How to compare an outcome across repeated measurements or conditions

01What repeated-measures ANOVA does

The paired t-test compares two measurements on the same people. Repeated-measures ANOVA applies the same logic to any number of measurements. With two it gives the same answer as the paired t-test, and its advantage shows with three or more. A course might measure students' confidence scores at the start, middle, and end of a term. Each student provides three scores, and those scores are linked because they come from the same student.

The test separates three sources of variation: differences between the time points, differences between students, and leftover variation that neither explains. Because stable differences between students are removed from the comparison, the test can pick up a pattern of change with far fewer students than an independent-groups design would need.

02When to use it

Use it when the outcome is numeric and approximately normal, each subject is measured under two or more conditions or time points, and you want to know whether the average changes across them. Conditions can be time points, or different treatments given to the same people, such as three versions of a message tested on the same participants.

With exactly two measurements, the paired t-test gives the same answer and is the usual choice. If the outcome is ordinal or strongly skewed, use the Friedman test, which is the rank-based counterpart. If subjects are nested in clusters such as schools and measured repeatedly, multilevel regression handles both layers.

03The sphericity assumption

In addition to the usual assumptions, repeated-measures ANOVA assumes sphericity: the differences between every pair of time points should have similar variability. This is easily violated in real data, particularly when time points are far apart, because measurements close together in time tend to correlate more strongly than those far apart.

Mauchly's W is a summary of how well sphericity holds. Values close to 1 indicate the assumption is reasonable. Lower values indicate a possible violation, which makes the F test too generous.

AnalyZ Solutions reports Mauchly's W with a plain note. When it is high, the note says sphericity is likely met. When it is low, the note warns you and the interpretation repeats the warning. The module does not apply a Greenhouse-Geisser correction, so when W is low, read a borderline result with caution or use the Friedman test as a cross-check, since it makes no sphericity assumption.

04Reading the result

The ANOVA table

The table has rows for the condition effect, the subjects, and the error. The row that matters is the condition effect. Its F statistic and significance level tell you whether the average differs across the time points. A significant result says at least one time point differs from another.

Partial eta-squared

Partial eta-squared shows the share of variation, after removing between-subject differences, that the time points explain. Below 1% is negligible, 1% to 6% small, 6% to 14% medium, and above 14% large.

Pairwise follow-ups

When the overall test is significant, and the post-hoc box is ticked, the module runs paired t-tests between every pair of time points with a Bonferroni adjustment. The adjustment multiplies each significance value by the number of comparisons, so the stated level already accounts for the number of pairs.

05How to run it in AnalyZ Solutions

Open Inferential Analysis, choose Compare Means, then Repeated-Measures ANOVA.

1. Arrange your data in wide format
   Each row is one subject, and each time point or condition is its own column, for example score_time1, score_time2, score_time3.
2. Select the conditions
   Pick three or more numeric columns in the order you want them compared. The module asks for at least three, so use the paired t-test when you have only two. The order does not change the overall test, but it sets how the descriptives and follow-ups are listed.
3. Add a subject ID if you have one
   This is optional. If you leave it blank, each row is treated as one subject.
4. Decide on follow-up comparisons
   The pairwise box is ticked by default and only produces output if the overall test is significant.
5. Apply a subsample filter if needed
   For example, to analyse one site only.
6. Click Run
   You receive descriptives per time point, the ANOVA table, the sphericity note, any pairwise comparisons, and a plain-language interpretation.

Only subjects with values for every selected time point are included. At least five complete subjects are required.

06Interpreting the result

The figures below are invented to show the steps, and they follow the results tables in AnalyZ Solutions, with a few columns left out to save space. Suppose 24 students rated their confidence on a 1 to 5 scale at the start, middle, and end of a term.

Reading it step by step

1. Check the sphericity note first
   Mauchly's W is 0.692, below 0.75, so the module warns of a possible violation. The F test may be slightly too generous. The result here is far from borderline, so the conclusion is safe, and a Friedman test on the same columns would confirm it.
2. Read the overall test
   The F test has a p-value below 0.001 with three stars, so confidence changed across the three rounds, and the change is significant at the 0.1% level.
3. Judge the size
   Partial eta-squared is 0.435, a large effect. After removing differences between students, the three rounds account for about 43% of the remaining variation.
4. Find where the change happened
   Confidence rose by 0.56 points from Round 1 to Round 2 and by 0.55 from Round 1 to Round 3, both significant after the Bonferroni adjustment. Rounds 2 and 3 barely differ (0.01 apart, adjusted p of 1.0000).

In plain words

Confidence rose across the term. Most of the increase happened between the first and second rounds, and scores then held steady until the end. The change is large and significant at the 0.1% level. The sphericity warning was noted and did not affect the conclusion. Without a comparison group, the rise cannot be credited to the course itself.

07Common mistakes

Ignoring sphericity. Report the note and be cautious when it signals a problem.

Using long-format data. The module expects one column per time point. Reshape the data if each row holds a single observation.

Reading a significant result as a steady trend. The test says the averages differ, not that they rise or fall in a straight line.

Assuming change was caused by a treatment. Without a comparison group, change across rounds may reflect season, inflation, or other events.