ANOVA Method

The ANOVA method first computes sums of squares and expected mean squares for all effects following the general linear model approach. Then a system of linear equations is established by equating the sums of squares of the random effects to their expected mean squares. The variables in the equations are the variance components and the residual variance. Any solution, if one exists, to this system of linear equations constitutes a set of estimates for the variance components.

This method is computationally less laborious and the estimates are statistically unbiased. However, negative variance estimates can happen and the variance-covariance matrix of the estimates is difficult to obtain even asymptotically.

The Variance Components procedure offers two types of sums of squares: Type I and Type III. Type III is the most commonly used and is the default.

Type I. This method is also known as the hierarchical decomposition of the sum-of-squares method. Each term is adjusted for only the term that precedes it in the model. The Type I sum-of-squares method is commonly used for:

A balanced ANOVA model in which any main effects are specified before any first-order interaction effects, any first-order interaction effects are specified before any second-order interaction effects, and so on.
A polynomial regression model in which any lower-order terms are specified before any higher-order terms.
A purely nested model in which the first-specified effect is nested within the second-specified effect, the second-specified effect is nested within the third, and so on. (This form of nesting can be specified only by using syntax.)

Type III. This method, the default, calculates the sums of squares of an effect in the design as the sums of squares adjusted for any other effects that do not contain it and orthogonal to any effects (if any) that contain it. The Type III sums of squares have one major advantage in that they are invariant with respect to the cell frequencies as long as the general form of estimability remains constant. Therefore, this type is often considered useful for an unbalanced model with no missing cells. In a factorial design with no missing cells, this method is equivalent to the Yates' weighted-squares-of-means technique. The Type III sum-of-squares method is commonly used for:

Any models listed in Type I.
Any balanced or unbalanced model with no empty cells.