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In statistics, a latent class model (LCM) relates a set of observed discrete multivariate variables to a set of latent variables. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Multivariate statistics or multivariate analysis in Statistics describes a collection of procedures which involve observation and analysis of more than In Statistics, latent variables (as opposed to Observable variables, are Variables that are not directly observed but are rather inferred (through a It is a type of latent variable model. A latent variable model is a Statistical model that relates a set of Variables (so-called manifest variables) to set of Latent variables It is called a latent class model because the latent variable is discrete. A class is characterized by a pattern of conditional probabilities that indicate the chance that variables takes on certain values. Conditional probability is the Probability of some event A, given the occurrence of some other event B.

For instance, the variables could be multiple choice items of a political questionnaire. For the 1974 John Wayne Crime drama movie see McQ. Multiple choice is a form of Assessment in which respondents The data in this case consists of a N-way contingency table with answers to the items for a number of respondents. In Statistics, contingency tables are used to record and analyse the relationship between two or more variables most usually Categorical variables Suppose 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. Conditional probability is the Probability of some event A, given the occurrence of some other event B.

Within each latent class, the observed variables are statistically independent. In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other 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). Local independence is the underlying assumption of Latent variable models The observed items are independent of each other given an individual score on the latent variable(s 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. p^n_{i_n, t} are the marginal or conditional probabilities.

For a two-way latent class model the form is

p_{ij} \approx \sum_t^T p_t \, p_{it} \, p_{jt}.

This two-way model is related to probabilistic latent semantic analysis and non-negative matrix factorization. Probabilistic latent semantic analysis (PLSA, also known as probabilistic latent semantic indexing ( PLSI, especially in information retrieval circles is a NMF redirects here For the bridge convention see New minor forcing.

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