Partitioning variance
SS between: variance between conditions
SS within: made up of SS subjects and SS error
SS subjects: variance within individuals
SS error: variance that we can’t explain
F ratio: difference between SS between and SS error
SS error
variance that we can’t explain
SS within - SS subject
Mean squares
Calculated to eliminate the bias associated with the number of scores used to calculate 𝑆𝑆
Sphericity
- Sphericity falls under the banner of what is referred to as compound symmetry
- Compound symmetry occurs when both the variances across conditions are equal, and when the covariances between pairs of conditions are equal
- Compound symmetry assumes that the variation within experimental conditions is fairly similar and that no 2 conditions are any more dependent or related than any other 2
- Compound symmetry itself is not a condition of the one-way repeated measures ANOVA
- Whereas, sphericity (which is a less restrictive form of compound symmetry) is a necessary condition
Assumptions for one-way, repeated measures ANOVA
- The accuracy of the F-test used in the one-way ANOVA depends on the assumption that scores in different conditions are independent
- Repeated-measures designs violate this assumption
- > This means that the conventional F-test will be inaccurate and biased
- In repeated measures designs, we use a different assumption:
- > Sphericity assumes that the relationship between pairs of experimental conditions is similar, or, that the level of dependence between experimental conditions is roughly equal
Testing sphericity in SPSS
- SPSS tests the severity of departures from sphericity using Mauchly’s test
- Tests that the variances of the differences between conditions are equal
- > If test statistic is significant then the assumption has been violated
- > If test statistic is not significant then the assumption has NOT been violated
Mauchly’s test
- Mauchly’s test is affected by sample size
- In very big samples, then only small deviations from sphericity can produce a significant test statistic
- In very small samples, then sometimes even quite large deviations from sphericity can produce a non-significant test statistic
Violation of sphericity
When the assumption of sphericity has been violated then there is a significant loss in the power of the F-test and inaccuracies in the F-ratio that is produced in the output
If data violate the sphericity assumption there are 3 alternatives
- Greenhouse-Geisser Correction
- Huynh-Feldt Correction
- Bonferroni Correction
Greenhouse-Geisser Correction
- Varies between 1/k – 1 and 1 (where k is the number of repeated-measures conditions)
- The closer the correction is to 1, the more homogeneous the variances of differences and the closer the data are to being spherical
- This correction can be over conservative
Huynh-Feldt Correction
- Is a less conservative correction than the Greenhouse-Geisser correction
- But it can overestimate sphericity
Bonferroni Correction
Divide alpha level by the number of comparisons
