Latent and observable variables: Difference between revisions

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{{Short description|Variable not directly observed}}

{{For|similar uses|Hidden variable (disambiguation){{!}}Hidden variable}}

In [[statistics]], '''latent variables''' (from [[Latin]]: [[present participle]] of ''{{wikt-lang|la|lateo'',}} “lie{{gloss|lie hidden”hidden}}{{cn|date=April 2025}}) are [[Variable (mathematics)|variables]] that can only be [[Statistical inference|inferred]] indirectly through a [[mathematical model]] from other '''observable variables''' that can be directly [[observation|observed]] or [[measurement|measured]].<ref>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. {{isbn|0-19-920613-9}}</ref> Such ''[[latent variable model]]s'' are used in many disciplines, including [[political science]], [[demography]], [[engineering]], [[medicine]], [[ecology]], [[physics]], [[machine learning]]/[[artificial intelligence]], [[natural language processing]], [[bioinformatics]], [[chemometrics]], [[naturaldemography]], language processing[[economics]], [[management]], [[political science]], [[psychology]] and the [[social sciences]].

Latent variables may correspond to aspects of physical reality. These could in principle be measured, but may not be for practical reasons. Among the earliest expressions of this idea is [[Francis Bacon]]'s [[polemic]] the ''[[Novum Organum]]'', itself a challenge to the more traditional logic expressed in [[Aristotle]]'s [[Organon]]:
{{quote|''But the latent process of which we speak, is far from being obvious to men’s minds, beset as they now are. For we mean not the measures, symptoms, or degrees of any process which can be exhibited in the bodies themselves, but simply a continued process, which, for the most part, escapes the observation of the senses.|''Sir [[Francis Bacon]]'', ''[[Novum Organum]]''<ref name=>{{cite book|author-last=Bacon|author-first=Francis|title=Novum Organum|chapter=APHORISMS—BOOK II: ON THE INTERPRETATION OF NATURE, OR THE REIGN OF MAN|url=https://www.gutenberg.org/files/45988/45988-h/45988-h.htm}}</ref>

}}

Latent variables may correspond to aspects of physical reality. These could in principle be measured, but may not be for practical reasons. Among the earliest definitions arises from Sir [[Francis Bacon]]'s eponymous ''[[Novum Organum]]'', [[Francis Bacon]] In this situation, the term ''hidden variables'' is commonly used, (reflecting the fact that the variables are meaningful, but not observable). Other latent variables correspond to abstract concepts, like categories, behavioral or mental states, or data structures. The terms ''hypothetical variables'' or ''hypothetical constructs'' may be used in these situations.

Latent variables may correspond to aspects of physical reality. These could in principle be measured, but may not be for practical reasons. Among the earliest definitions arises from Sir [[Francis Bacon]]'s eponymous ''[[Novum Organum]]'', [[Francis Bacon]] In this situation, the term ''hidden variables'' is commonly used (reflecting the fact that the variables are meaningful, but not observable). Other latent variables correspond to abstract concepts, like categories, behavioral or mental states, or data structures. The terms ''hypothetical variables'' or ''hypothetical constructs'' may be used in these situations.

The use of latent variables can serve to [[Dimensionality reduction|reduce the dimensionality]] of data. Many observable variables can be aggregated in a model to represent an underlying concept, making it easier to understand the data. In this sense, they serve a function similar to that of scientific theories. At the same time, latent variables link observable "[[sub-symbolic]]" data in the real world to symbolic data in the modeled world.

