An Isotonic Partial Credit Model for Ordering Subjects on the Basis of Their Sum Scores

• Published in: Psychometrika Vol. 77; no. 3; pp. 479 - 494
• Main Author: Ligtvoet, Rudy
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.07.2012; Springer; Springer Nature B.V
• Subjects: Ability tests; Algorithms; Assessment; Bayesian analysis; Bayesian Statistics; Behavioral Science and Psychology; Biological and medical sciences; Comparative Analysis; Computation; Data Analysis; Data processing; Evaluation Methods; Fundamental and applied biological sciences. Psychology; Humanities; Item Response Theory; Law; Mathematical models; Nonparametric Statistics; Observation; Psychology; Psychology. Psychoanalysis. Psychiatry; Psychology. Psychophysiology; Psychometrics; Psychometrics. Statistics. Methodology; Rasch model; Sampling; Scores; Serial Ordering; Simulation; Statistical Theory and Methods; Statistics for Social Sciences; Statistics. Mathematics; Stochasticity; Test Items; Testing and Evaluation; Tests; France; Somme; Picardie; Europe; ordinal inferences; sum score; partial credit model; stochastic ordering; monotone likelihood ratio; polytomously scored items; item response theory; Item response theory; Stochastic order; Likelihood ratio; Psychometrics; Inference; Statistical estimation; Ordering; Test scoring
• ISSN: 0033-3123; 1860-0980
• DOI: 10.1007/s11336-012-9272-6

In practice, the sum of the item scores is often used as a basis for comparing subjects. For items that have more than two ordered score categories, only the partial credit model (PCM) and special cases of this model imply that the subjects are stochastically ordered on the common latent variable. However, the PCM is very restrictive with respect to the constraints that it imposes on the data. In this paper, sufficient conditions for the stochastic ordering of subjects by their sum score are obtained. These conditions define the isotonic (nonparametric) PCM model. The isotonic PCM is more flexible than the PCM, which makes it useful for a wider variety of tests. Also, observable properties of the isotonic PCM are derived in the form of inequality constraints. It is shown how to obtain estimates of the score distribution under these constraints by using the Gibbs sampling algorithm. A small simulation study shows that the Bayesian p -values based on the log-likelihood ratio statistic can be used to assess the fit of the isotonic PCM to the data, where model-data fit can be taken as a justification of the use of the sum score to order subjects.