Repeated-measures refers to when the same entities (e.g. people) participate in all conditions of an experiment or provide data at multiple points of time. One of the assumptions of the standard linear model is that residuals are independent, which is not true for repeated-measures designs. The residuals are affected by both between-participant factors and within-participant factors. There are two solutions to this:Model within-participant variabilityApply additional assumptions to make a simpler, less flexible model fitOne of these assumptions is sphericity (circularity). This assumption states that the relationship between scores in pairs of treatment conditions is similar (e.g. the level of dependence between means is roughly equal). It states that variation within conditions are similar and that no two conditions are any more dependent than any other two. Local sphericity refers to when some conditions do have equal variance and some do not. Sphericity is not relevant if there are only two groups. It becomes relevant when there are at least three conditions. The assumption of sphericity can be tested using Mauchly’s test. The degree of sphericity can be estimated using the Greenhouse-Geisser estimate or the Huyn-Feldt estimate. If the assumption of sphericity is not met, then there is a loss of power and the F-statistic doesn’t have the distribution it is supposed to have. In order to do post-hoc tests when you worry about whether the assumption of sphericity is violated, Bonferonni method can be used, if it is not violated, Tukey’s test can be used.If the assumption of sphericity is violated, the degrees of freedom has to be adjusted. The degrees of freedom is multiplied by the estimate of sphericity to calculate the adjusted degrees of freedom. F-STATISTIC OF REPEATED MEASURES DESIGNIn repeated measured designs, the within-groups variance consists of within-participant variance, as there is only one group of participants. This...
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