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==Examples ==

[[File:Estimation of a mean height curve for boys from the Berkeley Growth Study with and without warping.gif|thumb|upright=2|Estimation of a mean height curve (black) for boys from the Berkeley Growth Study with and without warping. The warping is based on latent variables that maps age to a synchronized biological age using a [[nonlinear mixed-effects model]].<ref name="Raket_et_al_2014">{{cite journal |vauthors=Raket LL, Sommer S, Markussen B |year=2014 |title=A nonlinear mixed-effects model for simultaneous smoothing and registration of functional data |journal=Pattern Recognition Letters |volume=38|pages=1-71–7 |doi=10.1016/j.patrec.2013.10.018|bibcode=2014PaReL..38....1R }}</ref>]]

===Psychology===

Latent variables, as created by factor analytic methods, generally represent "shared" variance, or the degree to which variables "move" together. Variables that have no correlation cannot result in a latent construct based on the common [[Factor analysis|factor model]].<ref>{{cite book |last1=Tabachnick |first1=B.G. |last2=Fidell |first2=L.S. |title=Using Multivariate Analysis |publisher=Allyn and Bacon |___location=Boston |year=2001 |isbn=978-0-321-05677-1 }}{{Page needed|date=November 2010}}</ref>

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* extraversion<ref name="status">{{cite journal |last1=Borsboom |first1=D. |author2-link=Gideon J. Mellenbergh |last2=Mellenbergh, G.J. |last3=van Heerden |first3=J. |title=The Theoretical Status of Latent Variables |journal=Psychological Review |volume=110 |issue=2 |pages=203–219 |year=2003 |doi=10.1037/0033-295X.110.2.203 |pmid=12747522 |url=http://rhowell.ba.ttu.edu/BorsboomLatentvars2003.pdf |citeseerx=10.1.1.134.9704 |access-date=2008-04-08 |archive-url=https://web.archive.org/web/20130120044039/http://rhowell.ba.ttu.edu/BorsboomLatentvars2003.pdf |archive-date=2013-01-20 |url-status=dead }}</ref>

* spatial ability<ref name="status"/>

* wisdom: “Two of the more predominant means of assessing wisdom include wisdom-related performance and latent variable measures.”<ref name="wisdom">{{cite journal |last1=Greene |first1=Jeffrey A. |last2=Brown |first2=Scott C. |title=The Wisdom Development Scale: Further Validity Investigations |journal=International Journal of Aging and Human Development |volume=68 |issue=4 |pages=289–320 (at p. 291) |year=2009 |pmid=19711618 |doi=10.2190/AG.68.4.b }}</ref>

* [[Spearman's g]], or the [[g factor (psychometrics)|general intelligence factor]] in [[psychometrics]]<ref>{{Cite journal | last1 = Spearman | first1 = C. | author-link = Charles Spearman| title = "General Intelligence," Objectively Determined and Measured | journal = The American Journal of Psychology | volume = 15 | issue = 2 | pages = 201–292 | doi = 10.2307/1412107 | year = 1904 | jstor = 1412107 }}</ref>

===Economics===

{{unreferenced section|date=January 2024}}

Examples of latent variables from the field of [[economics]] include [[quality of life]], business confidence, morale, happiness and conservatism: these are all variables which cannot be measured directly. ButHowever, by linking these latent variables to other, observable variables, the values of the latent variables can be inferred from measurements of the observable variables. Quality of life is a latent variable which cannot be measured directly, so observable variables are used to infer quality of life. Observable variables to measure quality of life include wealth, employment, environment, physical and mental health, education, recreation and leisure time, and social belonging.

===Medicine===

{{unreferenced section|date=January 2024}}

Latent-variable methodology is used in many branches of [[medicine]]. A class of problems that naturally lend themselves to latent variables approaches are [[longitudinal studies]] where the time scale (e.g. age of participant or time since study baseline) is not synchronized with the trait being studied. For such studies, an unobserved time scale that is synchronized with the trait being studied can be modeled as a transformation of the observed time scale using latent variables. Examples of this include [[Nonlinear_mixed-effects_model#Example: Disease progression modeling|disease progression modeling]] and [[Nonlinear_mixed-effects_model#Example: Growth analysis|modeling of growth]] (see box).

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===Bayesian algorithms and methods===

{{unreferenced section|date=January 2024}}

[[Bayesian statistics]] is often used for inferring latent variables.