Latent Class Model - Related Methods

Related Methods

As in much of statistics, there are a large number of methods with distinct names and uses, which share a common relationship. Cluster analysis is, like LCA, used to discover taxon-like groups of cases in data. Multivariate mixture estimation (MME) is applicable to continuous data, and assumes that such data arise from a mixture of distributions: imagine a set of heights arising from a mixture of men and women. If a Multivariate mixture estimation is constrained so that measures must be uncorrelated within each distribution it termed latent profile analysis. Modified to handle discrete data, this constrained analysis is known as LCA. Discrete latent trait models further constrain the classes to formed from segments of a single dimension: essentially allocating members to classes on that dimension: an example would be assigning cases to social classes on a dimension of ability or merit.

As a practical instance, the variables could be multiple choice items of a political questionnaire. The data in this case consists of a N-way contingency table with answers to the items for a number of respondents. In this example, the latent variable refers to political opinion and the latent classes to political groups. Given group membership, the conditional probabilities specify the chance certain answers are chosen.

Within each latent class, the observed variables are statistically independent. This is an important aspect. Usually the observed variables are statistically dependent. By introducing the latent variable, independence is restored in the sense that within classes variables are independent (local independence). We then say that the association between the observed variables is explained by the classes of the latent variable (McCutcheon, 1987).

In one form the latent class model is written as

p_{i_1, i_2, \ldots, i_N} \approx \sum_t^T p_t \, \prod_n^N p^n_{i_n, t},

where T is the number of latent classes and pt are the so-called recruitment or unconditional probabilities that should sum to one. are the marginal or conditional probabilities.

For a two-way latent class model the form is

This two-way model is related to probabilistic latent semantic analysis and non-negative matrix factorization.

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