Three-way ANOVA
Test three grouping factors and their interactions on one numeric outcome.
When to use it
Use three-way ANOVA when three categorical factors might affect one numeric outcome. For example: do practice scores depend on sleep, caffeine, and practice schedule, including the ways those factors combine?
Read the three-way interaction first. If it is significant, the two-way patterns are not the same at every level of the third factor, and the main effects are a blunt summary.
Assumptions
Each factor needs at least two levels, and Tensr asks for at least eight complete rows. A balanced design (the same number of scores in every combination) uses Type II sums of squares. An unbalanced or incomplete design uses Type III.
Tensr does not run Levene’s test or Shapiro–Wilk here. η² in the table is each term’s sum of squares divided by the total sum of squares, not partial η². There is no switch for that.
If a sum of squares comes out negative, the report warns that the model may be overspecified. Tensr then drops the three-way term and refits the two-way interactions. If squares are still negative, the run stops.
Running it in Tensr
Options
Prop
Type
Reading the output
Method, time of day, and a north/south campus label on the same exam. Only method was built to move the scores.
| Source | SS | df | F | p | η² |
|---|---|---|---|---|---|
| method | 523.351 | 2 | 3.858 | .025 | 0.08 |
| time | 5.95 | 1 | 0.088 | .768 | 0.0009073 |
| campus | 1.426 | 1 | 0.021 | .885 | 0.0002174 |
| method × time | 166.676 | 2 | 1.229 | .298 | 0.025 |
| method × campus | 88.883 | 2 | 0.655 | .522 | 0.014 |
| time × campus | 46.343 | 1 | 0.683 | .411 | 0.007 |
| method × time × campus | 28.416 | 2 | 0.209 | .811 | 0.004 |
| Residual | 5697 | 84 | — | — | — |
| Factor | Level | Mean |
|---|---|---|
| method | Lecture | 68.209 |
| method | Online | 72.234 |
| method | Workshop | 73.741 |
| time | Afternoon | 71.644 |
| time | Morning | 71.146 |
| campus | North | 71.273 |
| campus | South | 71.517 |
Three-way ANOVA for score, n = 96. Three-way ANOVA for score, n = 96. Metrics: R² = 0.131.
Reporting (APA 7)
Three-way ANOVA for score, n = 96. Report the estimate in the table. This procedure is not summarised by one p-value.
Coming from SPSS
Analyze → General Linear Model → Univariate, with three fixed factors. Tensr’s stored path is Analyze → General Linear Model → Three-Way ANOVA. SPSS can print partial η² and post-hoc tests from that dialog. Tensr prints η² as SS / SS total and has no post-hoc option here.
Related
Two factors: two-way ANOVA. One factor: one-way ANOVA.