# 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 Edit

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 *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.

## Variance components Edit

The variance of *Y*_{ij} is the sum of the variances τ^{2} and σ^{2} of *U*_{i} and *W*_{ij} respectively.

Let

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".

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 that

and

These "expected mean squares" can be used as the basis for estimation of the "variance components" σ^{2} and τ^{2}.

## ReferencesEdit

- Random effect model at Bandolier (Oxford EBM website)
- Fixed and random effects models
- Distinguishing Between Random and Fixed: Variables, Effects, and Coefficients
- How to Conduct a Meta-Analysis: Fixed and Random Effect Models

## See alsoEdit

- zh:随机效应模型

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