Fit a mixed-subjects 1PL calibration via marginal maximum likelihood

Analogous to fit_mixed_subjects_mml() but estimates a shared discrimination parameter a across all items (1PL model). Posteriors are recomputed at every gradient evaluation — no frozen-posterior gradient asymmetry.

Usage

fit_mixed_subjects_mml_1pl(
  observed,
  predicted,
  generated,
  lambda = 1,
  n_quad = 31,
  initial_pars = NULL,
  quadrature = NULL,
  mml_pred_weights = c("own", "human"),
  slope_lower = 1e-04,
  slope_upper = NULL,
  control = list(maxit = 500),
  ...
)

Arguments

observed

Human response matrix, with rows for subjects and columns for items. Values must be binary when initial_pars is omitted.

predicted

Binary LLM responses (0/1) for the same rows and items as observed. Probabilities are not accepted: fractional values are not a valid likelihood input for the marginal IRT objective and break the PPI correction, so sample binary responses from any probabilities first (e.g. rbinom).

generated

Binary generated or unlabeled LLM responses (0/1) for the same item columns. Probabilities are not accepted (see predicted).

lambda

Power-tuning parameter in [0, 1].

n_quad

Number of standard-normal quadrature nodes.

initial_pars

Optional starting item parameters. If omitted, a 2PL model is fit to observed.

quadrature

Optional quadrature grid with theta and weight columns.

mml_pred_weights

How to compute posteriors for the paired predicted term. "own" uses posteriors from the predicted responses; "human" uses posteriors from the observed human responses. See Details.

slope_lower

Lower bound for discrimination parameters during optimization. Use NULL for no lower bound.

slope_upper

Upper bound on discrimination parameters. Unlike fit_mixed_subjects(), this function should not require capping for well-posed problems because the true marginal objective has no false minimum at large discrimination.

control

Control list passed to stats::optim().

...

Additional arguments passed to fit_1pl() when initial_pars is omitted.

Value

An object of class c("mixedsubjects_1pl_fit", "mixedsubjects_fit").

Details

Only scalar lambda is supported; per-item lambda is not meaningful for the 1PL because the discrimination is shared across items.

See also

fit_mixed_subjects_1pl() for the frozen expected-count version; fit_mixed_subjects_mml() for the 2PL version.

Examples

set.seed(1)
pars <- data.frame(a = 1, d = c(-0.5, 0, 0.5))
observed  <- simulate_2pl(rnorm(40), pars)
generated <- simulate_2pl(rnorm(100), pars)
fit <- fit_mixed_subjects_mml_1pl(
  observed, observed, generated,
  lambda = 0.5, initial_pars = pars, n_quad = 7,
  control = list(maxit = 100)
)
fit$item_pars
#>   item        a           d           b
#> 1    1 1.152006 -0.40800463  0.35416883
#> 2    2 1.152006 -0.06273155  0.05445419
#> 3    3 1.152006  0.37033652 -0.32147099