## Stats: ANOVA/ANCOVA: Type I, II, III SS

The different types of sums of squares then arise depending on the stage of model reduction at which they are carried out. In particular:

• Type I (“sequential”):
• SS(A) for factor A.
• SS(B | A) for factor B.
• SS(AB | B, A) for interaction AB.
• This tests the main effect of factor A, followed by the main effect of factor B after the main effect of A, followed by the interaction effect AB after the main effects.
• Because of the sequential nature and the fact that the two main factors are tested in a particular order, this type of sums of squares will give different results for unbalanced data depending on which main effect is considered first.
• For unbalanced data, this approach tests for a difference in the weighted marginal means. In practical terms, this means that the results are dependent on the realized sample sizes, namely the proportions in the particular data set. In other words, it is testing the first factor without controlling for the other factor .
• Note that this is often not the hypothesis that is of interest when dealing with unbalanced data.
• Type II:
• SS(A | B) for factor A.
• SS(B | A) for factor B.
• This type tests for each main effect after the other main effect.
• Note that no significant interaction is assumed (in other words, you should test for interaction first (SS(AB | A, B)) and only if AB is not significant, continue with the analysis for main effects).
• If there is indeed no interaction, then type II is statistically more powerful than type III (see Langsrud [3] for further details).
• Computationally, this is equivalent to running a type I analysis with different orders of the factors, and taking the appropriate output (the second, where one main effect is run after the other, in the example above).
• Type III:
• SS(A | B, AB) for factor A.
• SS(B | A, AB) for factor B.
• This type tests for the presence of a main effect after the other main effect and interaction. This approach is therefore valid in the presence of significant interactions.
• However, it is often not interesting to interpret a main effect if interactions are present (generally speaking, if a significant interaction is present, the main effects should not be further analyzed).
• If the interactions are not significant, type II gives a more powerful test.

When data is balanced, the factors are orthogonal, and types I, II and III all give the same results.

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