Optimizing information using the EM algorithm in item response theory

Abstract

Latent trait models such as item response theory (IRT) hypothesize a functional relationship between an unobservable, or latent, variable and an observable outcome variable. In educational measurement, a discrete item response is usually the observable outcome variable, and the latent variable is associated with an examinee’s trait level (e.g., skill, proficiency). The link between the two variables is called an item response function. This function, defined by a set of item parameters, models the probability of observing a given item response, conditional on a specific trait level.

Typically in a measurement setting, neither the item parameters nor the trait levels are known, and so must be estimated from the pattern of observed item responses. Although a maximum likelihood approach can be taken in estimating these parameters, it usually cannot be employed directly. Instead, a method of marginal maximum likelihood (MML) is utilized, via the expectation-maximization (EM) algorithm. Alternating between an expectation (E) step and a maximization (M) step, the EM algorithm assures that the marginal log likelihood function will not decrease after each EM cycle, and will converge to a local maximum.

Interestingly, the negative of this marginal log likelihood function is equal to the relative entropy, or Kullback-Leibler divergence, between the conditional distribution of the latent variables given the observable variables and the joint likelihood of the latent and observable variables. With an unconstrained optimization for the M-step proposed here, the EM algorithm as minimization of Kullback-Leibler divergence admits the convergence results due to Csiszár and Tusnády (Statistics & Decisions, 1:205–237, 1984), a consequence of the binomial likelihood common to latent trait models with dichotomous response variables. For this unconstrained optimization, the EM algorithm converges to a global maximum of the marginal log likelihood function, yielding an information bound that permits a fixed point of reference against which models may be tested. A likelihood ratio test between marginal log likelihood functions obtained through constrained and unconstrained M-steps is provided as a means for testing models against this bound. Empirical examples demonstrate the approach.