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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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Presentation Transcript
more on 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 sphericity1
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
more on sphericity2
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
more on sphericity3
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

technical side
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
slide7
Compound symmetry is the case where all variances are equal, and all covariances are equal
  • Not bloody likely
slide8
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
slide9

= 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
slide10
SPSS
  • You can produce the variance/ covariance matrix in SPSS repeated measures