Sphericity

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Sphericity
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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

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More on sphericity

• Homogeneity of variance assumption means we want
  to see similar variability from group to group
• In other words we dont want more or less
  variability in one groups scores relative to
  another

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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

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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
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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

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

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• 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

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• 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

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SPSS

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

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Output from our previous stress data