PROX Estimation for the Rasch Model — prox

This function implements the PROX routine for Rasch models, ported from Alexander Robitzsch's R implementation to C++ for speed.

Usage

prox_algorithm(dat, dat.resp = NULL, freq = NULL, conv = 0.001, maxiter = 30)

Arguments

dat: Numeric matrix or data frame of binary item responses.
dat.resp: Numeric matrix indicating missing data (1 = observed, 0 = missing). If missing, it is generated internally.
freq: Numeric vector of frequencies for each response pattern. Defaults to 1 for all respondents.
conv: Numeric value specifying the convergence criterion for iteration. Defaults to 0.001.
maxiter: Integer specifying the maximum number of iterations. Defaults to 30.

Value

A list with elements:

b: Estimated item difficulties (logits).
theta: Estimated person abilities (logits).
iter: Number of iterations performed.
sigma.i: Estimated standard deviations of abilities for persons responding to items.
sigma.n: Estimated standard deviations of difficulties for items responded to by persons.

Details

The function starts with initial estimates using standardized raw scores and applies a bias adjustment to item difficulties based on Robitzsch's formula: $$ d_i = \mu_i - \sqrt{1 + \frac{\sigma_i^2}{2.9}} \times \text{logit}(\text{item proportion}) $$ to correct bias in finite samples. Then, person and item parameters are updated iteratively using logistic transformations until convergence or the maximum number of iterations is reached.

This implementation is primarily for quick and robust approximate Rasch model estimation, useful as a starting point for joint maximum likelihood estimation.

References

Robitzsch, A. (2020). sirt: Supplementary Item Response Theory Models. R package version 3.10-9. https://CRAN.R-project.org/package=sirt

Linacre, J.M. (1994). Many-Facet Rasch Measurement. Chicago: MESA Press.

Examples

if (FALSE) { # \dontrun{
data <- matrix(sample(0:1, 100, replace = TRUE), nrow = 10, ncol = 10)
result <- prox_algorithm(data)
print(result$b)
} # }