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What is the difference between Type I SS and Type III SS?

The Type III SS will produce the same SS as a Type I SS for a data set in which the missing data are replaced by the leastsquares estimates of the values. The Type III SS correspond to Yates' weighted squares of means analysis.

md.psych.bio.uni-goettingen.de - Anova – Type I/II/III SS explained

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It’s from http://goanna.cs.rmit.edu.au/~fscholer/anova.php This link has gone, alternatively

ANOVA (and R) - The ANOVA Controversy

ANOVA is a statistical process for analysing the amount of variance that is contributed to a sample by different factors. It was initially derived by R. A. Fisher in 1925, for the case of balanced data (equal numbers of observations for each level of a factor). When data is unbalanced, there are different ways to calculate the sums of squares for ANOVA. There are at least 3 approaches, commonly called Type I, II and III sums of squares (this notation seems to have been introduced into the statistics world from the SAS package but is now widespread). Which type to use has led to an ongoing controversy in the field of statistics (for an overview, see Heer [2]). However, it essentially comes down to testing different hypotheses about the data. Type I, II and III Sums of Squares Consider a model that includes two factors A and B; there are therefore two main effects, and an interaction, AB. The full model is represented by SS(A, B, AB). Other models are represented similarly: SS(A, B) indicates the model with no interaction, SS(B, AB) indicates the model that does not account for effects from factor A, and so on. The influence of particular factors (including interactions) can be tested by examining the differences between models. For example, to determine the presence of an interaction effect, an F-test of the models SS(A, B, AB) and the no-interaction model SS(A, B) would be carried out. It is convenient to define incremental sums of squares to represent these differences. Let SS(AB | A, B) = SS(A, B, AB) – SS(A, B) SS(A | B, AB) = SS(A, B, AB) – SS(B, AB) SS(B | A, AB) = SS(A, B, AB) – SS(A, AB) SS(A | B) = SS(A, B) – SS(B) SS(B | A) = SS(A, B) – SS(A) The notation shows the incremental differences in sums of squares, for example SS(AB | A, B) represents “the sum of squares for interaction after the main effects”, and SS(A | B) is “the sum of squares for the A main effect after the B main effect and ignoring interactions” [1]. 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, also called “sequential” sum of squares:

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 (for further discussion and a worked example, see Zahn [4]). Note that this is often not the hypothesis that is of interest when dealing with unbalanced data.

Type II:

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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 analysed). If the interactions are not significant, type II gives a more powerful test. NOTE: when data is balanced, the factors are orthogonal, and types I, II and III all give the same results. Summary: Usually the hypothesis of interest is about the significance of one factor while controlling for the level of the other factors. This equates to using type II or III SS. In general, if there is no significant interaction effect, then type II is more powerful, and follows the principle of marginality. If interaction is present, then type II is inappropriate while type III can still be used, but results need to be interpreted with caution (in the presence of interactions, main effects are rarely interpretable).

md.psych.bio.uni-goettingen.de - Anova – Type I/II/III SS explained

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