- Random effects model
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 datawhen one assumes no fixed effects (i.e. no individual effects). The fixed effects model is a special case of the random effects model.
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
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 , that is, the deviations are normal with mean zero and variance , the value of which is unknown.
The variance of "Y""ij" is the sum of the variances τ2 and σ2 of "U""i" and "W""ij" respectively.
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".
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
expected mean squares" can be used as the basis for estimationof the "variance components" σ2 and τ2.
Random effects estimation
The estimation for the
coefficients in multiple comparisonsmodel 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 is the
dependent variable, is the vector of regressors, is the vector of coefficients, are the random effects, and is the error term, then should have a normal distribution with meanzero and a constant variance.
The coefficients can be estimated via
where "X" and "Y" are the matrix version of the
regressorand independent variable, respectively, is the identity matrix, is the varianceof and , and is the variance-covariance matrix.
Hierarchical linear modeling
* [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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