Joint Maximum Likelihood Estimation of the Rasch Model • jmleIRT

Overview

The jmleIRT package provides tools to estimate Rasch (1PL) model parameters using Joint Maximum Likelihood Estimation (JMLE) in R. The package supports bias correction and Warm’s Weighted Likelihood Estimates suitable for psychometric research.

Features

Joint Maximum Likelihood Estimation of person abilities $\theta_p$ and item difficulties $\beta_i$ .
Bias correction to reduce estimation bias, including epsilon bias stabilization.
Warm’s Weighted Likelihood Estimates (WLE) for person abilities.
Support for missing data (missing responses coded as NA).
Verbose mode to trace iteration progress.
Comprehensive unit tests for reliability.

Installation

Or from GitHub development version:

# install.packages("remotes")
remotes::install_github("jansteinfeld/jmleIRT")

Usage Example

library(jmleIRT)

# Simulate response  40 persons (p) × 5 items (i)
set.seed(123)
X <- matrix(rbinom(40 * 5, 1, 0.5), nrow = 40)

# Estimate Rasch model using JMLE with bias correction
fit <- jmle_estimation(X, max_iter = 200, conv = 1e-5,
                      center = "items", bias_correction = TRUE)

# Item difficulties (centered)
print(fit$beta)

# Compute Weighted Likelihood Estimates for persons
wle_res <- estimate_wle(X, fit$beta)

# Person ability estimates via WLE
print(wle_res$wle)

Mathematical Model

The Rasch Model

The Rasch model is a fundamental Item Response Theory (IRT) model used for dichotomous item data. The probability that person $p$ with ability $\theta_p$ answers item $i$ with difficulty $\beta_i$ correctly is modeled as:

$P(X_{pi} = 1 \mid \theta_p, \beta_i) = \frac{e^{\theta_p - \beta_i}}{1 + e^{\theta_p - \beta_i}}$

where $X_{pi} \in \{0,1\}$ indicates the correctness of response.

Joint Maximum Likelihood Estimation (JMLE)

JMLE simultaneously estimates person abilities $\theta = (\theta_1, \ldots, \theta_P)$ and item difficulties $\beta = (\beta_1, \ldots, \beta_I)$ by maximizing the joint likelihood of observed responses:

$L(\theta, \beta) = \prod_{p=1}^P \prod_{i=1}^I P(X_{pi} \mid \theta_p, \beta_i)$

This is typically solved via an iterative algorithm alternating updates for $\theta_p$ and $\beta_i$ , until convergence criteria are met.

Although easy to implement and computationally efficient, JMLE estimates suffer from bias, especially in small samples or with extreme response patterns.

Here is the updated version of the Bias Correction Methods section, formatted for integration into your README or vignette:

Bias Correction Methods

The Joint Maximum Likelihood Estimation (JMLE) for the Rasch model is known to produce biased parameter estimates, especially in finite samples and with extreme response patterns. The jmleIRT package implements two key bias correction techniques:

Epsilon adjustment: Proposed by Bertoli Barsotti, Punzo, and Lando, this method adds a small positive constant $\varepsilon > 0$ during the estimation process. This adjustment prevents infinite log-odds for perfect or zero scores, ensuring finite item difficulty and ability estimates. It also helps alleviate bias related to extreme response patterns by smoothing the estimation.
Analytical bias correction: This approach involves modifying the item difficulty estimates based on theoretical bias expressions derived from asymptotic expansions or minimum divergence estimators. It aims to reduce the structural bias inherent in the JMLE estimates as described in psychometric literature (e.g., Bertoli Barsotti & Punzo, 2012; Lando & Bertoli Barsotti, 2014).

In combination, the epsilon adjustment promotes numerical stability and improved bias behavior at the estimation level, while analytical bias correction provides targeted, theoretically grounded refinement of item parameter estimates to enhance overall accuracy and robustness of the JMLE algorithm.

Package Features

Estimation of person and item parameters using bias-corrected JMLE
optional ability estimation based on Warm’s Weighted Likelihood Estimates (WLE)
Handling of missing data
documentation and examples
Open-source development on GitHub for transparency and collaboration
Optimized C++ backend for performance

References

Bertoli Barsotti, L., & Punzo, A. (2012). Comparison of two bias reduction techniques for the Rasch model. Electronic Journal of Applied Statistical Analysis, 5(3), 360-366.
Lando, T., & Bertoli Barsotti, L. (2014). A modified minimum divergence estimator: some preliminary results for the Rasch model. Electronic Journal of Applied Statistical Analysis, 7(1), 37-57.
Wright, B. D., & Panchapakesan, N. (1969). A Procedure for Sample-Free Item Analysis. Educational and Psychological Measurement, 29(1), 23-48.

License

GPL-3

Contributions welcome via GitHub: [jansteinfeld/jmleIRT].

jmleIRT: Rasch Model Joint Maximum Likelihood Estimation in R