# 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.

Sometimes latent variables correspond to aspects of physical reality, which 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 "really there", but hidden). Other times, 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.

One advantage of using latent variables is that they 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 of latent variables

### 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.

## Common methods for inferring latent variables

- 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.

## See also

## 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. - 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.

## Further reading

- Kmenta, Jan (1986). "Latent Variables".
*Elements of Econometrics*(Second ed.). New York: Macmillan. pp. 581–587. ISBN 978-0-02-365070-3.