Semiparametric Estimation in the Rasch Model and Related Exponential Response Models, Including a Simple Latent Class Model for Item Analysis

Abstract

The Rasch model for item analysis is an important member of the class of exponential response models in which the number of nuisance parameters increases with the number of subjects, leading to the failure of the usual likelihood methodology. Both conditional-likelihood methods and mixture-model techniques have been used to circumvent these problems. In this article, we show that these seemingly unrelated analyses are in fact closely linked to each other, despite dramatic structural differences between the classes of models implied by each approach. We show that the finite-mixture model for J dichotomous items having T latent classes gives the same estimates of item parameters as conditional likelihood on a set whose probability approaches one if T 2: (J + 1)/2. Unconditional maximum likelihood estimators for the finite-mixture model can be viewed as Keifer-Wolfowitz estimators for the random-effects version of the Rasch model. Latent-class versions of the model are especially attractive when T is small relative to J. We analyze several sets of data, propose simple diagnostic checks, and discuss procedures for assigning scores to subjects based on posterior means. A flexible and general methodology for item analysis based on latent class techniques is proposed.

Feature‐Specific Penalized Latent Class Analysis for Genomic Data

“…In this article we have proposed a latent class model where we have penalized the likelihood to address high dimensionality. Our methodology generalizes the parameter constraints proposed by Lazarsfeld and Henry (1968), Agresti and Lang (1993), and Lindsay et al (1991) in the sense that these prior methods essentially impose either a 0 or an infinite penalty on the conditional probabilities transformed as h(U β) for some appropriate matrix U. Figure 2 depicts the results of a threeclass model fit to the LOH data with a LASSO penalty, thus shrinking a majority of the coefficients completely to 0.…”

“…Early approaches fixed some of the conditional probabilities of a success, given class membership, to given values or constrained them to be equal (Lazarsfeld and Henry, 1968). Along the same lines, Agresti and Lang (1993) and Lindsay, Clogg, and Grego (1991) considered models in which the associations between the latent class and the observed variables are the same for all variables. Meulders et al (2002) considered constrained latent class models in which the conditional probabilities are a nonlinear function of a smaller set of basic parameters.…”

Testing the Rasch model by means of the mixture fit index

“…It has been known for some years that -apart from degenerate cases -the Rasch (1960) model and specifically restricted latent class models become equivalent in that the conditional maximum likelihood (ML) method for the Rasch model and the semiparametric ML method realized by latent class models result in the same item parameter estimates (cf. De Leeuw & Verhelst, 1986;Follman, 1988;Lindsay, Clogg, & Grego, 1991). This equivalence is attained if two conditions are met.…”

Evaluating psychological distress data

“…and The Lindsay, Clogg, and Crego (1991) score associated with the IRT model, which is the conditional expectation of θ v given the item outcomes is defined as…”

“…In addition there is a score which can be derived directly from an IRT model which is the Bayesian conditional mean, θ * first proposed by Lindsay, Clogg, and Crego (1991). This an alternative to k 6 and can be compared to k 6 in terms of its ability to explain which respondents get treatments as well as which regressors are important in explaining psychological distress.…”