In latent class analysis, the latent variable is a discrete latent class to which each respondent is assumed to belong. Thus, in latent class analysis the latent variable is categorical rather than continuous, and it may or may not have an implied order among its values. Latent class models do not necessarily assume any particular form for the connection between the manifest variables and the latent class variable. (This is a source of the problem of identifiability discussed in a later section.)

For IRT models, the latent variable is continuous and univariate (or multivariate). In educational applications, the latent variable indicates the underlying latent proficiency of each test taker that in turn influences the likelihood of correct responses to the test. In the application to food insecurity measurement, the latent variable represents the degree of food insecurity experienced by a given household that in turn influences the likelihood of endorsing or affirming responses to questions about lack of food due to economic constraints. There is a close connection between IRT models and latent class models. If the latent variable in an IRT model is assumed to have a discrete distribution concentrated on a few points, it becomes a latent class model with ordered latent classes.

Deciding whether a latent variable is more appropriately thought of as discrete or continuous cannot really be based on data, and in fact it is often impossible to assess any difference between the fit of the two types of models (Lindsey, Clogg, and Grego, 1991). More usually, this decision is based on other considerations. For example, in the case of food insecurity, it seems plausible that varies in a continuous way across households rather than only having a few possible values that it can take on.

It is evident that, of these different types of latent variable models, IRT models are particularly appropriate for modeling the measurement of food insecurity using survey data of the type collected in the CPS. The manifest variables or indicators of food insecurity in the FSS are all either binary or polytomous and ordered. In addition, food insecurity may be viewed as an underlying continuous, unidimensional, but not directly observable quantity that varies from household to household. Higher values of latent food insecurity are indicated by higher probabilities of endorsing or affirming survey items that indicate higher degrees of not being able to obtain sufficient food due to a lack of economic resources.

Returning to the structure of latent variable models, they all involve the notion of conditional statistical independence, so the panel first reviews this important idea.



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