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.
Contrast this to the biostatistics definitions,[1][2][3][4][5] as biostatisticians use "fixed" and "random" effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables).
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.
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.
pupils of the same age are chosen randomly at each selected school.
Their scores on a standard aptitude test are ascertained.
is the average test score for the entire population.
is the school-specific random effect: it measures the difference between the average score at school
and the average score in the entire country.
is the individual-specific random effect, i.e., it's the deviation of the
The model can be augmented by including additional explanatory variables, which would capture differences in scores among different groups.
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 fixed-effects terms for Sex and Parents' Education.
-th school that are included in the random sample.
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 [citation needed] that and These "expected mean squares" can be used as the basis for estimation of the "variance components"
parameter is also called the intraclass correlation coefficient.
For random effects models the marginal likelihoods are important.
[7] Random effects models used in practice include the Bühlmann model of insurance contracts and the Fay-Herriot model used for small area estimation.