1 G Lect 10M Contrasting coefficients: a review ANOVA and Regression software Interactions of categorical predictors Type I, II, and III sums of squares G Multiple Regression Week 10 (Monday)

2 G Lect 10M Contrasting Coefficients Suppose dummy codes are used to represent categories »And group k is the reference »B i -B j contrasts the means of groups i and j The standard error of the contrast can be computed two different ways »By recognizing that the groups i and j are independent, and using the usual se of the mean to compute a t test. »By using the se of the B estimates and the estimated correlation of the two estimates to compute a general contrast.

3 G Lect 10M Numerical Example The approach using the standard errors of the regression weights has to correct for the common reference group (the 12 year olds in the example) which makes the b's correlated.

4 G Lect 10M Two crossed categorical independent variables Suppose subjects can be classified into one of six categories according to a 2x3 crossed design. A main effects ANOVA model attempts to represent the six means with four degrees of freedom: a grand mean, an effect for Factor A and two effects for Factor B. Main effects suggest that the difference between levels of Factor B are consistent in both levels of Factor A. cdefghcdefgh Factor B Factor A

5 G Lect 10M Main effects and Interactions c d e f g h Example of a Main Effect Result c d e f g h cd e f g h Examples of Interaction Results

6 G Lect 10M Six Cells with Dummy Variables Note that the products of the dummy variables allow cells c and d to be fit exactly. This flexibility allows all patterns of six means to be fit perfectly. »The effect of A can be moderated in one level of B. In general, if there are J levels of Factor A and K levels of Factor B, then there will be (J-1)(K-1) interaction terms in the model.

7 G Lect 10M Weighted and Unweighted Means When the cell n's are different, the marginal means are confounded with the cell means. Example of depression among PR adolescents (age group by gender)

8 G Lect 10M Type I Sums of Squares Suppose we have factor A, B and A*B When the numbers are representative of a population, then a hierarchical regression approach is appropriate »Sets for A, B and A*B are entered »The first is entered ignoring the others »The second set is adjusted for the first, but ignores the later sets »The last set is adjusted for all before it.

9 G Lect 10M Type II Sums of Squares When the numbers are representative of a population, and when there is strong belief that interactions are not important, Type II Sums of squares might be right. »All sets, A, B and A*B are considered »A is adjusted for B, but not for A*B »B is adjusted for A, but not for A*B »A*B is adjusted for both A and B Type II SS is not much used in practice.

10 G Lect 10M Type III Sums of Squares When the cell means are constructed by design, and are not representative, Type III SS are appropriate »Conceptually, the Type III SS contrast the marginal means in the unweighted mean table »In practice, this is accomplished by fitting a specific regression model Unweighted effect codes are used A is adjusted for B and A*B B is adjusted for A and A*B A*B is adjusted for A and B Type III is the default in ANOVA programs