Tune lambda by downstream ability-score risk for a 1PL model

Selects the lambda minimizing E[g' Sigma_1pl g] — the propagated ability-score risk in the 1PL parameterization — using fit_mixed_subjects_mml_1pl() by default. As in the 2PL tune_lambda_ability_risk(), lambda is chosen by direct 1-D optimization (method = "optimize", the default) or over lambda_grid (method = "grid").

Usage

tune_lambda_ability_risk_1pl(
  lambda_grid = seq(0, 1, by = 0.1),
  observed,
  predicted,
  generated,
  target_resp = NULL,
  theta_true = NULL,
  n_quad = 31,
  initial_pars = NULL,
  fit_fn = fit_mixed_subjects_mml_1pl,
  method = c("optimize", "grid"),
  bounds = c(-6, 6),
  max_discrimination = 10,
  control = list(maxit = 500),
  ...
)

Arguments

lambda_grid

Numeric vector of candidate lambda values in [0, 1]. For method = "grid" these are the evaluated candidates; for method = "optimize" only range(lambda_grid) matters and bounds the search (e.g. lambda_grid = c(0, 0.8) caps lambda at 0.8). Defaults to seq(0, 1, by = 0.1).

observed, predicted, generated

Response matrices passed to fit_mixed_subjects().

target_resp

Response matrix defining the target scoring population. If omitted, observed is used.

theta_true

Optional true theta values for target_resp, used in simulation studies to add squared scoring error to the risk. When omitted, mean_squared_error in the summary is NA; only mean_param_var is computed.

n_quad

Number of quadrature nodes.

initial_pars

Optional starting item parameters.

fit_fn

Fitting function. Defaults to fit_mixed_subjects_mml_1pl().

method

How lambda is chosen: "optimize" (default, direct 1-D optimization over range(lambda_grid), continuous lambda) or "grid" (evaluate every value in lambda_grid and take the argmin).

bounds

Bounds passed to score_theta().

max_discrimination

Upper bound on plausible item discrimination. Any candidate fit whose maximum |a| exceeds this value is treated as degenerate and excluded from selection. This guards against runaway discrimination fits, which can "converge" with a spuriously low model-based risk (huge discrimination collapses the item-parameter covariance). The default of 10 is far above any realistic 2PL discrimination.

control

Control list passed to stats::optim().

...

Additional arguments passed to fit_fn.

Value

A list with summary, best_lambda, best_fit, fits, risks.

Details

Passes fit_fn to allow switching between the frozen expected-count estimator (fit_mixed_subjects_1pl()) and the marginal-MML estimator (fit_mixed_subjects_mml_1pl()).

See also

tune_lambda_ability_risk() for the 2PL version; tune_lambda_ppi_score_1pl() for the PPI++ score diagnostic.

Examples

set.seed(1)
pars <- data.frame(a = 1, d = c(-0.5, 0, 0.5))
obs  <- simulate_2pl(rnorm(40), pars)
gen  <- simulate_2pl(rnorm(100), pars)
tuned <- tune_lambda_ability_risk_1pl(
  c(0, 0.5), obs, obs, gen,
  initial_pars = pars, n_quad = 5, control = list(maxit = 30)
)
tuned$best_lambda
#> [1] 0.5