Repeated Measures ANOVA

When you measure the same people under several conditions (the same patients before, during, and after treatment), you have a repeated-measures design. Just as the paired t-test beats the independent one, repeated-measures ANOVA has a superpower: it can subtract out the fact that people just differ from each other.

The problem with big individual differences

Suppose you test three conditions and people vary enormously to begin with: some always score high, some always low. In a between-subjects design, that person-to-person variability becomes noise that swamps the condition effect. You might have a perfectly real effect and still fail to find it, buried under "well, people are just different."

The repeated-measures trick

Because every person is measured in every condition, each person serves as their own control. The analysis can split off a "subjects" chunk of variation and set it aside, leaving a much cleaner comparison of the conditions. In the subject-centered view above, where each person's own average is subtracted out, the tangle of lines collapses into a clear shared pattern. That's exactly what repeated-measures ANOVA does mathematically. It's also the reason the honest picture of a within-subjects result is one line per person rather than a row of bars: the chart chooser calls that a slopegraph, and Tables & Figures That Don't Lie shows what goes wrong when you draw the bars instead.

The catch: sphericity

Repeated measures come with their own assumption, sphericity: roughly, that the differences between every pair of conditions have similar variance. When it's violated (common with time-based data, where adjacent timepoints correlate more than distant ones), the test becomes too liberal. Standard corrections (Greenhouse-Geisser, Huynh-Feldt) adjust the degrees of freedom to compensate.

What ε is, and what it costs

Both corrections work through a single number, epsilon, written ε. It measures how far the pattern of condition-to-condition differences has drifted from sphericity, and it runs from 1 down to a hard floor of 1/(k − 1). Perfect sphericity gives ε = 1. Three conditions can fall no lower than .5, four no lower than .33. Multiply both degrees of freedom by ε and re-read the F table; the F ratio itself never changes, only the distribution it is compared against.

Problem 16 of the practice problems hands you a case with 12 typists and 3 keyboard layouts, F = 5.84, and a Greenhouse-Geisser ε of .71. Four ways of reading the same ratio:

The correction more than doubles the p-value here without touching a single data point. Even the floor leaves this result standing, but a study reporting an uncorrected .04 would not survive the trip.

Greenhouse-Geisser is known to under-estimate ε when the violation is mild, so Huynh-Feldt applies a correction to the correction (that .785 is computed from the .71, not measured separately) and can reach 1, at which point software caps it. The convention most textbooks give: take Greenhouse-Geisser when its ε falls below .75 and Huynh-Feldt when it does not. Report which one you used and print the ε beside it, because fractional degrees of freedom tell a reader that something was corrected without telling them by how much. The assumption-checks sheet lists this one alongside every other check a comparison of means owes you.

There is also a more modern route. A mixed (multilevel) model treats each participant as a random effect and estimates the correlation structure directly, so it needs no sphericity correction, tolerates unequal spacing between timepoints, and keeps a participant who missed one session instead of dropping them from every condition. Repeated-measures ANOVA is still the clearer way to understand what within-subjects designs buy you, which is why it comes first. When the repeated outcome is a category rather than a number, the same reasoning arrives as McNemar's test: pair each person with themselves and count only the ones who changed.

Problem 16 of the practice problems works the case above from the sums of squares up, so you build the F yourself before correcting it, and Problem 27 asks the prior question: whether a study is between, within, or neither.