Three-step estimation of cognitive diagnosis models with covariate

Abstract

Applied researchers often seek to relate covariates or external variables to latent group classification. The literature on latent class models offers two approaches, the one-step or three-step procedure. The former incorporates the covariates in the latent class model, estimating the parameters for the measurement model and the structural model simultaneously. The latter estimates the parameters of the measurement model using only the item responses, then proceeds to use the examinee classification as observed dependent variables in a multinomial logistic regression. The three-step approach leads to downward bias in the parameter estimates (Bolck, Croon, & Hagenaars, 2004). In a 2010 paper, Vermunt proposed a new three-step maximum likelihood procedure that used the examinee classifications from the second step as the dependent variable in a multinomial logistic regression that incorporated the measurement error probabilities. These measurement error probabilities were calculated as the conditional probability of the estimated classification, given the true classification. Including this matrix of conditional probabilities in the objective function corrects the bias in the third-step estimates. An improvement to this correction procedure is proposed for the cognitive diagnosis framework that can be applied at the level of the attribute vector or the individual attributes. A simulation study is designed to investigate the ability of the revised correction matrix to estimate the parameters of the structural model. Factors manipulated include sample size, test length, the number of attributes, and item quality. Preliminary results suggest that the proposed procedure returns incremental improvements over a variety of conditions.