Plug-in PPI++ optimal tuning parameter for a 1PL model — tune_lambda_ppi_score

Applies the PPI++ Proposition 2 formula using (J+1)-dimensional score vectors for the 1PL parameterization (a_shared, d_1, ..., d_J).

Usage

tune_lambda_ppi_score_1pl(
  observed,
  predicted,
  item_pars,
  n_generated,
  quadrature = NULL,
  n_quad = 31
)

Arguments

observed: Human response matrix.
predicted: Paired binary LLM responses (0/1) for the same rows as observed. Probabilities are not accepted; sample binary responses first.
item_pars: Item parameters in slope-intercept form at which to evaluate the score vectors. Typically the human 2PL MLE from fit_2pl().
n_generated: Number of generated (unpaired) LLM subjects, used to compute the ratio r (n / n_generated).
quadrature: Optional quadrature grid. If omitted, a standard-normal grid with n_quad nodes is created.
n_quad: Number of quadrature nodes when quadrature is omitted.

Value

A list with lambda, n, n_generated, r, C_hf, V_f.

Details

This is the item-parameter variance objective — it minimizes Tr(Sigma_1pl). For practical scoring applications use tune_lambda_ability_risk_1pl() instead.

Examples

set.seed(1)
pars <- data.frame(a = 1, d = c(-0.5, 0, 0.5))
obs  <- simulate_2pl(rnorm(40), pars)
tune_lambda_ppi_score_1pl(obs, obs, pars, n_generated = 100, n_quad = 7)$lambda
#> [1] 0.7142857