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Sphericity

Sphericity. More on sphericity. 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. More on sphericity.

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Sphericity

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  1. Sphericity

  2. More on sphericity • 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

  3. More on sphericity • 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

  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 • 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) • Difference scores (bottom) • 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). • In practice, we'd expect the observed sample variances of the differences to be similar if the sphericity assumption was met. Var1-2 Var1-3 Var1-4

  6. Technical side • We can check sphericity assumption using the covariance matrix • A1-A4 equals time1-time4 or what have you • Variances for individual treatments in red

  7. Compound symmetry is the case where all variances are equal, and all covariances are equal • Not bloody likely

  8. 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 • The constant is equal to the variance of their difference scores

  9. = 10 + 20 - 2(5) = 20 • = 10 + 30 - 2(10) = 20 • = 10 + 40 - 2(15) = 20 • = 20 + 30 - 2(15) = 20 • = 20 + 40 - 2(20) = 20 • = 30 + 40 - 2(25) = 20

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

  11. Output from our previous stress data

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