Per-item ability-risk lambda tuning via coordinate descent

Finds a per-item vector of lambda values lambda_j in [0, 1] that minimizes propagated ability-score risk E[g' Sigma_gamma g] using coordinate descent on the items. Each coordinate step holds the other lambda_{j'} fixed and selects lambda_j by direct 1-D optimization (method = "optimize", the default, continuous) or over lambda_grid (method = "grid").

Usage

tune_lambda_ability_risk_item(
  lambda_grid = seq(0, 1, by = 0.1),
  observed,
  predicted,
  generated,
  target_resp = NULL,
  theta_true = NULL,
  n_quad = 31,
  initial_pars = NULL,
  n_pass = 1,
  init_lambda = 0,
  method = c("optimize", "grid"),
  bounds = c(-6, 6),
  max_discrimination = 10,
  control = list(maxit = 300),
  ...
)

Arguments

lambda_grid: Numeric vector of candidate lambda values in [0, 1] to try for each item independently.
observed, predicted, generated: Response matrices passed to fit_mixed_subjects_mml().
target_resp: Target scoring population. If omitted, observed is used.
theta_true: Optional true theta values, used to add squared scoring error to the risk.
n_quad: Number of quadrature nodes.
initial_pars: Optional starting item parameters.
n_pass: Number of coordinate-descent passes (default 1).
init_lambda: Starting lambda vector for coordinate descent. Supply the global scalar optimum from tune_lambda_ability_risk() (e.g. init_lambda = 0.5) to start the search around a useful operating point. Starting from all-zeros is not recommended: each single-item improvement is too small to detect when other items are at zero. A scalar is broadcast to all items; a vector of length n_items sets per-item starting values.
method: How each item's lambda is chosen at a coordinate step: "optimize" (default, direct 1-D optimization over range(lambda_grid), continuous) or "grid" (evaluate every value in lambda_grid).
bounds: Bounds passed to score_theta().
max_discrimination: Upper bound on plausible item discrimination; any candidate fit whose maximum |a| exceeds it is treated as degenerate and skipped. See tune_lambda_ability_risk() for the rationale. Default 10.
control: Control list passed to stats::optim().
...: Additional arguments passed to fit_mixed_subjects_mml().

Value

A list with lambda (per-item vector), item (item names), n_pass, method, and final_fit (the fit_mixed_subjects_mml() fit at the selected lambda).

Details

Calls fit_mixed_subjects_mml() with a per-item lambda vector at each candidate evaluation. Because the lambda is a vector, that function switches to its frozen expected-count Q-function path — posteriors are frozen at initial_pars, not recomputed continuously. This is an approximation; see the @note below. The resulting lambda vector can be used directly with fit_mixed_subjects_mml().

Computational cost. Each pass refits per item per candidate lambda: method = "grid" does n_items × length(lambda_grid) fits; method = "optimize" does roughly n_items × 12 (the optimizer's evaluations plus the endpoints). Use n_pass = 1 (the default) for a single greedy sweep, which is usually sufficient.

Note

Approximation status. The coordinate descent fits use the frozen expected-count Q-function (not the full marginal-MML objective) because the IRT marginal likelihood integrates over the joint response pattern and does not decompose item-wise. The approach is approximately correct when initial_pars is close to the converged parameters. Report per-item results as experimental / approximate.

Examples

set.seed(1)
pars <- data.frame(a = c(1, 1.2, 0.9), d = c(0, -0.5, 0.3))
observed  <- simulate_2pl(rnorm(40), pars)
generated <- simulate_2pl(rnorm(100), pars)
tuned <- tune_lambda_ability_risk_item(
  c(0, 0.5), observed, observed, generated,
  initial_pars = pars, n_quad = 5, control = list(maxit = 30)
)
tuned$lambda
#> [1] 0.5 0.5 0.5