Sums of Squares

1. Sums of Squares

2. Sums of squares • Besides the unweighted means solution, sums of squares can be calculated in various ways depending on the situation and desired result of the analysis • Different methods correct for overlap of main effects in different ways • For this reason they do not always generate the same set of sums of squares values for a particular data set • With no rationale to select among different sets of estimates, they should all be seen as equally correct • Most analysis options for unbalanced designs in contemporary computer programs for factorial ANOVA are based on regression methods

3. Sums of squares • Although not exhaustive, this typology by Overall and Spiegel (1969) is helpful: • Method 1 estimates effect sums of squares controlling for all other effects—these sums of squares may be labeled “Type III” or “unique” in program output • Method 2 adjusts sums of squares for the main effects for overlap with each other—these sums of squares may be labeled “Type II” or “classical experimental” in program output • Method 3 does not remove shared variance from the sums of squares of one main effect (e.g., A) but adjusts the sums of squares of the other main effect for overlap with the first (e.g., B adjusted for A)—these sums of squares may be labeled “Type I,” “sequential,” or “hierarchical” in program output

4. Sums of squares • Some suggestions: • Method 1 does not generally give greater weight to cells with more observations, so this method may be optimal when unequal cell sizes result from random data loss from a few cells • Method 2 and Method 3 may be a better choice for nonexperimental designs where unequal cell sizes reflect unequal group sizes in the population—this is because they permit the actual cell sizes to contribute to the analysis but with different priorities given to certain main effects

5. Sums of squares • So again in SPSS/SAS etc: • Type I (hierarchical decomposition). • Each term is adjusted only for the terms that precede it • If the design is balanced (if there are equal ns in each cell and there are no missing cells) then the sums of squares in the model add up to the total sums of squares. • Type II • Calculates the sums of squares of an effect in the model adjusted for all other "appropriate" effects where an appropriate effect is an effect that does not contain the effect being examined. For example, in a three way ANOVA, A x B x C, the main effect of A would be adjusted by the B and C main effects and by the B by C interaction. • Type III • Calculates the sum of squares of an effect adjusted for all other effects regardless of order • It is the unweighted means approach for unequal cell sizes • Type IV • Designed for the situation in which there are missing cells. • Unfortunately much research shows it doesn’t do its job well

6. Sums of squares • It’s important to check what your stat program is doing • SPSS default is type III, which takes into account unbalanced designs • S-plus, type I • Note that Type III is probably the preferred sums of squares type if you have unequal ns. • Hypotheses tested with TYPE III sums of squares are hypotheses about the unweighted means. So technically you should report unweighted means rather than weighted means when you have an unequal n design.