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."
🎮 Removing the Individual Differences
Eight people, three conditions — each line is one person. Crank up how different people are, then flip to the subject-centered view to see the condition pattern snap into focus.
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.
The payoff is power. By removing stable individual differences from the error term, a repeated-measures design can detect a real effect with far fewer participants than a between-subjects design would need. Fewer people, more sensitivity — that's why it's so popular.
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.
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.
Why it matters: within-subjects designs are everywhere in psychology, medicine, and UX research — anywhere you can measure the same unit repeatedly. Knowing that they trade a sphericity assumption for a big gain in power helps you design more efficient studies. Plan My Analysis will lay one out for you, assumptions and write-up included.
Problem 16 of the practice problems hands you a sphericity test that has failed and asks you to apply the Greenhouse-Geisser correction yourself, and Problem 27 asks the prior question: whether a study is between, within, or neither.
Common questions
What is sphericity and why does it matter?
Sphericity is the assumption that the variance of the difference between every pair of conditions is roughly equal. When it fails (typical for time-based measurements, where neighboring timepoints correlate more than distant ones), the F-test's p-values come out too small, inflating false positives. Mauchly's test flags it, and corrections fix it.
What do I do if sphericity is violated?
Apply the Greenhouse–Geisser correction (or Huynh–Feldt when the violation is mild) — both shrink the degrees of freedom to restore an honest p-value, and every stats package offers them next to the uncorrected test. The modern alternative is a mixed model, which sidesteps sphericity entirely.
What happens if a participant misses one condition?
A classical repeated-measures ANOVA drops that person from the analysis entirely, in every condition, because the computation needs a complete row per participant. Lose a handful of people that way and you can give back most of the power the design was chosen for. Two respectable routes out: a mixed model, which keeps every observation a person did provide, or a principled treatment of the missing values before the ANOVA. Substituting the person's own mean for the gap is not one of them, since it invents data with zero variance and shrinks your error term.