Latent variable

In statistics, latent variables (from Latin: present participle of lateo (“lie hidden”), as opposed to observable variables) are variables that are not directly observed but are rather inferred (through a mathematical model) from other variables that are observed (directly measured). Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models. Latent variable models are used in many disciplines, including psychology, demography, economics, engineering, medicine, physics, machine learning/artificial intelligence, bioinformatics, chemometrics, natural language processing, econometrics, management 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. 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 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.

Examples

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 model.[1]

The "Big Five personality traits" have been inferred using factor analysis.
extraversion[2]
spatial ability[2]
wisdom “Two of the more predominant means of assessing wisdom include wisdom-related performance and latent variable measures.”[3]
Spearman's g, or the general intelligence factor in psychometrics[4]

Economics

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. 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.

Inferring latent variables

Various techniques allow latent variables to be inferred:

Hidden Markov models
Factor analysis
Principal component analysis
Partial least squares regression
Latent semantic analysis and probabilistic latent semantic analysis
EM algorithms
Metropolis-Hastings algorithm

Bayesian algorithms and methods

Bayesian statistics is often used for inferring latent variables.

Latent Dirichlet Allocation
The Chinese Restaurant Process is often used to provide a prior distribution over assignments of objects to latent categories.
The Indian buffet process is often used to provide a prior distribution over assignments of latent binary features to objects.

References

Tabachnick, B.G.; Fidell, L.S. (2001). Using Multivariate Analysis. Boston: Allyn and Bacon. ISBN 978-0-321-05677-1.
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.
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.
Spearman, C. (1904). ""General Intelligence," Objectively Determined and Measured". The American Journal of Psychology. 15 (2): 201–292. doi:10.2307/1412107. JSTOR 1412107.