Regularized Latent Class Analysis with Application in Cognitive Diagnosis

Abstract

Diagnostic classification models are confirmatory in the sense that the relationship between the latent attributes and responses to items is specified or parameterized. Such models are readily interpretable with each component of the model usually having a practical meaning. However; parameterized diagnostic classification models are sometimes too simple to capture all the data patterns, resulting in significant model lack of fit. In this paper, we attempt to obtain a compromise between interpretability and goodness of fit by regularizing a latent class model. Our approach starts with minimal assumptions on the data structure, followed by suitable regularization to reduce complexity, so that readily interpretable, yet flexible model is obtained. An expectation–maximization-type algorithm is developed for efficient computation. It is shown that the proposed approach enjoys good theoretical properties. Results from simulation studies and a real application are presented.

Regularized Latent Class Analysis for Polytomous Item Responses: An Application to SPM-LS Data

“…Regularized latent class models (RLCMs; Chen et al 2017 ) estimate item response probabilities under the presupposition that similar item response probabilities in these models are grouped and receive the same value. The main idea of using the regularization technique (see Hastie et al 2015 for an overview) to LCMs is that by subtracting an appropriate penalty term from the log-likelihood function, some simpler structure on item response probabilities is posed.…”

“…Different penalty terms typically result in different estimated parameter structures. In a recent Psychometrika paper, Chen et al ( 2017 ) proposed the RLCM for dichotomous item responses. Related work for dichotomous data can be found in Wu ( 2013 ) and Yamamoto and Hayashi ( 2015 ).…”

“…In an RLCM, not all item response probabilities ( ) are assumed to be unique. Chen et al ( 2017 ) subtracted a penalty term from the log-likelihood function that penalizes differences in ordered item response probabilities. In more detail, denote by ordered item response probabilities of the original probabilities such that , and collect all parameters in .…”

“…If denotes the matrix of observed data and denotes the vector of all ordered item response probability and the vector that represents the skill class probabilities, the following function is maximized in Chen et al ( 2017 ): where l denotes the log-likelihood function of the data. By employing a penalty function in the estimation, some item response probabilities are merged, which, in turn, eases the interpretation of resulting latent classes.…”

“…The maximization of ( 7 ) is conducted using an expectation-maximization (EM) algorithm (see Section 3.3 for general description). The estimation approach of Chen et al ( 2017 ) is implemented in the R package CDM ( George et al 2016 ; Robitzsch and George 2019 ).…”