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Mixed-Subjects 1PL Calibration

The 1PL (one-parameter logistic) model estimates a single shared discrimination \(a\) across all items together with per-item intercepts \(d_j\):

\[P(x_j = 1 \mid \theta) = \text{logistic}(a\,\theta + d_j)\]

The parameter vector has length \(J+1\) rather than \(2J\). The package provides exact analogues of the 2PL mixed-subjects functions for the 1PL case.

When to prefer 1PL over 2PL: - Ability-focused tests where the items are designed to be equally discriminating. - Tests built from a single item pool with homogeneous item characteristics. - When the 2PL discrimination estimates are very noisy (small \(n\)).

Note on vcov. vcov_mixed_subjects_1pl() currently uses the EM complete-data Hessian (not Louis’ marginal-information correction). The uncertainty estimates are slightly over-precise. A Louis-corrected 1PL bread is planned for a future release.

Simulate a 1PL test

library(mixedsubjectsirt)
library(ggplot2)

set.seed(2026)

n_human     <- 400
n_generated <- 1200
n_items     <- 8

# True 1PL: shared discrimination a = 1.2, varying difficulties
true_1pl <- data.frame(
  item = paste0("Item", seq_len(n_items)),
  a    = 1.2,
  d    = seq(-1.1, 1.1, length.out = n_items)
)
true_1pl$b <- -true_1pl$d / true_1pl$a

theta_human <- rnorm(n_human)
observed    <- simulate_2pl(theta_human, true_1pl)

# LLM: same 1PL structure, small intercept shift
llm_1pl   <- true_1pl
llm_1pl$d <- true_1pl$d + 0.25
llm_1pl$b <- -llm_1pl$d / llm_1pl$a

predicted <- simulate_2pl(theta_human, llm_1pl)
generated <- simulate_2pl(rnorm(n_generated), llm_1pl)

Step 1: Fit the 1PL baseline

fit_1pl() estimates \(a\) and \(d_1, \ldots, d_J\) by maximizing the IRT marginal likelihood under a standard-normal ability prior.

fit1 <- fit_1pl(observed, n_quad = 15)
cat("Shared a:", round(fit1$pars$a[1], 3), " (true:", true_1pl$a[1], ")\n")
#> Shared a: 1.287  (true: 1.2 )
cat("Convergence:", fit1$convergence, "\n\n")
#> Convergence: 0
fit1$pars
#>    item        a          d           b
#> 1 Item1 1.286555 -1.1603124  0.90187542
#> 2 Item2 1.286555 -0.8618391  0.66988122
#> 3 Item3 1.286555 -0.1642846  0.12769343
#> 4 Item4 1.286555 -0.4177364  0.32469371
#> 5 Item5 1.286555  0.1133618 -0.08811268
#> 6 Item6 1.286555  0.5983198 -0.46505573
#> 7 Item7 1.286555  0.8240620 -0.64051824
#> 8 Item8 1.286555  1.3131182 -1.02064664

All items in the output have the same a value, confirming the 1PL constraint.

Step 2: Fit mixed-subjects MML (1PL)

fit_mixed_subjects_mml_1pl() uses the true marginal likelihood with a 1PL-specific gradient: the shared discrimination gradient accumulates contributions from all \(J\) items, while each intercept has its own gradient.

fit_mml_1pl <- fit_mixed_subjects_mml_1pl(
  observed     = observed,
  predicted    = predicted,
  generated    = generated,
  lambda       = 0.5,
  initial_pars = fit1$pars,
  n_quad       = 15,
  control      = list(maxit = 300)
)

print(fit_mml_1pl)
#> mixedsubjectsirt 1PL fit
#>   items:      8
#>   a (shared): 1.327
#>   lambda:     0.5
#>   loss:       4.72331
#>   convergence: 0 
#>   estimator:  marginal MML PPI++ (1PL)
fit_mml_1pl$item_pars
#>    item        a          d          b
#> 1 Item1 1.327115 -1.2025380  0.9061297
#> 2 Item2 1.327115 -0.8728234  0.6576850
#> 3 Item3 1.327115 -0.1832574  0.1380871
#> 4 Item4 1.327115 -0.4426497  0.3335429
#> 5 Item5 1.327115  0.2317895 -0.1746567
#> 6 Item6 1.327115  0.6473713 -0.4878036
#> 7 Item7 1.327115  0.8889582 -0.6698428
#> 8 Item8 1.327115  1.5297000 -1.1526510

Step 3: Correct covariance — \((J+1) \times (J+1)\) sandwich

vcov() dispatches to vcov_mixed_subjects_1pl() for 1PL fits, returning a \((J+1) \times (J+1)\) matrix with a_shared and per-item d_j as rows/columns.

