1 G Lect 14b G Lecture 14b Within subjects contrasts Types of Within-subject contrasts Individual vs pooled contrasts Example Between subject effects in repeated measures design Mapping the repeated measures contrasts on to the mixed model ANOVA

2 G Lect 14b Within subjects contrasts When we compute a paired t-test with repeated measures data, we are calculating a planned contrast on the within-subjects design. ► The average and the difference completely represent the paired data lists. ► The difference score completely represents the time (occasion) effect When we have more than two times (T>2), we can consider a broader class of contrasts. These can be used to partition the (T-1) degrees of freedom for occasion just as we have partitioned the between subjects degrees of freedom in factorial ANOVA

3 G Lect 14b Types of within-subject contrasts FACTORIAL DESIGN: Suppose there are four measurements reflecting a factorial within-subjects design: A 1 B 1, A 1 B 2, A 2 B 1, A 2 B 2. There would be (4-1)=3 degrees of freedom and they would be partitioned into main effects for A and B and an AB interaction. These three degrees of freedom can be thought of as three contrasts of the four ordered conditions: A 1 B 1, A 1 B 2, A 2 B 1, A 2 B 2 ► ( ) A1 vs A2 ► ( ) B1 vs B2 ► ( ) AB interaction

4 G Lect 14b Types of within-subject contrasts TREND ANALYSIS: Suppose there are four measurements reflecting four equally spaced measurements over time. The 3 occasion degrees of freedom can be partitioned into a linear trend, a quadratic curve form and a cubic curve form. ► ( ) linear trend ► ( ) quadratic trend ► ( ) cubic trend ► E.g. for data points ( ) Linear: 8; Quadratic 2; Cubic

5 G Lect 14b Individual vs pooled contrasts Individual contrasts could be analyzed just as we have analyzed the difference scores in paired t tests. ► Use COMPUTE statement in SPSS to compute the contrast for each subject and then check to see if the average of the contrasts is zero. With N subjects, each test would be on N-1 degrees of freedom If we are willing to consider the ANOVA Mixed Model as correct, we can pool the 3 degrees of freedom and test an overall occasion effect. In this case the degrees of freedom for the error term are 3(N-1), and the concerns about the sphericity assumption holds.

6 G Lect 14b Example We previously considered an example of four time points measured on 15 subjects: If we computed separate contrasts we would have found the following: Source df c MS C df e MS E F(1,14) linear quadr cubic Note that the average of the MS C equal the overall MS C shown in the mixed model analysis. The average of the MS E is the error term in the mixed model. This works because contrasts were scaled to length 1 in SPSS.

7 G Lect 14b Between subject effects in repeated measures design Suppose we have two or more groups of subjects that have been measured T times. If we wanted to test group effects at any given time, t, we would have the usual one way ANOVA model Following our discussion of contrasts, we know we could transform the sequence of measures over time, into a set of specific contrasts. If we call the k th contrast then we could compute the one way ANOVA on this new variable. ► This would allow us to determine if there are group differences in the contrasts, such as slope, quadratic effects. ► The degrees of freedom for error would be (N-1)g, where g is the number of groups.

8 G Lect 14b Full accounting for repeated measures effects when comparing between-subject groups. If there are T repeated measures, then we can construct T-1 contrasts that fully represent the original data when considered with the average of the T measures. SPSS and other programs will construct this set of contrasts when instructed to do so. Orthogonal contrasts, such as polynomial coefficients or factorial design contrasts, split the information about the repeated measures into distinct partitions. Consider the T measures [Y 1ij Y 2ij... Y Tij ] being represented as for each subject i in group j. ► Z 0ij is the scaled (length 1) average of the Y values ► Z 1 to Z (T-1) are the (T-1) occasion contrasts. We carry out the ANOVA on

9 G Lect 14b Mixed Model ANOVA Model The mixed model ANOVA provides a framework for pooling subject-time error terms into a global mean square term. The structural model for the between groups repeated measures analysis is where µ is the overall mean  ij is the effect of the i th subject in group j  t is the effect of the t th time point  j is the effect of the j th group  tj is the group by time interaction  tij is the time by subject interaction  tij is the residual term

10 G Lect 14b Mapping the repeated measures contrasts on to the mixed model ANOVA When we compute the one way anova on each individual contrast term in we get answers to both the contrast-based approach and the mixed model approach to repeated measures. The overall average of gives information about ► the grand mean, µ ► and the overall time effects,  t The comparison of group means on gives information about ► the average group effects,  j ► and the group by time effects,  tj The within group variance gives information about ► The overall subject effects (averaged across time),  ij ► and the residual variances,  tij +  tij