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A nested expectation-maximization algorithm for latent class models with covariates

Abstract

Latent class models with covariates express the joint distribution of a multivariate categorical random variable under an assumption of conditional independence, given a covariate-dependent latent class variable. These models are popular in many fields, and current computational procedures for point estimation rely either on stepwise routines, or combine the expectation-maximization (EM) algorithm with Newton-Raphson methods to facilitate the derivations for the maximization steps. Although these algorithms are routinely implemented, the stepwise strategies do not maximize the full log-likelihood associated with the statistical model, whereas the Newton-Raphson steps within the EM algorithm do not provide monotone log-likelihood sequences, thereby leading to routines which may not guarantee reliable maximization. Motivated by these issues, we define a nested EM algorithm, which relies on a sequence of conditional expectation-maximizations for the regression coefficients associated with the covariate-dependent latent class variables. Leveraging a recently developed P\`olya-gamma data augmentation for logistic regression, the conditional expectation-maximizations reduce to a set of generalized least squares minimization problems. We show that the proposed nested EM provides a monotone log-likelihood sequence, and highlight the substantial performance gains in two real data applications.

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