Random effects model

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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. The fixed effects model is a special case.

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.

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. Then it can be shown that

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

and

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

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

References

Acknowledgement and Attribution Regarding Sources of Content

Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .

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Category: Statistics