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In [[statistics]], a '''latent class model (LCM)''' relates a set of observed (usually discrete) [[multivariate random variable|multivariate]] variables to a set of [[latent variable]]s. It is a type of [[latent variable model]]. It is called a latent class model because the latent variable is discrete. A class is charterized by a pattern of [[conditional probabilities]] that indicate the chance that variables take on certain values.

'''Latent Class Analysis (LCA)''' is a subset of [[structural equation modeling]], used to find groups or subtypes of cases in multivariate categorical data. These subtypes are called "latent classes".<ref name="Lazarsfeld">Lazarsfeld P.F. and Henry, N.W. (1968)''Latent structure analysis''. Boston: Houghton Mifflin</ref>

Confronted with a situation as follows, a researcher might choose to use LCA to understand the data: Imagine that symptoms a-d have been measured in a range of patients with diseases X Y and Z, and that disease X is associated with the presence of symptoms a, b, and c, disease Y with symptoms b, c, d, and disease Z with symptoms a, c and d.

The LCA will attempt to detect the presence of latent classes (the disease entities), creating patterns of association in the symptoms. As in factor analysis, the LCA can also be used to classify case according to their [[maximum likelihood]] class membership.<ref name="Lazarsfeld"/>

Because the criterion for solving the LCA is to achieve latent classes within which there is no longer any association of one symptom with another (because the class is the disease which causes their association, and the set of diseases a patient has (or class a case is a member of) causes the symptom association, the symptoms will be "conditionally independent", i.e., conditional on class membership, they are no longer related.<ref name="Lazarsfeld"/>

== 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
:  <math>p_{i_1, i_2, \ldots, i_N} \approx \sum_t^T p_t \, \prod_n^N p^n_{i_n,
t},</math>
where T is the number of latent classes and pt are the so-called recruitment
or unconditional probabilities that should sum to one. <math>p^n_{i_n, t}</math> are the
marginal or conditional probabilities.

For a two-way latent class model the form is
: <math>p_{ij} \approx \sum_t^T p_t \, p_{it} \, p_{jt}.</math>
This two-way model is related to [[probabilistic latent semantic analysis]] and [[non-negative matrix factorization]].

== Application ==
LCA may be used in many fields, such as: [[Collaborative Filtering]]
<ref name="Cheung2004">{{cite journal
 |last1=Cheung |first1=Kwok-Wai
 |last2=Tsui |first2=Kwok-Ching
 |last3=Liu |first3=Jiming
 |title=Extended latent class models for collaborative recommendation
 |journal=IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans
 |volume=34
 |issue=1|pages=143&ndash;148 |year=2004
 |doi=10.1109/TSMCA.2003.818877
}}</ref> and [[Behavior Genetics]]
<ref name="Eaves2013">{{cite journal
 |author = Eaves, L. J., Silberg, J. L., Hewitt, J. K., Rutter, M., Meyer, J. M., Neale, M. C., & Pickles, A
 |title=Analyzing twin resemblance in multisymptom data: genetic applications of a latent class model for symptoms of conduct disorder in juvenile boys
 |journal=Behavior Genetics
 |volume=23
 |issue=1|pages=5&ndash;19 |year=1993
}}</ref>

==External links==
* The Methodology Center, [http://methodology.psu.edu/ra/lcalta Latent Class Analysis], a research center at [[Penn State]], free software, FAQ
* John Uebersax, [http://www.john-uebersax.com/stat/index.htm Latent Class Analysis], 2006. A web-site with bibliography, software, links and FAQ for latent class analysis

==References==
{{Reflist}}
* {{Cite book
 | author = Linda M. Collins, Stephanie T. Lanza
 | title = Latent class and latent transition analysis for the social, behavioral, and health sciences
 | location = New York
 | publisher = [[John Wiley & Sons|Wiley]]
 | year = 2010
 | isbn = 978-0-470-22839-5
}}
* {{Cite book
 | author = A. L. McCutcheon
 | title = Latent class analysis
 | series = Quantitative Applications in the Social Sciences Series No. 64.
 | location = Thousand Oaks, California
 | publisher = [[Sage Publications]]
 | year = 1987
 | isbn = 0-521-59451-0
}}
* {{Cite journal
 | author = [[Leo A. Goodman]]
 | title = Exploratory latent structure analysis using both identifiable and unidentifiable models
 | journal = [[Biometrika]]
 | volume = 61
 | pages = 215&ndash;231
 | year = 1974
 | doi = 10.1093/biomet/61.2.215
 | issue = 2
}}
* {{Cite book
 | author = [[Paul F. Lazarsfeld]], Neil W. Henry
 | title = Latent Structure Analysis
 | year = 1968
}}

{{DEFAULTSORT:Latent Class Model}}
[[Category:Categorical data]]
[[Category:Latent variable models]]