Assessment |
Biopsychology |
Comparative |
Cognitive |
Developmental |
Language |
Individual differences |
Personality |
Philosophy |
Social |

Methods |
Statistics |
Clinical |
Educational |
Industrial |
Professional items |
World psychology |

**Statistics:**
Scientific method ·
Research methods ·
Experimental design ·
Undergraduate statistics courses ·
Statistical tests ·
Game theory ·
Decision theory

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

- $ 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 *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

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

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

- $ \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" 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

This page uses Creative Commons Licensed content from Wikipedia (view authors). |