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:

1. Model within-participant variability
2. Apply additional assumptions to make a simpler, less flexible model fit

One 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 DESIGN
In repeated measured designs, the within-groups variance consists of within-participant variance, as there is only one group of participants. This consists of the effect of the experiment and the error (variance not explained by the experiment). The between-groups variance now consists of the between-participant variance.

The formula for the within-entity (groups) variance is the following:

The n represents the number of scores within the person (e.g. number of experimental conditions). The total amount of variance that is explained by the experimental manipulation can be calculated by comparing the condition mean to the grand mean for all the conditions. It uses the following formula:

The total error variance (residual sum of squares), the amount of variance that cannot be explained by the experimental manipulation can be calculated in the following way:

In order to calculate the F-statistic, the mean squares have to be calculated and this can be done by dividing both the SSR and the SSM by the degrees of freedom:

This leads to the following formula for the F-statistic:

It is the ratio of the systematic variation to the unsystematic variation. A repeated contrast compares each category against the previous one.