Latent class model

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

${\displaystyle p_{i_{1},i_{2},\ldots ,i_{N}}\approx \sum _{t}^{T}p_{t}\,\prod _{n}^{N}p_{i_{n},t}^{n},}$

where ${\displaystyle T}$ is the number of latent classes and ${\displaystyle p_{t}}$ are the so-called recruitment or unconditional probabilities that should sum to one. ${\displaystyle p_{i_{n},t}^{n}}$ are the marginal or conditional probabilities.

For a two-way latent class model, the form is

${\displaystyle p_{ij}\approx \sum _{t}^{T}p_{t}\,p_{it}\,p_{jt}.}$

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

There are a number of methods with distinct names and uses that 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 is 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 form 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.

LCA may be used in many fields, such as: collaborative filtering,[4] Behavior Genetics[5] and Evaluation of diagnostic tests.[6]

• Linda M. Collins; Stephanie T. Lanza (2010). Latent class and latent transition analysis for the social, behavioral, and health sciences. New York: Wiley. ISBN 978-0-470-22839-5.
• Allan L. McCutcheon (1987). Latent class analysis. Quantitative Applications in the Social Sciences Series No. 64. Thousand Oaks, California: SAGE Publications. ISBN 978-0-521-59451-6.
• Leo A. Goodman (1974). "Exploratory latent structure analysis using both identifiable and unidentifiable models". Biometrika. 61 (2): 215–231. doi:10.1093/biomet/61.2.215.
• Paul F. Lazarsfeld, Neil W. Henry (1968). Latent Structure Analysis.

• Statistical Innovations, Home Page, 2016. Website with latent class software (Latent GOLD 5.1), free demonstrations, tutorials, user guides, and publications for download. Also included: online courses, FAQs, and other related software.
• The Methodology Center, Latent Class Analysis, a research center at Penn State, free software, FAQ
• John Uebersax, Latent Class Analysis, 2006. A web-site with bibliography, software, links and FAQ for latent class analysis