Latent Class Models

The Mathematical Model

The LC model explains the relationship among manifest items by positing the assumption that the population comprises different classes related to the construct of interest. In other words, the population is assumed to consist of mutually exclusive and exhaustive groups, called latent classes, and the distribution of the items varies across classes. The LC model comprises two types of parameters: the relative size of each class and the probability of a particular response to each item within a class. To specify an LC model, let Y = (Y1, …, YM)) be M discrete items measuring latent classes, where variable Ym takes possible values from 1 to rm: Let C = 1, 2, …, L be the variable of LC membership, and let I(y = k) denote the indicator function that takes the value 1 if y = k and 0 otherwise.

If the class membership were observed, the joint probability that an individual belongs to Class 1 and provides responses y = (y1, …, yM) would be

where

γl = P(C = 1) represents the probability of belonging to Latent Class 1 and

represents the probability of response k to the mth item given a class membership in 1.

Therefore, the marginal probability of a particular response pattern y = (y1, …, yM) without regard for the unseen class membership is

Here, we have assumed local independence—that is, the items are assumed to be unrelated within each class. This assumption is the crucial feature of the LC model that allows us to draw inferences about the unseen class variable.

Model Identifiability

Model identification is imperative in estimating parameters of an LC model. The parameters of an LC model are said to be locally identifiable if the likelihood function is uniquely determined by the parameters within some neighborhood of a particular value of parameters. A necessary (but not sufficient) condition to make the LC model locally identifiable is that the number of possible response patterns of manifest items must be greater than the number of free parameters in the LC model. Even if this necessary condition is satisfied, it is impossible to say a priori whether or not this model is indeed identifiable. A necessary and sufficient condition for local identifiability is that the first derivative matrix of the log-likelihood function with respect to the parameters evaluated at a particular value must have full column rank. When an LC model is not identifiable, the simplest way to achieve identification is to reduce the number of parameters to be estimated by fixing or constraining parameters.

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