Factor analysis, latent class analysis, and item response theory when viewed as statistical models all share a basic mathematical structure. Examples from the measurement of food insecurity, when appropriate, are used to make the ideas concrete. These three types of statistical models all involve several observed variables or measurements and one (or more) latent, unobserved variable. These models have contact with data because they may be used to describe the distribution of the observed variables over a population of respondents. In addition, they allow users to draw inferences about the unobserved latent variable (e.g., food insecurity) based on the observed data (e.g., the FSS questions).
In general, the observed data consist of a set of p variables that are observed for each respondent in the study. These are called the manifest variables. Denote them by X1, X2, … Xp. In factor analysis, the X’s are the observed test scores from p tests for each person in the study. In latent class analysis, the X’s are the observed categorical responses of each respondent to p questions on a survey instrument. In IRT, the X’s are the dichotomous/binary or ordered polytomous responses of respondents to p questions/items on a test or survey instrument. In factor analysis, the manifest variables are continuous variables. In latent class analysis, the manifest variables may be dichotomous or polytomous nominal variables whose values are unordered categories. In IRT, the manifest variables are typically categorical and ordered and may be dichotomous/binary (e.g., “wrong/right” or “affirmed/not affirmed”) or polytomous (e.g., “never, sometimes, often”), as opposed to the continuous manifest variables of factor analysis or the unordered nominal manifest variables of latent class analysis.
In addition to the manifest variables, all latent variable models also assume the existence of a latent variable, the value of which varies from respondent to respondent but that is not directly observable for any respondent. The value of the latent variable affects the distribution of each manifest variable for each respondent—for example, the probability of endorsing each food insecurity question. This chapter uses the symbol to denote the latent variable to remind us that the main application of interest here is the measurement of food insecurity.
The three different types of latent variable models make different assumptions about the nature of the latent variable and how they are connected to the manifest variables. In factor analysis, each latent factor is continuous and univariate and the mean or expected value of each manifest variable is a linear combination of the latent factors. The weights on these linear combinations indicate the influence of each underlying latent factor on each test score.