Two-Way ANOVA: Main Effects, Interactions, and Worked Examples

Two-way ANOVA tests the effect of two categorical independent variables (factors) on one continuous dependent variable. Unlike running two separate one-way ANOVAs, two-way ANOVA also tests whether the factors interact — whether the effect of one factor depends on the level of the other. Example setup: a researcher tests whether plant growth (dependent variable, continuous) depends on fertilizer type (Factor A: three types) AND sunlight level (Factor B: two levels, full/partial). The design has 3 × 2 = 6 treatment combinations. Two-way ANOVA answers three questions at once: (1) does fertilizer affect growth? (2) does sunlight affect growth? (3) does the fertilizer effect depend on sunlight level (interaction)? Why not two one-way ANOVAs: running separate tests ignores the interaction, inflates Type I error across the multiple tests, and misses variance that the two-factor model captures. Two-way ANOVA is more statistically efficient and more interpretively rich.

Main effect of Factor A: the effect of Factor A averaged across all levels of Factor B. Significant main effect of fertilizer means plants grow differently across fertilizer types, on average, ignoring sunlight. Main effect of Factor B: the effect of Factor B averaged across all levels of Factor A. Significant main effect of sunlight means plants grow differently across sunlight levels, on average, ignoring fertilizer. Interaction effect A×B: whether the effect of one factor depends on the level of the other. Significant interaction means fertilizer affects growth differently in full sun than in partial sun — for example, fertilizer 1 might be best in full sun but worst in partial sun. Interpretation priority rule: always check the interaction first. If the interaction is significant, the main effects alone are misleading. You must describe effects separately at each level of the other factor (simple main effects). If the interaction is not significant, main effects can be interpreted on their own. Visual shortcut: plot group means on a line graph with one factor on the x-axis and the other shown as different lines. Parallel lines suggest no interaction. Non-parallel lines suggest an interaction (crossing lines indicate a strong interaction; spreading lines indicate a weaker interaction).

Two-way ANOVA partitions total variability into components: SS_total = SS_A + SS_B + SS_AB + SS_error SS_A: variability explained by Factor A (differences between A-level means). SS_B: variability explained by Factor B (differences between B-level means). SS_AB: variability explained by interaction (how much cell means differ from what main effects alone predict). SS_error: unexplained variability within cells (residual). Degrees of freedom: - df_A = (levels of A) − 1 - df_B = (levels of B) − 1 - df_AB = df_A × df_B - df_error = N − (levels of A × levels of B), where N is total observations - df_total = N − 1 Mean squares (MS) = SS / df for each component. F-statistics (each main effect and the interaction has its own F-test): - F_A = MS_A / MS_error, with df_A and df_error - F_B = MS_B / MS_error, with df_B and df_error - F_AB = MS_AB / MS_error, with df_AB and df_error Compare each F to the critical F-value at chosen alpha (typically 0.05) or use p-values to decide significance.

