Item Response Theory

5Item Parameter Estimation – Polytomous Data

It is rather surprising that systematic studies of human abilities were not undertaken until the second half of the last century An accurate method was available for measuring the circumference of the earth 2,000 years before the first systematic measures of human ability were developed.

(Source: Nunnally (1967))

In the following we present the results essential to marginal maximum likelihood (MML) estimation of parameters for polytomous item response models. They concisely unify, clarify, and extend scattered results now in the item response theory (IRT) literature. The treatment includes the following polytomous models now in wide use: nominal categories, graded categories, generalized partial credit in both its original and rating scale versions. For all models, derivations of likelihood equations are given for both the EM item‐by‐item solution and the Newton–Gauss solution for all items jointly. Computing formulas for the gradient vectors and information matrices are included. Also considered are boundary problems in the solutions and issues of failure of assumptions.

5.1 General Results

Suppose that the data record of respondent , consists of integers, , indicating assignment of a response to one of mutually exclusive categories ...