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Variance components analysis usually applies to a mixed effects model - that is, one in which there are random and fixed effects, differences in either of which might account for variance in the dependent variable. There must be at least one random effects variable. For instance, a researcher might study time-to-promotion for a random sample of fire stations, also looking at hours of training of the firemen. Stations would be a random effect. Training would be a fixed effect. Variance components analysis would reveal if the between-stations random effect accounted for an important or a trivial amount of the variance in time-to-promotion, based on a model which included random-effects variables, fixed-effects variables, covariates, and interactions among them. In general, variance components analysis is often used in split plot, univariate repeated measures, random block, and other mixed effects designs.
Actual hypothesis testing is normally done using linear mixed models (which also analyzes variance components), univariate general linear modeling (GLM), or some other procedure. The variance components procedure is often an adjunct to these procedures, particularly linear mixed modeling. Note that unlike the linear mixed models procedure in SPSS, the variance components procedure estimates only variance components, not model regression coefficients.
Variance components analysis is an adaptation of general linear model (GLM) univariate procedures, with the same models as in the GLM Univariate procedure in SPSS. GLM in SPSS does support analysis of variance associated with random effects but estimates their parameters as if they were fixed, calculating variance components based on expected mean squares. In contrast, the variance components procedure, like the linear mixed models procedure, uses maximum likelihood estimation to estimate these parameters. In fact, the variance components procedure supports four methods of estimation, each of which gives somewhat different estimates: analysis of variance (ANOVA), maximum likelihood (ML), restricted maximum likelihood (REML), and the minimum norm quadratic unbiased estimator (MINQUE) method. The Variance Components procedure in SPSS also contrasts in this way with the linear mixed modeling procedure, which only supports ML and REML estimation of variance components.
In SPSS, select Analyze, General Linear Model, Variance Components. One must have the Advanced Models option installed.
There is a second reason to choose Type I sums of squares, however, and this reason is why Type I is the default in SPSS and SAS. Type I sums of squares, as the figure below illustrates, are additive, whereas Type III are not. Total and error sums of squares, and mean square error, are the same under Type I and Type III, but sums of squares for other model terms are not. The result is that the VARCOMP procedure will produce the same variance components estimate for error under Type I and III, but other random effect component estimates will differ, except for the highest-order term (here, storeid*gender*usecoup). This is because the highest order term in a factorial model is confounded with residual (random) error.
In the example above, storeid is a random factor identifying stores selected at random from a population of stores. The dependent variable is amount spent. The fixed factors are gender and source of coupon used. There is no covariate. As can be seen, the four methods do not differ sharply on the estimate of the unexplained variance component, reflected in the Var(Error) term. In all four methods, the unexplained variance component is the largest component. The highest-order interaction term (here, storeid*gender*usecoup) is confounded with residual (random) error (or one may say residual error is confounded with the highest order interaction). The researcher may be interested, for instance, in whether the variation in estimates of the random effect variable's contribution to the random variation in the dependent variable are large or small compared to the error variance component and random error. In the example above, they are relatively small.
However, in this example the partition of other variance components of the random-effects variable storeid differ considerably by method, and in the case of the ANOVA method even lead to a negative variance component estimate. When estimates contradict each other (ex., differ in sign), a common research strategy is to re-run the analysis using alternative methods, then discount the aberrant estimate.
VARCOMP dependent variable BY factor list [WITH covariate list]
/RANDOM = factor [factor ...]
[/METHOD = {MINQUE({1})**}]
{0}
{ML }
{REML }
{SSTYPE({3}) }
{1}
[/INTERCEPT = {INCLUDE**}]
{EXCLUDE }
[/MISSING = {EXCLUDE**}]
{INCLUDE }
[/REGWGT = varname]
[/CRITERIA = [CONVERGE({1.0E-8**})] [EPS({1.0E-8**})] [ITERATE({50**})]]
{n } {n } {n }
[/PRINT = [EMS] [HISTORY({1**})] [SS]]
{n }
[/OUTFILE = [VAREST] [{COVB}] ('savfile'|'dataset') ]
{CORB}
[/DESIGN = {[INTERCEPT] [effect effect ...]}]
** Default if the subcommand or keyword is omitted.
Example:
VARCOMP Y1 BY B C WITH X1 X2
/RANDOM = C.