- Random effects model
In

statistics , a**random effect(s) model**, also called a**variance components model**is a kind ofhierarchical linear model . It assumes that the data describe a hierarchy of different populations whose differences are constrained by the hierarchy. Ineconometrics , random effects models are used in analysis of hierarchical orpanel 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 τ^{2}and σ^{2}of "U"_{"i"}and "W"_{"ij"}respectively.Let

:$overline\{Y\}\_\{iullet\}\; =\; frac\{1\}\{n\}sum\_\{j=1\}^n\; Y\_\{ij\}$

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:$overline\{Y\}\_\{ulletullet\}\; =\; frac\{1\}\{mn\}sum\_\{i=1\}^msum\_\{j=1\}^n\; Y\_\{ij\}$

be the "grand average".

Let

:$SSW\; =\; sum\_\{i=1\}^msum\_\{j=1\}^n\; (Y\_\{ij\}\; -\; overline\{Y\}\_\{iullet\})^2\; ,$

:$SSB\; =\; nsum\_\{i=1\}^m\; (overline\{Y\}\_\{iullet\}\; -\; overline\{Y\}\_\{ulletullet\})^2\; ,$

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 square s" can be used as the basis forestimation of the "variance components" σ^{2}and τ^{2}.**Random effects estimation**The estimation for the

coefficient s inmultiple comparisons model in which the effects of different classes are random can be done viageneralized least squares (GLS). If we assume random effects the error term in the model:$y\_\{it\}=x\_\{it\}eta+alpha\_\{i\}+u\_\{it\},,$

where $y\_\{it\}$ is the

dependent variable , $x\_\{it\}$ is the vector ofregressor s, $eta$ is the vector ofcoefficient s, $alpha\_\{i\}=alpha$ are the random effects, and $u\_\{it\}$ is the error term, then $alpha\_\{i\}$ should have a normal distribution withmean zero and a constant variance.The coefficients can be estimated via

:$widehat\{eta\}=(X\text{'}Omega^\{-1\}\; X)^\{-1\}(X\text{'}Omega^\{-1\}Y),$:$widehat\{Omega\}^\{-1\}=Iota\; otimes\; Sigma,$

where "X" and "Y" are the matrix version of the

regressor andindependent variable , respectively, $Iota$ is theidentity matrix , $Sigma$ is thevariance of $u\_\{it\}$ and $alpha$, and $Omega$ is thevariance-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*]

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