Research question: does teaching method (Factor A: Lecture, Discussion, Flipped) interact with class size (Factor B: Small, Large) to affect exam score? Data structure: 3 methods × 2 sizes = 6 cells. 5 students per cell. N = 30 total. Hypothetical cell means and cell variances: Cell means (rows = method, columns = size): Small Large Lecture 78 72 Discussion 82 74 Flipped 88 71 Row means (main effect A, averaging across B): Lecture: (78+72)/2 = 75 Discussion: (82+74)/2 = 78 Flipped: (88+71)/2 = 79.5 Grand mean (GM): (75 + 78 + 79.5)/3 = 77.5 Column means (main effect B, averaging across A): Small: (78+82+88)/3 = 82.67 Large: (72+74+71)/3 = 72.33 Step 1: compute SS_A (between-row variability, scaled by n per row): Each row has 2 × 5 = 10 observations. SS_A = 10 × [(75 − 77.5)² + (78 − 77.5)² + (79.5 − 77.5)²] = 10 × [6.25 + 0.25 + 4] = 10 × 10.5 = 105 df_A = 3 − 1 = 2 MS_A = 105 / 2 = 52.5 Step 2: compute SS_B (between-column variability): Each column has 3 × 5 = 15 observations. SS_B = 15 × [(82.67 − 77.5)² + (72.33 − 77.5)²] = 15 × [26.73 + 26.73] = 15 × 53.47 = 802.0 df_B = 2 − 1 = 1 MS_B = 802.0 / 1 = 802.0 Step 3: compute SS_AB (interaction): Interaction SS comes from how much each cell mean deviates from what the main effects alone would predict. Expected cell mean = grand mean + (row effect) + (column effect) For Lecture-Small: 77.5 + (75 − 77.5) + (82.67 − 77.5) = 77.5 − 2.5 + 5.17 = 80.17 Actual cell mean: 78 Deviation: 78 − 80.17 = −2.17, squared: 4.71 Repeat for all 6 cells, sum the squared deviations, and multiply by n per cell (5): After computing for all cells, suppose SS_AB = 5 × 8.3 = 41.5 df_AB = 2 × 1 = 2 MS_AB = 41.5 / 2 = 20.75 Step 4: compute SS_error (within-cell variability): Sum of squared deviations of each observation from its cell mean. With the raw data, suppose SS_error = 288. df_error = N − (3 × 2) = 30 − 6 = 24 MS_error = 288 / 24 = 12 Step 5: compute F-statistics and compare to critical values (α = 0.05): F_A = MS_A / MS_error = 52.5 / 12 = 4.38, df(2, 24), critical F ≈ 3.40 → significant F_B = MS_B / MS_error = 802.0 / 12 = 66.83, df(1, 24), critical F ≈ 4.26 → significant F_AB = MS_AB / MS_error = 20.75 / 12 = 1.73, df(2, 24), critical F ≈ 3.40 → NOT significant Conclusion: main effects of teaching method and class size are both significant; interaction is not. We can interpret main effects directly: discussion and flipped classrooms produce higher exam scores than lecture, on average; small classes produce higher exam scores than large, on average. The effect of teaching method does not depend meaningfully on class size in this dataset.

When F_AB is significant, the main effects alone are misleading. You must describe effects at each level of the other factor — called simple main effects (or simple effects). Example where interaction is significant: imagine a different dataset where flipped classroom was best in small classes (mean = 92) but worst in large classes (mean = 65). Other methods were more consistent. Here, averaging across class size would hide that the flipped method is either the best or the worst depending on context. Main effect of teaching method would look small because the strong small-class performance is canceled by the weak large-class performance. How to report: rather than saying "flipped classrooms produce higher scores," you must say: "in small classes, flipped produces the highest scores; in large classes, flipped produces the lowest scores." You are describing the effect of Factor A separately at each level of Factor B. Statistical procedures: use simple main effect F-tests (one-way ANOVA within each level of the other factor), often with a corrected error term from the full model. Post-hoc tests (Tukey HSD, Bonferroni) identify which specific pairs differ within each level. This is why the interaction test matters. A non-trivial percentage of two-way ANOVA results has significant interactions. Reporting only the main effects in those cases is factually incorrect.

Two-way ANOVA assumes: 1. Normality of residuals within each cell (check with Q-Q plots; more important for small n). 2. Homogeneity of variance across cells (check with Levene's test; ratio of largest to smallest cell variance should be less than 4:1). 3. Independence of observations (design issue; can't test statistically — must be satisfied by how data was collected). 4. Interval or ratio dependent variable. Diagnostics: - Plot residuals vs fitted values (should show random scatter, no funnel shape) - Q-Q plot of residuals (should be approximately linear) - Levene's test p-value > 0.05 supports homogeneity - Shapiro-Wilk test for normality within cells Common pitfalls: 1. Running two one-way ANOVAs instead of a two-way. This misses the interaction and can produce contradictory results. 2. Interpreting main effects when interaction is significant. Reread the interaction first. If significant, the main effect story is wrong or incomplete. 3. Unequal cell sizes (unbalanced design). Requires Type III sum of squares (the default in SPSS, not always in R without specification). Type I and Type II SS give different results when cells are unbalanced; Type III is usually what you want. 4. Ignoring cell size requirements. Power depends on cell size, not total N. 30 observations in 10 cells (3 per cell) has very low power. Aim for at least 10 per cell for meaningful tests; ideally 20+. 5. Using two-way ANOVA on repeated measures. If the same subjects are measured under both conditions, you need repeated-measures ANOVA (accounts for within-subject correlation). Two-way ANOVA assumes independent observations. 6. Treating a continuous predictor as categorical just to use ANOVA. If one of your "factors" is continuous, ANCOVA or multiple regression is usually more appropriate. 7. Reporting only the F-statistic without effect size. Report η² (eta squared) or ω² (omega squared): SS_effect / SS_total tells you what proportion of variability the effect explains. A significant F with η² = 0.01 is statistically significant but practically trivial.