Latent and observable variables

<h2 id="examples">Examples</h2>

<h3>Psychology</h3>
<p>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 <a href="/facts/Factor_analysis/LT6B9I7D">factor model</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<ul><li>The "<a href="/facts/Big_Five_personality_traits/tQfRewjk">Big Five personality traits</a>" have been inferred using <a href="/facts/Factor_analysis/LT6B9I7D">factor analysis</a>.</li>
<li>extraversion<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li>
<li>spatial ability<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li>
<li>wisdom “Two of the more predominant means of assessing wisdom include wisdom-related performance and latent variable measures.”<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li>
<li><a href="/facts/Spearman%2527s_g/1Bp8fFHr">Spearman's g</a>, or the <a href="/facts/G_factor_(psychometrics)/1Bp8fFHr">general intelligence factor</a> in <a href="/facts/Psychometrics/56X26MDI">psychometrics</a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li></ul>
<h3>Economics</h3>

<p>Examples of latent variables from the field of <a href="/facts/Economics/HYctA2Dn">economics</a> include <a href="/facts/Quality_of_life/37ljmgAH">quality of life</a>, business confidence, morale, happiness and conservatism: these are all variables which cannot be measured directly. But 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.
</p>
<h3>Medicine</h3>

<p>Latent-variable methodology is used in many branches of <a href="/facts/Medicine/2fnp90Je">medicine</a>. A class of problems that naturally lend themselves to latent variables approaches are <a href="/facts/Longitudinal_studies/snyWEsXK">longitudinal studies</a> 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 <a href="/facts/Nonlinear_mixed-effects_model/o7DLN38n">disease progression modeling</a> and <a href="/facts/Nonlinear_mixed-effects_model/o7DLN38n">modeling of growth</a> (see box).
</p>
<h2 id="inferring-latent-variables">Inferring latent variables</h2>
<p>There exists a range of different model classes and methodology that make use of latent variables and allow inference in the presence of latent variables.  Models include:
</p>
<ul><li><a href="/facts/Mixed_model/PovxPrAy">linear mixed-effects models</a> and <a href="/facts/Nonlinear_mixed-effects_model/o7DLN38n">nonlinear mixed-effects models</a></li>
<li><a href="/facts/Hidden_Markov_model/ur1zTAhP">Hidden Markov models</a></li>
<li><a href="/facts/Factor_analysis/LT6B9I7D">Factor analysis</a></li>
<li><a href="/facts/Item_response_theory/4WmQgExu">Item response theory</a></li></ul>
<p>Analysis and inference methods include:
</p>
<ul><li><a href="/facts/Principal_component_analysis/pCbVTqdR">Principal component analysis</a></li>
<li>Instrumented principal component analysis<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li>
<li><a href="/facts/Partial_least_squares_regression/jYIcInPw">Partial least squares regression</a></li>
<li><a href="/facts/Latent_semantic_analysis/IugrMyMl">Latent semantic analysis</a> and <a href="/facts/Probabilistic_latent_semantic_analysis/RehsIdWb">probabilistic latent semantic analysis</a></li>
<li><a href="/facts/EM_algorithm/32P2bdBc">EM algorithms</a></li>
<li><a href="/facts/Metropolis%25E2%2580%2593Hastings_algorithm/8K7bz3LN">Metropolis–Hastings algorithm</a></li></ul>
<h3>Bayesian algorithms and methods</h3>

