Random effects model

In statistics, a random effect(s) model, also called a variance components model is a kind of hierarchical linear model. It assumes that the data describe a hierarchy of different populations whose differences are constrained by the hierarchy. In econometrics, random effects models are used in analysis of hierarchical or panel data when one assumes no fixed effects (i.e. no individual effects). The fixed effects model is a special case of the random effects model.

Simple example

Suppose "m" large elementary schools are chosen randomly from among millions in a large country. Then "n" pupils are chosen randomly at each selected school. Their scores on a standard aptitude test are ascertained. Let "Y"_"ij" be the score of the "j"th pupil at the "i"th school. Then

:$Y\_\{ij\}\; =\; mu\; +\; U\_i\; +\; W\_\{ij\},,$

where μ is the average of all scores in the whole population, "U"_"i" is the deviation of the average of all scores at the "i"th school from the average in the whole population, and "W"_"ij" is the deviation of the "j"th pupil's score from the average score at the "i"th school. It is assumed that $W\_\{ij\}sim\; N(0,sigma^2)$, that is, the deviations are normal with mean zero and variance $sigma^2$, the value of which is unknown.

Variance components

The variance of "Y"_"ij" is the sum of the variances τ² and σ² of "U"_"i" and "W"_"ij" respectively.

Let

:

be the average, not of all scores at the "i"th school, but of those at the "i"th school that are included in the random sample. Let

:

be the "grand average".

Let

:

:

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 that

:$frac\{1\}\{m(n\; -\; 1)\}E(SSW)\; =\; sigma^2$

and

:$frac\{1\}\{n\}E(SSB)\; =\; frac\{sigma^2\}\{n\}\; +\; au^2.$

These "expected mean squares" can be used as the basis for estimation of the "variance components" σ² and τ².

Random effects estimation

The estimation for the coefficients in multiple comparisons model in which the effects of different classes are random can be done via generalized least squares (GLS). If we assume random effects the error term in the model

:

where $y\_\{it\}$ is the dependent variable, $x\_\{it\}$ is the vector of regressors, is the vector of coefficients, $alpha\_\{i\}=alpha$ are the random effects, and $u\_\{it\}$ is the error term, then $alpha\_\{i\}$ should have a normal distribution with mean zero and a constant variance.

The coefficients can be estimated via

::$widehat\{Omega\}^\{-1\}=Iota\; otimes\; Sigma,$

where "X" and "Y" are the matrix version of the regressor and independent variable, respectively, $Iota$ is the identity matrix, $Sigma$ is the variance of $u\_\{it\}$ and $alpha$, and $Omega$ is the variance-covariance matrix.

ee also

*Bühlmann model
*Meta analysis
*Hierarchical linear modeling

References

* [http://www.jr2.ox.ac.uk/bandolier/booth/glossary/random.html Random effect model at Bandolier (Oxford EBM website)]
* [http://teaching.sociology.ul.ie/DCW/confront/node45.html Fixed and random effects models]
* [http://www.ioa.pdx.edu/newsom/mlrclass/ho_randfixd.doc Distinguishing Between Random and Fixed: Variables, Effects, and Coefficients]
* [http://www.pitt.edu/~SUPER1/lecture/lec1171/012.htm How to Conduct a Meta-Analysis: Fixed and Random Effect Models]
* [http://www.uwyo.edu/aadland/classes/econ5350/ch13.pdf ECON 5350 Class Notes: Chapter 13. Panel Data]