Factorial ANOVA#

The factorial ANOVA extends the one-way analysis of variance by simultaneously examining multiple independent factors. Beyond the main effects of each factor, it can also detect interaction effects between factors. It is one of the most important methods in experimental research when multiple influences are studied at the same time.

When to Use#

You have two or more categorical independent variables (factors) and a continuous dependent variable
You want to analyze both the individual effect of each factor and their interplay (interaction)
The study design is fully crossed (every factor level occurs with every other factor level)
Samples are independent of each other (between-subjects design)
You want to go beyond additive effects — the interaction is often the most interesting research question

Assumptions#

Normality of residuals (Shapiro-Wilk test, QQ plot)
Homogeneity of variances across all cell combinations (Levene's test)
Independence of observations (study design)
Continuous (interval or ratio scaled) dependent variable

Note: The factorial ANOVA is robust against moderate violations of normality when designs are balanced (equal cell sizes). For unbalanced designs, Type III sums of squares should be used to avoid biased results.

Formula#

The total variance is decomposed into several components. For a two-way ANOVA with factors A and B:

SS_{total} = SS_A + SS_B + SS_{AB} + SS_{error}

The F-statistic for the main effect of Factor A:

F_A = \frac{MS_A}{MS_{error}} = \frac{SS_A / (a - 1)}{SS_{error} / (N - a \cdot b)}

The F-statistic for the main effect of Factor B:

F_B = \frac{MS_B}{MS_{error}} = \frac{SS_B / (b - 1)}{SS_{error} / (N - a \cdot b)}

The F-statistic for the A × B interaction:

F_{AB} = \frac{MS_{AB}}{MS_{error}} = \frac{SS_{AB} / ((a-1)(b-1))}{SS_{error} / (N - a \cdot b)}

Here, $a$ is the number of levels of Factor A, $b$ is the number of levels of Factor B, and $N$ is the total sample size.

Example#

Practical Example: Teaching Method × Gender on Exam Performance

An education researcher investigates whether teaching method (lecture, group work, blended learning) and gender (female, male) affect exam scores. The design is 3 × 2 with 20 students per cell (N = 120).

Results:

Main effect of teaching method: $F(2, 114) = 8.45$ , $p < .001$ — teaching method significantly affects exam performance
Main effect of gender: $F(1, 114) = 1.22$ , $p = .272$ — no significant gender difference
Interaction teaching method × gender: $F(2, 114) = 4.67$ , $p = .011$ — the effect of teaching method differs between genders

The significant interaction reveals that blended learning leads to particularly better scores for female students, while male students show little difference across methods.

Effect Size#

The most commonly reported effect size is partial eta-squared ( $\eta_p^2$ ):

\eta_p^2 = \frac{SS_{effect}}{SS_{effect} + SS_{error}}

Effect Size	$\eta_p^2$
Small	0.01
Medium	0.06
Large	0.14

Partial eta-squared indicates the proportion of variance explained by an effect after removing the variance attributable to other effects. It is calculated separately for each main effect and the interaction.

Factorial ANOVA (Two-Way ANOVA)