An Extension of the Partial Credit Model with an Application to the Measurement of Change

Abstract

The partial credit model is considered under the assumption of a certain linear decomposition of the item × category parameters δ ih into “basic parameters” α j . This model is referred to as the “linear partial credit model”. A conditional maximum likelihood algorithm for estimation of the α j is presented, based on (a) recurrences for the combinatorial functions involved, and (b) using a “quasi-Newton” approach, the so-called Broyden-Fletcher-Goldfarb-Shanno (BFGS) method; (a) guarantees numerically stable results, (b) avoids the direct computation of the Hesse matrix, yet produces a sequence of certain positive definite matrices B k , k = 1, 2, ..., converging to the asymptotic variance-covariance matrix of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat \alpha _j $$ \end{document} . The practicality of these numerical methods is demonstrated both by means of simulations and of an empirical application to the measurement of treatment effects in patients with psychosomatic disorders.