Random effects model
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Background 

In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy. A random effects model is a special case of a mixed model.
Contrast this to the biostatistics definitions,^{[1]}^{[2]}^{[3]}^{[4]}^{[5]} as biostatisticians use "fixed" and "random" effects to respectively refer to the populationaverage and subjectspecific effects (and where the latter are generally assumed to be unknown, latent variables).
Qualitative description
Random effect models assist in controlling for unobserved heterogeneity when the heterogeneity is constant over time and not correlated with independent variables. This constant can be removed from longitudinal data through differencing, since taking a first difference will remove any time invariant components of the model.^{[6]}
Two common assumptions can be made about the individual specific effect: the random effects assumption and the fixed effects assumption. The random effects assumption is that the individual unobserved heterogeneity is uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables.^{[6]}
If the random effects assumption holds, the random effects estimator is more efficient than the fixed effects model.
Simple example
Suppose m large elementary schools are chosen randomly from among thousands in a large country. Suppose also that n pupils of the same age are chosen randomly at each selected school. Their scores on a standard aptitude test are ascertained. Let Y_{ij} be the score of the jth pupil at the ith school. A simple way to model this variable is
 [math]\displaystyle{ Y_{ij} = \mu + U_i + W_{ij} + \epsilon_{ij},\, }[/math]
where μ is the average test score for the entire population. In this model U_{i} is the schoolspecific random effect: it measures the difference between the average score at school i and the average score in the entire country. The term W_{ij} is the individualspecific random effect, i.e., it's the deviation of the jth pupil's score from the average for the ith school.
The model can be augmented by including additional explanatory variables, which would capture differences in scores among different groups. For example:
 [math]\displaystyle{ Y_{ij} = \mu + \beta_1 \mathrm{Sex}_{ij} + \beta_2 \mathrm{ParentsEduc}_{ij} + U_i + W_{ij}+ \epsilon_{ij},\, }[/math]
where Sex_{ij} is a binary dummy variable and ParentsEduc_{ij} records, say, the average education level of a child's parents. This is a mixed model, not a purely random effects model, as it introduces fixedeffects terms for Sex and Parents' Education.
Variance components
The variance of Y_{ij} is the sum of the variances τ^{2} and σ^{2} of U_{i} and W_{ij} respectively.
Let
 [math]\displaystyle{ \overline{Y}_{i\bullet} = \frac{1}{n}\sum_{j=1}^n Y_{ij} }[/math]
be the average, not of all scores at the ith school, but of those at the ith school that are included in the random sample. Let
 [math]\displaystyle{ \overline{Y}_{\bullet\bullet} = \frac{1}{mn}\sum_{i=1}^m\sum_{j=1}^n Y_{ij} }[/math]
be the grand average.
Let
 [math]\displaystyle{ SSW = \sum_{i=1}^m\sum_{j=1}^n (Y_{ij}  \overline{Y}_{i\bullet})^2 \, }[/math]
 [math]\displaystyle{ SSB = n\sum_{i=1}^m (\overline{Y}_{i\bullet}  \overline{Y}_{\bullet\bullet})^2 \, }[/math]
be respectively the sum of squares due to differences within groups and the sum of squares due to difference between groups. Then it can be shown^{[citation needed]} that
 [math]\displaystyle{ \frac{1}{m(n  1)}E(SSW) = \sigma^2 }[/math]
and
 [math]\displaystyle{ \frac{1}{(m  1)n}E(SSB) = \frac{\sigma^2}{n} + \tau^2. }[/math]
These "expected mean squares" can be used as the basis for estimation of the "variance components" σ^{2} and τ^{2}.
The σ^{2} parameter is also called the intraclass correlation coefficient.
Applications
Random effects models used in practice include the Bühlmann model of insurance contracts and the FayHerriot model used for small area estimation.
See also
 Bühlmann model
 Hierarchical linear modeling
 Fixed effects
 MINQUE
 Covariance estimation
 Conditional variance
Further reading
 Baltagi, Badi H. (2008). Econometric Analysis of Panel Data (4th ed.). New York, NY: Wiley. pp. 17–22. ISBN 9780470518861.
 Hsiao, Cheng (2003). Analysis of Panel Data (2nd ed.). New York, NY: Cambridge University Press. pp. 73–92. ISBN 0521522714. https://archive.org/details/analysispaneldat00chsi.
 Wooldridge, Jeffrey M. (2002). Econometric Analysis of Cross Section and Panel Data. Cambridge, MA: MIT Press. pp. 257–265. ISBN 0262232197. https://archive.org/details/econometricanaly0000wool.
 Gomes, Dylan G.E. (20 January 2022). "Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixedeffects model?". PeerJ 10: e12794. doi:10.7717/peerj.12794.
References
 ↑ Diggle, Peter J.; Heagerty, Patrick; Liang, KungYee; Zeger, Scott L. (2002). Analysis of Longitudinal Data (2nd ed.). Oxford University Press. pp. 169–171. ISBN 0198524846. https://archive.org/details/analysislongitud00digg_730.
 ↑ Fitzmaurice, Garrett M.; Laird, Nan M.; Ware, James H. (2004). Applied Longitudinal Analysis. Hoboken: John Wiley & Sons. pp. 326–328. ISBN 0471214876.
 ↑ Laird, Nan M.; Ware, James H. (1982). "RandomEffects Models for Longitudinal Data". Biometrics 38 (4): 963–974. doi:10.2307/2529876.
 ↑ Gardiner, Joseph C.; Luo, Zhehui; Roman, Lee Anne (2009). "Fixed effects, random effects and GEE: What are the differences?". Statistics in Medicine 28 (2): 221–239. doi:10.1002/sim.3478. PMID 19012297.
 ↑ Gomes, Dylan G.E. (20 January 2022). "Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixedeffects model?". PeerJ 10: e12794. doi:10.7717/peerj.12794.
 ↑ ^{6.0} ^{6.1} Wooldridge, Jeffrey (2010). Econometric analysis of cross section and panel data (2nd ed.). Cambridge, Mass.: MIT Press. pp. 252. ISBN 9780262232586. OCLC 627701062.
External links
Original source: https://en.wikipedia.org/wiki/Random effects model.
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