Sigma_1pl <- vcov(fit_mml_1pl)
dim(Sigma_1pl)
#> [1] 9 9
rownames(Sigma_1pl)
#> [1] "a_shared" "d_Item1"  "d_Item2"  "d_Item3"  "d_Item4"  "d_Item5"  "d_Item6" 
#> [8] "d_Item7"  "d_Item8"

Step 4: Ability-score risk and lambda tuning

tune_lambda_ability_risk_1pl() uses the 1PL-parameterized gradient \(\partial\hat\theta / \partial (a_\text{shared}, d_1, \ldots, d_J)\) for the ability-score risk. The chain rule gives \(\partial\hat\theta / \partial a_\text{shared} = \sum_j \partial\hat\theta / \partial a_j\). As in the 2PL version, \(\lambda\) is chosen by direct 1-D optimization by default (pass method = "grid" to scan a grid instead).

tuned_1pl <- tune_lambda_ability_risk_1pl(
  observed     = observed,
  predicted    = predicted,
  generated    = generated,
  initial_pars = fit1$pars,
  n_quad       = 15,
  control      = list(maxit = 300)
)

tuned_1pl$best_lambda
#> [1] 0.2784741

Step 5: Verify — F = Y gives lambda > 0

With predicted = observed (perfect paired predictor), the ability-risk criterion should select a positive lambda.

tuned_fy <- tune_lambda_ability_risk_1pl(
  observed     = observed,
  predicted    = observed,     # F = Y
  generated    = simulate_2pl(rnorm(n_generated), true_1pl),
  initial_pars = fit1$pars,
  n_quad       = 15,
  control      = list(maxit = 300)
)

cat("F=Y best lambda:", tuned_fy$best_lambda,
    " (theory: N/(n+N) =", round(n_generated / (n_human + n_generated), 3), ")\n")
#> F=Y best lambda: 0.7544813  (theory: N/(n+N) = 0.75 )

Compare 1PL and 2PL

On a well-specified 1PL test, how do the 1PL and 2PL estimators compare?

fit_2pl_mml <- fit_mixed_subjects_mml(
  observed     = observed,
  predicted    = predicted,
  generated    = generated,
  lambda       = tuned_1pl$best_lambda,
  initial_pars = fit_2pl(observed, technical = list(NCYCLES = 500))$pars,
  n_quad       = 15,
  control      = list(maxit = 300)
)

rmse <- function(x, y) sqrt(mean((x - y)^2))
cat("1PL RMSE(a):", round(rmse(tuned_1pl$best_fit$item_pars$a, true_1pl$a), 4), "\n")
#> 1PL RMSE(a): 0.1085
cat("2PL RMSE(a):", round(rmse(fit_2pl_mml$item_pars$a, true_1pl$a), 4), "\n")
#> 2PL RMSE(a): 0.3108

# Difficulty recovery
cat("1PL RMSE(d):", round(rmse(tuned_1pl$best_fit$item_pars$d, true_1pl$d), 4), "\n")
#> 1PL RMSE(d): 0.1989
cat("2PL RMSE(d):", round(rmse(fit_2pl_mml$item_pars$d, true_1pl$d), 4), "\n")
#> 2PL RMSE(d): 0.2077

The 1PL uses fewer parameters (\(J+1\) vs \(2J\)), which can give lower RMSE on a test generated from a true 1PL DGP — especially for small \(n\).

Ability-score risk: 1PL vs 2PL parameterization

The 1PL ability-score risk is smaller in the \((J+1)\)-parameter space because the shared \(a\) concentrates all discrimination information in a single parameter.

Sigma_2pl <- vcov(fit_2pl_mml)  # 2J × 2J Louis-corrected

risk_1pl <- ability_risk_1pl(observed, tuned_1pl$best_fit)
risk_2pl <- ability_risk(observed, fit_2pl_mml, vcov = Sigma_2pl)

cat("1PL mean param_var:", round(risk_1pl$summary$mean_param_var, 5), "\n")
#> 1PL mean param_var: 0.00297
cat("2PL mean param_var:", round(risk_2pl$summary$mean_param_var, 5), "\n")
#> 2PL mean param_var: 0.04299

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