Multi-Factorial ANOVA
Hypothesis TestingTests the effects of three or more categorical factors and their interactions on a continuous dependent variable in a single model, with user-controlled interaction depth.
When to Use
Use this test when you have a continuous outcome and at least three categorical factors, and you need to evaluate main effects and higher-order interactions in one analysis. For example, testing treatment, dose, and sex effects on tumour size in the same model.
Assumptions
- The dependent variable is continuous.
- Observations are independent.
- The dependent variable is approximately normally distributed within each cell of the design.
- Homogeneity of variances across cells.
- For unbalanced designs, Type III sums of squares should be used.
Required Inputs
| Input | Type | Notes |
|---|---|---|
| Dependent Variable | Numeric | Continuous outcome variable |
| Factors | Categorical (3+ columns) | Three or more grouping factors included in one model |
| Parameter | Default | Options |
|---|---|---|
| Max Interaction Depth | 3 | Limits the highest interaction order included in the fitted model. For example, 2 keeps only main effects and two-way interactions; 3 includes up to three-way interactions. |
| Significance Level | 0.05 | Alpha level used for p-values, confidence limits, and post-hoc inference. |
Output Metrics
| Metric | What it means |
|---|---|
| Source | Each main effect and interaction term included in the model, plus Error and Total rows. |
| DF | Degrees of freedom for each main effect, interaction, and error term. |
| Sum of Squares | Sum of squared deviations attributable to each source. |
| Mean Square | Sum of squares divided by degrees of freedom. |
| F Value | F-statistic for each main effect and interaction. |
| Pr > F | P-value for each model term. |
| Partial Eta-Squared | Effect size for each main effect and interaction. |
| Omega-Squared | Less biased variance-explained estimate for each model term. |
| Means Type | Balanced designs report observed cell means; unbalanced designs report LS means (estimated marginal means). |
| Means Table | Per-cell means with standard error and confidence limits using the selected means type. |
| Simple Effects | Conditional effects for selected factors within levels of other factors. |
Interpretation
- Interpret higher-order interactions before lower-order terms. If a three-way interaction is significant, lower-order main effects may be misleading on their own.
- If Max Interaction Depth is set below the total number of factors, higher-order interactions are intentionally omitted from the model and cannot be interpreted from that run.
- Use simple-effects outputs to identify where effects occur across factor combinations.
- For balanced designs, means are observed cell averages; for unbalanced designs, LS means are used for fairer between-cell comparisons.
- Report effect sizes with p-values; statistically significant but tiny effects may be practically unimportant.
- Complex models require sufficient cell sample sizes to stabilize estimates and improve power.
Common Pitfalls
- Sparse or empty cells produce unstable or non-estimable interactions.
- Changing Max Interaction Depth changes the model being fit. Results from a depth-limited run are not equivalent to a full-interaction model.
- Unbalanced designs can change inference depending on sums-of-squares type; keep Type III handling consistent.
- Overly complex factor structures with small samples reduce power and can overfit.
- Interpreting only main effects when interactions are significant can be incorrect.
How It Works
- Fit a general linear model including all specified main effects and all interactions up to the selected maximum depth.
- Compute observed cell means and check whether the design is balanced from per-cell sample counts.
- If balanced, report observed cell means; if unbalanced, derive LS means (estimated marginal means) from model predictions.
- Partition total variance into model terms and residual error.
- Compute mean squares and F-ratios for each term against the residual mean square.
- Derive p-values from the F-distribution and report effect sizes per term.
Citations
References
- Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver & Boyd.
- Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society A, 135(3), 370-384.