Random effects model

From Wikipedia, the free encyclopedia

It has been suggested that this article or section be merged with random effects estimation. (Discuss)

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. It is an opposite of fixed effects model.

1 Simple example
2 Variance components
3 References
4 See also

[edit] Simple example

Suppose m elementary large schools are chosen randomly from among millions in a large country. Then n pupils are chosen randomly from among those at each such school. Their scores on a standard aptitude test are ascertained. Let Y_ij be the score of the jth pupil at the ith 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 ith school from the average in the whole population, and W_ij is the deviation of the jth pupil's score from the average score at the ith school.

[edit] Variance components

The variance of Y_ij is the sum of the variances τ² and σ² of U_i and W_ij respectively.

Let

$\overline{Y}_{i\bullet} = \frac{1}{n}\sum_{j=1}^n Y_{ij}$

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

$\overline{Y}_{\bullet\bullet} = \frac{1}{mn}\sum_{i=1}^m\sum_{j=1}^n Y_{ij}$

be the "grand average".

Let

$SSW = \sum_{i=1}^m\sum_{j=1}^n (Y_{ij} - \overline{Y}_{i\bullet})^2 \,$

$SSB = n\sum_{i=1}^m (\overline{Y}_{i\bullet} - \overline{Y}_{\bullet\bullet})^2 \,$

be respectively the sum of squares due to differences within groups and the sum of squares due to difference between groups. The it can be shown that

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