Every Trait Counts: Marginal Maximum Likelihood Estimation for Novel Multidimensional Count Data Item Response Models with Rotation or l1-Regularization for Simple Structure

Abstract

The framework of multidimensional item response theory (MIRT) offers psychometric models for various data settings, most popularly for dichotomous and polytomous data. Less attention has been devoted to count responses. A recent growth in interest in count item response models (CIRM)---perhaps sparked by increased occurrence of psychometric count data, e.g., in the form of process data, clinical symptom frequency, number of ideas or errors in cognitive ability assessment---has focused on unidimensional models. A few recently proposed unidimensional CIRMs rely on the Conway-Maxwell-Poisson distribution as the conditional response distribution which allows to model conditionally over-, under-, and equidispersed responses. In this article, we generalize one of those CIRMs to the multidimensional case, introducing the Multidimensional Two-Parameter Conway-Maxwell-Poisson Model (M2PCMPM) class. Using the Expectation-Maximization (EM) algorithm, we develop marginal maximum likelihood estimation methods, primarily for exploratory M2PCMPMs. The resulting discrimination matrices are rotationally indeterminate. We pursue the goal of obtaining a simple structure for them by (1) rotating and (2) regularizing the discrimination matrix. Recent IRT research has successfully used regularization of the discrimination matrix to obtain a simple structure (i.e., a sparse solution) for dichotomous and polytomous data. We develop an EM algorithm with lasso ($\ell_1$) regularization for the M2PCMPM and compare (1) and (2) in a simulation study. We illustrate the proposed model with an empirical example using intelligence test data.

Smooth Information Criterion for Regularized Estimation of Item Response Models

“…In this article, we confined ourselves to analyzing dichotomous item responses and continuous factor variables. Future research could investigate the application of these techniques to polytomous item response, count item response data [55], or cognitive diagnostic models that involve multivariate binary factor variables [56]. More generally, smooth information criteria can be used in all modeling approaches that involve regularized estimation.…”