Multilevel factor mixture modeling (FMM) is a hybrid of multilevel confirmatory factor analysis (CFA) and multilevel latent class analysis (LCA). It allows researchers to examine population heterogeneity at the within level, between level, or both levels. This tutorial focuses on explicating the model specification of multilevel FMM that considers…

Notwithstanding a large body of literature on log-linear models and odds ratios, no general marginal-free index of the association in a contingency table has gained a wide acceptance. Building on a framework developed by L. A. Goodman, we put into light the direct links between odds ratios, the Altham index, the intrinsic association coefficient,…

In this digital ITEMS module, we introduce the framework of nonparametric item response theory (IRT), in particular Mokken scaling, which can be used to evaluate fundamental measurement properties with less strict assumptions than parametric IRT models. We walk through the key distinction between parametric and nonparametric models, introduce the…

Mokken scale analysis (MSA) is a probabilistic-nonparametric approach to item response theory (IRT) that can be used to evaluate fundamental measurement properties with less strict assumptions than parametric IRT models. This instructional module provides an introduction to MSA as a probabilistic-nonparametric framework in which to explore…

Item response models typically assume that the item characteristic (step) curves follow a logistic or normal cumulative distribution function, which are strictly monotone functions of person test ability. Such assumptions can be overly-restrictive for real item response data. We propose a simple and more flexible Bayesian nonparametric IRT model…

Education systems are increasingly creating rich, longitudinal data sets with frequent, and even real-time, data updates of many student measures, including daily attendance, homework submissions, and exam scores. These data sets provide an opportunity for district and school staff members to move beyond an indicators-based approach and instead…

In contrast to dichotomous item response theory (IRT) models, most well-known polytomous IRT models do not imply stochastic ordering of the latent trait by the total test score (SOL). This has been thought to make the ordering of respondents on the latent trait using the total test score questionable and throws doubt on the justifiability of using…

The (univariate) isotonic psychometric (ISOP) model (Scheiblechner, 1995) is a nonparametric IRT model for dichotomous and polytomous (rating scale) psychological test data. A weak subject independence axiom W1 postulates that the subjects are ordered in the same way except for ties (i.e., similarly or isotonically) by all items of a psychological…

Describes relationships among 20 polytomous item response theory (IRT) models (parametric and nonparametric) and 8 measurement properties relevant to polytomous IRT. Provides three tables to assist in the choice of an appropriate model and demonstrates the use of the models in test construction. (SLD)

Presents a general class of ordinal logit models that specifies equality and inequality constraints on sums of conditional response probabilities. Uses maximum likelihood to estimate these models, making their assumptions testable with likelihood-ratio statistics. Illustrates the proposed models with an example using reported adult crying…

Discusses usability and interpretation issues for single-strategy cognitive assessment models that posit a stochastic, conjunctive relationship between a set of cognitive attributes to be assessed and performance on particular items/tasks of the assessment. Also discusses stochastic ordering and monotonicity properties that enhance the…

Discusses relationships between a mathematical measurement model and its real-world applications. Makes a distinction between large-scale data matrices commonly found in educational measurement and smaller matrices found in attitude and personality measurement. Also evaluates nonparametric methods for estimating item response functions and…

In the present paper, a new family of item response theory (IRT) models for dichotomous item scores is proposed. Two basic assumptions define the most general model of this family. The first assumption is local independence of the item scores given a unidimensional latent trait. The second assumption is that the odds-ratios for all item-pairs are…

The authors discuss the applicability of nonparametric item response theory (IRT) models to the construction and psychometric analysis of personality and psychopathology scales, and they contrast these models with parametric IRT models. They describe the fit of nonparametric IRT to the Depression content scale of the Minnesota Multiphasic…