1000 resamples
)

# View 95% CI (values will vary)
result$recommend_n_ci
# $ci_lower: 400   # Lower bound of 95% CI
# $ci_upper: 800   # Upper bound of 95% CI
# $ci_median: 600  # Bootstrap median

Interpretation: “The recommended sample size is n* = 600 (95% CI: 400-800).”

Example 5: Comparing Multiple Models

Compare sample requirements across algorithms:

result <- ml_sample_size(
  task = "classification",
  model = c("rf", "xgboost", "glm"),  # Compare 3 models
  metric = "auc",
  target = 0.80,
  seed = 2026,
  reps = 50
)

# View recommendations by model (values will vary)
result$recommend_n
# $rf:      600   # or higher depending on data
# $xgboost: 600   # similar to RF
# $glm:     200   # typically needs less data

# GLM needs less data but may have lower asymptotic performance
# RF/XGBoost need more data but achieve higher maximum AUC

Trade-off: Simpler models (GLM) reach targets with less data but have lower ceilings. Complex models (RF, XGBoost) need more data but achieve better asymptotic performance.

Example 6: Checking Fit Quality Before Extrapolation

Always verify the power-law fit before extrapolating:

# Check fit diagnostics
result$curve_diagnostics
# $pseudo_r2: 0.967    # Good: > 0.95
# $rmse_fit: 0.008     # Low fitting error
# $fit_quality: "good" # Automatic classification

# Fit quality interpretation:
# - "good":       R² ≥ 0.95 (safe to extrapolate)
# - "acceptable": R² ≥ 0.80 (extrapolate with caution)
# - "poor":       R² < 0.80 (do not extrapolate)

Example 7: Extrapolation Functions

Estimating N for a Specific Target

# What n is needed for AUC = 0.85?
estimate_n(result, target_metric = 0.85)
# [1] 1250

# Warnings are issued if:
# - Target exceeds asymptote
# - n exceeds 2x max observed
# - Fit quality is poor

Finding the Plateau

# Where does performance plateau?

# Absolute delta (within 0.01 of asymptote)
estimate_n_plateau(result, delta = 0.01)
# [1] 8500

# Relative delta (within 1% of target)
estimate_n_plateau(result, delta = 0.01, relative = TRUE)
# Using relative delta: 0.01 x 0.8 = 0.008
# [1] 12000

Interpretation: Beyond n = 8,500, additional data yields < 0.01 AUC improvement.

Example 8: Regression Tasks

The method works for regression with appropriate metrics:

result <- ml_sample_size(
  task = "regression",
  metric = "rmse",        # Or "mae", "r2"
  target = 1.0,           # Target RMSE ≤ 1.0
  target_dir = "lower_better",  # Auto-detected for RMSE
  seed = 2026,
  reps = 100,
  sim_params = list(
    p = 15,
    noise_sd = 1.5  # Controls signal-to-noise ratio
  )
)

Best Practices

1. Start with Exploration

# Quick exploration with low R
result_explore <- ml_sample_size(
  task = "classification",
  target = 0.80,
  seed = 2026,
  reps = 30,
  n_grid = seq(100, 1000, by = 100)
)

# Refine with higher R after identifying the region
result_final <- ml_sample_size(
  task = "classification",
  target = 0.80,
  seed = 2026,
  reps = 200,
  n_grid = c(300, 400, 500, 600, 700, 800)  # Focus on relevant range
)

2. Always Check Diagnostics

# Before extrapolating:
if (result$curve_diagnostics$fit_quality == "poor") {
  warning("Do not extrapolate - use only observed values")
}

3. Report Uncertainty

Good reporting practice:

“Using Monte Carlo simulation with R = 100 repetitions, we estimated n* = 600 (95% CI: 400-800) to achieve AUC ≥ 0.80 with probability ≥ 0.80. The power-law fit quality was good (pseudo-R² = 0.97).”

4. Consider Hyperparameter Tuning

The \(n^*\) estimate assumes fixed hyperparameters. If you plan extensive tuning:

# Add a safety margin (e.g., 20-30%)
n_star <- result$recommend_n
n_with_tuning <- ceiling(n_star * 1.25)

Troubleshooting

Curve Fit Fails

# If curve_fit is NULL or fit_quality is "poor":
# 1. Increase reps
result <- ml_sample_size(..., reps = 200)

# 2. Expand n_grid range
result <- ml_sample_size(..., n_grid = c(50, 100, 200, 500, 1000, 2000, 5000))

# 3. Use only interpolation (don't extrapolate)

High Variability in n*

# If CV > 50% with multiple partitions:
# 1. This is expected for small datasets - report the range
# 2. Consider using the upper bound for safety
# 3. Acknowledge uncertainty in your report

Target Seems Unachievable

# If recommend_n is NA or Inf:
# 1. Check if target exceeds the asymptote
result$curve_fit  # Look at coefficient 'a'

# 2. Lower your target or increase signal strength
result <- ml_sample_size(..., target = 0.75)  # Lower target

# 3. Use a different algorithm
result <- ml_sample_size(..., model = "glm")  # Simpler model

Scenario	Recommendation
Exploratory	R = 50, single partition
Standard	R = 100, check diagnostics
Publication	R ≥ 200, bootstrap CI
Small N (<1000)	n_outer_splits = 5-10
Extrapolation	Verify R² > 0.95, n < 2× max observed
With tuning	Add 20-30% safety margin