Testing Manifest Monotonicity Using Order-Constrained Statistical Inference

PSYCHOMETRIKAVOL. 78, NO. 1, 8397

JANUARY 2013

DOI: 10.1007/S11336-012-9297-X

TESTING MANIFEST MONOTONICITY USING ORDER-CONSTRAINED STATISTICAL INFERENCE

JESPER TIJMSTRA, DAVID J. HESSEN, AND PETER G.M. VAN DER HEIJDEN

UTRECHT UNIVERSITY

KLAAS SIJTSMA

TILBURG UNIVERSITY

Most dichotomous item response models share the assumption of latent monotonicity, which states that the probability of a positive response to an item is a nondecreasing function of a latent variable intended to be measured. Latent monotonicity cannot be evaluated directly, but it implies manifest monotonicity across a variety of observed scores, such as the restscore, a single item score, and in some cases the total score. In this study, we show that manifest monotonicity can be tested by means of the order-constrained statistical inference framework. We propose a procedure that uses this framework to determine whether manifest monotonicity should be rejected for specic items. This approach provides a likelihood ratio test for which the p-value can be approximated through simulation. A simulation study is presented that evaluates the Type I error rate and power of the test, and the procedure is applied to empirical data.

Key words: item response theory, latent monotonicity, manifest monotonicity, monotone homogeneity model, order-constrained statistical inference.

1. Introduction

A very general dichotomous item response theory (IRT) model for the ordinal measurement of a continuous latent variable is Mokkens non-parametric monotone homogeneity model (Mokken, 1971). This model is characterized by three assumptions: unidimensionality, local independence, and latent monotonicity. The rst two assumptions state that all possible associations between the items can be explained by a single latent variable representing a trait or an ability. Latent monotonicity species that the probability of observing a positive response to an item increases monotonously as the latent variable increases. Special cases of the model are the well-known one-, two-, and three-parameter logistic models (Rasch, 1960; Birnbaum, 1968) and Mokkens non-parametric double monotonicity model (Mokken, 1971).

From the assumptions of the monotone homogeneity model it follows that the total score the unweighted sum of the item scoreshas the property of monotone likelihood ratio with respect to the latent variable (Grayson, 1998; Huynh, 1994; nl, 2008), which ensures that the total score stochastically orders persons on the latent variable (Hemker, Sijtsma, Molenaar & Junker, 1997). As such, the model captures the idea that the...