Latent variable model: Difference between revisions

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{{Short description|Statistical model relating manifest and latent variables}}

{{Unreferenced|date=December 2009}}

{{Multiple issues|

A '''latent variable model''' is a [[statistical model]] that relates a set of [[Variable (mathematics)|variables]]

{{refimprove|date=April 2011}}

(so-called ''manifest variables'') to set of [[latent variable]]s.

{{more footnotes|date=April 2011}}

}}

A '''latent variable model''' is a [[statistical model]] that relates set of [[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 [[latent variable]]s. Latent variable models are applied across a wide range of fields such as biology, computer science, and social science.<ref>{{Cite journal |last=Blei |first=David M. |date=2014-01-03 |title=Build, Compute, Critique, Repeat: Data Analysis with Latent Variable Models |url=https://www.annualreviews.org/doi/10.1146/annurev-statistics-022513-115657 |journal=Annual Review of Statistics and Its Application |language=en |volume=1 |issue=1 |pages=203–232 |doi=10.1146/annurev-statistics-022513-115657 |bibcode=2014AnRSA...1..203B |issn=2326-8298}}</ref> Common use cases for latent variable models include applications in [[psychometrics]] (e.g., summarizing responses to a set of survey questions with a [[factor analysis]] model positing a smaller number of psychological attributes, such as the trait [[Extraversion and introversion|extraversion]], that are presumed to cause the survey question responses),<ref>{{Cite journal |last1=Borsboom |first1=Denny |last2=Mellenbergh |first2=Gideon J. |last3=van Heerden |first3=Jaap |date=April 2003 |title=The theoretical status of latent variables. |url=https://doi.apa.org/doi/10.1037/0033-295X.110.2.203 |journal=Psychological Review |language=en |volume=110 |issue=2 |pages=203–219 |doi=10.1037/0033-295X.110.2.203 |pmid=12747522 |issn=1939-1471|url-access=subscription }}</ref> and [[natural language processing]] (e.g., a [[topic model]] summarizing a corpus of texts with a number of "topics").<ref>{{Cite journal |last1=Blei |first1=David M. |last2=Ng |first2=Andrew Y. |last3=Jordan |first3=Michael I. |date=2003 |title=Latent dirichlet allocation |url=https://dl.acm.org/doi/10.5555/944919.944937 |journal=J. Mach. Learn. Res. |volume=3 |issue=3/1/2003 |pages=993–1022 |issn=1532-4435}}</ref>

It is assumed that 1) the responses on the indicators or manifest variables are the result of

an individual's position on the latent variable(s), and 2) that the manifest variables have nothing

in common after controlling for the latent variable ([[local independence]]).

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]]).

Different types of the latent variable model can be grouped according to whether the manifest and

latent variables are categorical or continuous: <ref>David J. Bartholomew, Fiona Steel, Irini Moustaki, Jane I. Galbraith (2002), ''The Analysis and Interpretation of Multivariate Data for Social Scientists'', Chapman & Hall/CRC, p. 145</ref>

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 |first1=David J. |last1=Bartholomew, |authorlink=D. J. Bartholomew |first2=Fiona |last2=Steel, |authorlink2=Fiona Steele |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, p. |page=145 |isbn=1-58488-295-6 }}</ref>

<center>

{| class="wikitable" style="margin: 1em auto;"

!

! colspan="2" | Manifest variables

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! Continuous

| [[Factor analysis]]

| [[LatentItem traitresponse analysistheory]]

|- align="center"

! Categorical

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| [[Latent class analysis]]

|}

</center>

AnotherThe name[[Rasch formodel]] latentrepresents traitthe analysissimplest isform of [[item response theory. [[Mixture models]] (IRT).are central to latent profile analysis.

The most simple IRT model is the [[Rasch model]].

An important part of the latent profile analysis is the [[mixture model]].

In [[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]].<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.

In [[factor analysis]] and [[latent trait analysis]] 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]].

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.

==See also==

Because the distribution of a continuous latent variable can be approximated by a discrete distribution,

* [[Confirmatory factor analysis]]

the distinction between continuous and discrete variables turns out not to be fundamental at all.

* [[Hidden Markov model]]

Therefore there may be a psychometrical latent variable, but not a [[psychology|psychological]] psychometric variable.

* [[Partial least squares path modeling]]

* [[Structural equation modeling]]

== Notes ==

{{reflist|group=note}}

== References ==

{{Reflist}}

== Further reading ==

<references />

* {{cite book |first1=Anders |last1=Skrondal |first2=Sophia |last2=Rabe-Hesketh |authorlink2=Sophia Rabe-Hesketh |title=Generalized Latent Variable Modeling |___location= |publisher=Chapman & Hall |year=2004 |isbn=1-58488-000-7 }}

{{DEFAULTSORT:Latent Variable Model}}

[[Category:StatisticalLatent variable models| Latent variable model]]

[[Category:Latent variable models]]

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