1. Sphericity

2. More on sphericity With our previous between groups Anova we had the assumption of homogeneity of variance With our previous between groups Anova we had the assumption of homogeneity of variance With repeated measures design we still have this assumption albeit in a different form With repeated measures design we still have this assumption albeit in a different form

3. More on sphericity Homogeneity of variance assumption means we want to see similar variability from group to group Homogeneity of variance assumption means we want to see similar variability from group to group In other words we don’t want more or less variability in one group’s scores relative to another In other words we don’t want more or less variability in one group’s scores relative to another

4. More on sphericity We are still worried about this problem, except now it applies to difference scores between pairs of the treatment (repeated measures) under consideration We are still worried about this problem, except now it applies to difference scores between pairs of the treatment (repeated measures) under consideration In other words the variances of the differences scores created by comparing any two treatments should be roughly the same for all pairs creating difference scores In other words the variances of the differences scores created by comparing any two treatments should be roughly the same for all pairs creating difference scores

5. More on sphericity Raw data (top) Raw data (top) Difference scores (bottom) Difference scores (bottom) We could then calculate variances for each of these sets of differences We could then calculate variances for each of these sets of differences The sphericity assumption is that the all these variances of the differences are equal (in the population sampled). The sphericity assumption is that the all these variances of the differences are equal (in the population sampled). In practice, we'd expect the observed sample variances of the differences to be similar if the sphericity assumption was met. In practice, we'd expect the observed sample variances of the differences to be similar if the sphericity assumption was met. A 1 A 2 A 3 A 4 Participant 1 89124 Participant 2 611163 Participant 3 98125 etc.... A 1 -A 2 A 1 -A 3 A 1 -A 4 etc. Participant 1 -4+4 Participant 2 -5-10+3 Participant 3 +1-3+4 etc.... Var 1-2 Var 1-3 Var 1-4

6. Technical side We can check sphericity assumption using the covariance matrix We can check sphericity assumption using the covariance matrix –A1-A4 equals time1- time4 or what have you Variances for individual treatments in red Variances for individual treatments in red Samples:A 1 A 2 A 3 A 4 A1A1 105 15 A2A2 5201520 A3A3 10153025 A4A4 15202540

7. Compound symmetry is the case where all variances are equal, and all covariances are equal Compound symmetry is the case where all variances are equal, and all covariances are equal Not bloody likely Not bloody likely Samples:A 1 A 2 A 3 A 4 A1A1 105 15 A2A2 5201520 A3A3 10153025 A4A4 15202540

8. Sphericity is a relaxed form of the assumption of compound symmetry Sphericity is a relaxed form of the assumption of compound symmetry It is that the sum of any two treatments’ variances minus their covariance equals a constant It is that the sum of any two treatments’ variances minus their covariance equals a constant The constant is equal to the variance of their difference scores The constant is equal to the variance of their difference scores

9. = 10 + 20 - 2(5) = 20 = 10 + 20 - 2(5) = 20 = 10 + 30 - 2(10) = 20 = 10 + 30 - 2(10) = 20 = 10 + 40 - 2(15) = 20 = 10 + 40 - 2(15) = 20 = 20 + 30 - 2(15) = 20 = 20 + 30 - 2(15) = 20 = 20 + 40 - 2(20) = 20 = 20 + 40 - 2(20) = 20 = 30 + 40 - 2(25) = 20 = 30 + 40 - 2(25) = 20 Samples:A 1 A 2 A 3 A 4 A1A1 105 15 A2A2 5201520 A3A3 10153025 A4A4 15202540

10. SPSS You can produce the variance/ covariance matrix in SPSS repeated measures You can produce the variance/ covariance matrix in SPSS repeated measures

11. Output from our previous stress data