Fit a 1PL (one-parameter logistic) model — fit

Estimates a shared discrimination parameter a (equal across all items) and per-item intercepts d_j by maximizing the IRT marginal likelihood under a standard-normal ability prior using L-BFGS-B.

Usage

fit_1pl(
  resp,
  n_quad = 31,
  initial_pars = NULL,
  quadrature = NULL,
  slope_lower = 1e-04,
  slope_upper = NULL,
  control = list(maxit = 500)
)

Arguments

resp: Binary response matrix.
n_quad: Number of standard-normal quadrature nodes.
initial_pars: Optional starting item parameters (data frame with a and d columns). If omitted, a = 1 and d_j = qlogis(p_j) where p_j is the observed proportion correct for item j.
quadrature: Optional quadrature grid.
slope_lower, slope_upper: Bounds on the shared discrimination.
control: Control list passed to stats::optim().

Value

A list with pars (item parameter data frame with all a equal), par (the raw parameter vector), and optimizer details.

Details

The response probability is P(x_j = 1 | theta) = plogis(a * theta + d_j). The parameter vector has length J + 1: one shared discrimination followed by J per-item intercepts.

Examples

set.seed(1)
pars <- data.frame(a = 1, d = c(-0.5, 0, 0.5))
resp <- simulate_2pl(rnorm(60), pars)
fit <- fit_1pl(resp, n_quad = 7)
fit$pars
#>   item         a           d          b
#> 1    1 0.1013721 -0.40647830  4.0097660
#> 2    2 0.1013721  0.06686741 -0.6596236
#> 3    3 0.1013721  0.84933748 -8.3784165