# Variance Components Analysis

The Variance Components procedure, for mixed-effects models, estimates the contribution of each random effect to the variance of the dependent variable. This procedure is particularly interesting for analysis of mixed models such as split plot, univariate repeated measures, and random block designs. By calculating variance components, you can determine where to focus attention in order to reduce the variance.

Four different methods are available for estimating the variance components: minimum norm quadratic unbiased estimator (MINQUE), analysis of variance (ANOVA), maximum likelihood (ML), and restricted maximum likelihood (REML). Various specifications are available for the different methods.

Default output for all methods includes variance component estimates. If the ML method or the REML method is used, an asymptotic covariance matrix table is also displayed. Other available output includes an ANOVA table and expected mean squares for the ANOVA method and an iteration history for the ML and REML methods. The Variance Components procedure is fully compatible with the GLM Univariate procedure.

WLS Weight allows you to specify a variable used to give observations different weights for a weighted analysis, perhaps to compensate for variations in precision of measurement.

**Example.** At an agriculture school, weight gains for pigs
in six different litters are measured after one month. The litter
variable is a random factor with six levels. (The six litters studied
are a random sample from a large population of pig litters.) The investigator
finds out that the variance in weight gain is attributable to the
difference in litters much more than to the difference in pigs within
a litter.

Variance Components Data Considerations

**Data.** The dependent variable is quantitative. Factors are
categorical. They can have numeric values or string values of up to
eight bytes. At least one of the factors must be random. That is,
the levels of the factor must be a random sample of possible levels.
Covariates are quantitative variables that are related to the dependent
variable.

**Assumptions.** All methods assume that model parameters of
a random effect have zero means and finite constant variances and
are mutually uncorrelated. Model parameters from different random
effects are also uncorrelated.

The residual term also has a zero mean and finite constant variance. It is uncorrelated with model parameters of any random effect. Residual terms from different observations are assumed to be uncorrelated.

Based on these assumptions, observations from the same level of a random factor are correlated. This fact distinguishes a variance component model from a general linear model.

ANOVA and MINQUE do not require normality assumptions. They are both robust to moderate departures from the normality assumption.

ML and REML require the model parameter and the residual term to be normally distributed.

**Related procedures.** Use the Explore procedure to examine
the data before doing variance components analysis. For hypothesis
testing, use GLM Univariate, GLM Multivariate, and GLM Repeated Measures.

Obtaining Variance Components Tables

This feature requires SPSS® Statistics Standard Edition or the Advanced Statistics Option.

- From the menus choose:
- Select a dependent variable.
- Select variables for Fixed Factor(s), Random Factor(s), and Covariate(s), as appropriate for your data. For specifying a weight variable, use WLS Weight.

This procedure pastes VARCOMP command syntax.