Does something change over several time points? (repeated-measures analysis of variance)

In brief

The same people were measured at three or more time points (for example a reading test at the start, middle and end of the school year); the aim is to check whether the means change over time. Choose the columns of the time points in the right order, at least three; with two time points, "Has something changed?" is the right question. Statable calculates the repeated-measures analysis of variance itself (PSPP cannot do this; checked against R), checks sphericity and corrects the degrees of freedom if needed.

With groups (mixed design)

Optionally a grouping column with two to four groups can be chosen. The main effect of group and the time × group interaction are then added; the interaction shows whether the groups develop differently. If it is significant, the main effects must be read with care.

Requirements

• Number columns of the same measurement on the same scale, one row per person (for long data, "Reshape" helps). Only people with all measurements are included, at least 4.
• Sphericity: all differences between any two time points vary equally (see below).
• Values without clear skewness or outliers; a departure from normality is then usually uncritical even with few people. The time points are treated as equally spaced.

Sphericity and correction

• Mauchly's test: W near 1 means met. ε (epsilon) measures the departure: 1 means met, smaller means violated. If violated, the degrees of freedom are multiplied by ε.
• The default is Greenhouse-Geisser, always applied: with few people Mauchly's test hardly detects departures, with many it responds to small ones. On the page you can switch to Huynh-Feldt (somewhat less strict) or "No correction".

Reading the result

• Sum of squares, degrees of freedom, F and p per effect (time; with groups also group and time × group). The degrees of freedom of time are already corrected with ε, hence values like 1.54.
• Effect size: generalized η² (ηG²), which is comparable with η² from group comparisons, and partial η² (ηp²) with a 95% confidence interval. The classification after Cohen (.01 small, .06 medium, .14 large) follows ηG²; with repeated measures ηp² is usually considerably larger (Olejnik and Algina 2003; Bakeman 2005).
• Pairwise comparisons: paired t tests for all pairs of time points (later minus earlier) with dz, Bonferroni-adjusted; with groups averaged over the groups.
• The chart shows the mean at each time point with a 95% confidence interval, with groups one line per group.
• Without a comparison group the result shows only that something changed, not why.
• If the values are skewed, have outliers or only express an order, the Friedman test can be chosen below the result. If all chosen columns are yes/no, Statable calculates Cochran's Q by itself.

F(1.54, 23.16) = 14.85, p < .001, ηp² = .50, 95% CI [.22, .67], ηG² = .18 (Greenhouse-Geisser)

For specialists

Type III sums of squares (as in SPSS, jamovi and R afex; with equal group sizes all types are the same). Lower bound of ε: 1 / (time points − 1). ηG² after Olejnik and Algina (2003). Checked against R afex::aov_ez and emmeans (sums of squares, F, p, Mauchly, ε, ηp², ηG², pairwise comparisons).

Typical examples

• Reading performance at three time points, training against control class.
• Pain before, during and after a treatment in the same people.