View source for Latent variable model

A '''latent variable model''' is a [[Statistical model|statistical model]] that relates a set of [[Philosophy:Observable variable|observable variable]]s (also called ''manifest variables'' or ''indicators'')<ref>{{Cite web |title=Latent Variable Models |url=https://www.statistics.com/glossary/latent-variable-models/ |url-status=live |archive-url=https://web.archive.org/web/20221101060559/https://www.statistics.com/glossary/latent-variable-models/ |archive-date=2022-11-01 |access-date=2022-11-01 |website=Statistics.com: Data Science, Analytics & Statistics Courses |language=en-US}}</ref> to a set of [[Social:Latent variable|latent variable]]s.

It is assumed that the responses on the indicators or manifest variables are the result of an individual's position on the latent variable(s), and that the manifest variables have nothing in common after controlling for the latent variable ([[Local independence|local independence]]).

Different types of the latent variable models can be grouped according to whether the manifest and latent variables are categorical or continuous:<ref>{{cite book |first=David J. |last=Bartholomew |first2=Fiona |last2=Steel |first3=Irini |last3=Moustaki |first4=Jane I. |last4=Galbraith |year=2002 |title=The Analysis and Interpretation of Multivariate Data for Social Scientists |location= |publisher=Chapman & Hall/CRC |page=145 |isbn=1-58488-295-6 }}</ref>

{| class="wikitable" style="margin: 1em auto;"
! 
! colspan="2" | Manifest variables
|-
! Latent variables
! Continuous
! Categorical
|- align="center"
! Continuous
| [[Factor analysis]]
| [[Item response theory]]
|- align="center"
! Categorical
| Latent profile analysis
| Latent class analysis
|}

The [[Rasch model]] represents the simplest form of item response theory.  Mixture models are central to latent profile analysis.

In [[Factor analysis|factor analysis]] and latent trait analysis{{refn|group=note|name=LTAandIRT| The terms "latent trait analysis" and "item response theory" are often used interchangeably.<ref>{{Cite web |first=John |last=Uebersax |title=Latent Trait Analysis and Item Response Theory (IRT) Models |url=http://www.john-uebersax.com/stat/lta.htm |url-status=live |archive-url=https://web.archive.org/web/20221101072029/http://www.john-uebersax.com/stat/lta.htm |archive-date=2022-11-01 |access-date=2022-11-01 |website=John-Uebersax.com |language=en-US}}</ref>}} the latent variables are treated  as continuous [[Normal distribution|normally distributed]] variables, and in latent profile analysis  and latent class analysis as from a [[Multinomial distribution|multinomial distribution]].<ref>{{cite book |last=Everitt |first=BS |title=An Introduction to Latent Variables Models |year=1984 |publisher=Chapman & Hall |isbn=0-412-25310-0 }}</ref>  The manifest variables in factor analysis and latent profile analysis  are continuous and in most cases, their conditional distribution given the latent variables  is assumed to be normal. In latent trait analysis and latent class analysis,  the manifest variables are discrete. These variables could be dichotomous, ordinal or nominal variables.  Their conditional distributions are assumed to be binomial or multinomial.

Because the distribution of a continuous latent variable can be approximated by a discrete distribution,  the distinction between continuous and discrete variables turns out not to be fundamental at all.{{Citation needed|date=September 2023}} Therefore, there may be a psychometrical latent variable, but not a [[Social:Psychology|psychological]] psychometric variable.

==See also==
* [[Confirmatory factor analysis]]
* [[Hidden Markov model]]
* [[Partial least squares path modeling]]
* [[Structural equation modeling]]

== Notes ==
{{reflist|group=note}}

== References ==
{{Reflist}}

== Further reading ==
* {{cite book |first=Anders |last=Skrondal |first2=Sophia |last2=Rabe-Hesketh |title=Generalized Latent Variable Modeling |location= |publisher=Chapman & Hall |year=2004 |isbn=1-58488-000-7 }}

{{DEFAULTSORT:Latent Variable Model}}
[[Category:Latent variable models| Latent variable model]]

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