<p><a href="/facts/Bayesian_statistics/9w7b1Bw4">Bayesian statistics</a> is often used for inferring latent variables.
</p>
<ul><li><a href="/facts/Latent_Dirichlet_allocation/BUWlwKWt">Latent Dirichlet allocation</a></li>
<li>The <a href="/facts/Chinese_restaurant_process/FqmwpAJ0">Chinese restaurant process</a> is often used to provide a prior distribution over assignments of objects to latent categories.</li>
<li>The <a href="/facts/Indian_buffet_process/ScCeGqJK">Indian buffet process</a> is often used to provide  a prior distribution over assignments of latent binary features to objects.</li></ul>
<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Confounding/G4TVx1dt">Confounding</a></li>
<li><a href="/facts/Dependent_and_independent_variables/dINfzIF2">Dependent and independent variables</a></li>
<li><a href="/facts/Errors-in-variables_models/PRDddry0">Errors-in-variables models</a></li>
<li><a href="/facts/Evidence_lower_bound/cANNQv7y">Evidence lower bound</a></li>
<li><a href="/facts/Factor_analysis/LT6B9I7D">Factor analysis</a></li>
<li><a href="/facts/Intervening_variable/UrpjgGVc">Intervening variable</a></li>
<li><a href="/facts/Latent_variable_model/FCcgtsmc">Latent variable model</a></li>
<li><a href="/facts/Item_response_theory/4WmQgExu">Item response theory</a></li>
<li><a href="/facts/Partial_least_squares_path_modeling/tvJyY2P1">Partial least squares path modeling</a></li>
<li><a href="/facts/Partial_least_squares_regression/jYIcInPw">Partial least squares regression</a></li>
<li><a href="/facts/Proxy_(statistics)/W5V2eqaT">Proxy (statistics)</a></li>
<li><a href="/facts/Rasch_model/8NzYrqmR">Rasch model</a></li>
<li><a href="/facts/Structural_equation_modeling/g5nWG6zy">Structural equation modeling</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Jan_Kmenta/RKLurjPH">Kmenta, Jan</a> (1986). <a href="https://books.google.com/books?id=Bxq7AAAAIAAJ&pg=PA581">"Latent Variables"</a>. <a href="https://archive.org/details/elementsofeconom0003kmen/page/581"><i>Elements of Econometrics</i></a> (Second ed.). New York: Macmillan. pp. <a href="https://archive.org/details/elementsofeconom0003kmen/page/581">581–587</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-02-365070-3.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Bacon, Francis. "APHORISMS—BOOK II: ON THE INTERPRETATION OF NATURE, OR THE REIGN OF MAN". Novum Organum. <a href="https://www.gutenberg.org/files/45988/45988-h/45988-h.htm" target="_blank">https://www.gutenberg.org/files/45988/45988-h/45988-h.htm</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Tabachnick, B.G.; Fidell, L.S. (2001). Using Multivariate Analysis. Boston: Allyn and Bacon. ISBN 978-0-321-05677-1.[page needed] <a href="978-0-321-05677-1" target="_blank">978-0-321-05677-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Borsboom, D.; Mellenbergh, G.J.; van Heerden, J. (2003). "The Theoretical Status of Latent Variables" (PDF). Psychological Review. 110 (2): 203–219. CiteSeerX 10.1.1.134.9704. doi:10.1037/0033-295X.110.2.203. PMID 12747522. Archived from the original (PDF) on 2013-01-20. Retrieved 2008-04-08. <a href="/wiki/Gideon_J._Mellenbergh" target="_blank">/wiki/Gideon_J._Mellenbergh</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Borsboom, D.; Mellenbergh, G.J.; van Heerden, J. (2003). "The Theoretical Status of Latent Variables" (PDF). Psychological Review. 110 (2): 203–219. CiteSeerX 10.1.1.134.9704. doi:10.1037/0033-295X.110.2.203. PMID 12747522. Archived from the original (PDF) on 2013-01-20. Retrieved 2008-04-08. <a href="/wiki/Gideon_J._Mellenbergh" target="_blank">/wiki/Gideon_J._Mellenbergh</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Greene, Jeffrey A.; Brown, Scott C. (2009). "The Wisdom Development Scale: Further Validity Investigations". International Journal of Aging and Human Development. 68 (4): 289–320 (at p. 291). doi:10.2190/AG.68.4.b. PMID 19711618. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Spearman, C. (1904). ""General Intelligence," Objectively Determined and Measured". The American Journal of Psychology. 15 (2): 201–292. doi:10.2307/1412107. JSTOR 1412107. <a href="/wiki/Charles_Spearman" target="_blank">/wiki/Charles_Spearman</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Kelly, Bryan T. and Pruitt, Seth and Su, Yinan, Instrumented Principal Component Analysis (December 17, 2020). Available at SSRN: https://ssrn.com/abstract=2983919 or http://dx.doi.org/10.2139/ssrn.2983919 <a href="https://ssrn.com/abstract=2983919" target="_blank">https://ssrn.com/abstract=2983919</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Latent and observable variables open-in-new

Latent and observable variables