latent variable model

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

{{Multiple issues|

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A latent variable model is a statistical model that relates a set of observable variables (also called manifest variables or indicators){{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}} to a set of latent variables. Latent variable models are applied across a wide range of fields such as biology, computer science, and social science.{{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}} 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, that are presumed to cause the survey question responses),{{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 }} and natural language processing (e.g., a topic model summarizing a corpus of texts with a number of "topics").{{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}}

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 models can be grouped according to whether the manifest and latent variables are categorical or continuous:{{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 |page=145 |isbn=1-58488-295-6 }}

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 and latent trait analysis{{refn|group=note|name=LTAandIRT| The terms "latent trait analysis" and "item response theory" are often used interchangeably.{{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}}}} the latent variables are treated as continuous normally distributed variables, and in latent profile analysis and latent class analysis as from a multinomial distribution.{{cite book |last=Everitt |first=BS |title=An Introduction to Latent Variables Models |year=1984 |publisher=Chapman & Hall |isbn=0-412-25310-0 }} 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.