Sums of Squares - Interpreting Type I, II, III Hypotheses

Type III sums of squares are most easily interpreted. Remember that Type III sums of squares are tests of effects after controlling for all other effects. For example, after finding a statistically significant Type III effect for factor A, you can say that after controlling for all other effects (factors) in the experimental design, there is a unique significant effect for factor A. In terms of marginal means, you can simply look at the (unweighted) marginal means for factor A as reported by ANOVA/MANOVA or GLM and interpret the effect accordingly. Probably, in 99% of all ANOVA applications this is the type of test that interests the researcher.

Significant effects based on Type I or Type II sums of squares cannot be interpreted as easily. One best considers such results in the context of stepwise multiple regression (e.g., as discussed in Multiple Regression). If, using Type I sums of squares, a main effect B is significant (after A was already entered into the model, but before the AxB interaction was added), you should conclude that there is a significant main effect for factor B, provided that there is no interaction between A and B. (Of course, if the Type III test for factor B is also significant, then you can conclude that there is a significant unique main effect for B, after controlling for all other factors and their interactions.)

In terms of marginal means, Type I and Type II hypotheses usually have no simple interpretation. That is to say, you cannot interpret significant effects by looking simply at marginal means. Rather, the reported p-values pertain to complex hypotheses that combine means and sample sizes. For example, the Type II hypothesis for factor A in the simple 2 x 2 example shown in the Unbalanced and Balanced Designs topic would be (see Woodward, Bonett, and Brecht, 1990, p. 219):

Sj(nij -nji2/n.j)*uij = Si'¹j[Sj(nij*ni'j/n.j*ui'j)]

Here the nij's are the cell n's, the uij's are the cell means, and the dots (e.g., n.j) refer to the respective marginal elements (e.g., marginal n's or marginal means). Without going into further detail (for a thorough discussion see Milliken & Johnson, 1984, Chapter 10), it is clear that this is not a simple hypothesis, and in most cases not one that is of particular interest to the researcher. However, there are cases when Type I hypotheses may be of interest.