Two-Way ANOVA Calculator

Two-way ANOVA decomposes total variability into four components: the main effect of Factor A, the main effect of Factor B, the interaction between A and B, and residual (error) variability.

For a balanced 2×2 design with \(n\) observations per cell:

$$SS_A = bn \sum_{i=1}^{a} (\bar{X}_{i\cdot} - \bar{X}_{\cdot\cdot})^2$$

$$SS_B = an \sum_{j=1}^{b} (\bar{X}_{\cdot j} - \bar{X}_{\cdot\cdot})^2$$

$$SS_{AB} = n \sum_{i=1}^{a} \sum_{j=1}^{b} (\bar{X}_{ij} - \bar{X}_{i\cdot} - \bar{X}_{\cdot j} + \bar{X}_{\cdot\cdot})^2$$

Each mean square is divided by the MSE to obtain F-statistics:

$$F_A = \frac{MS_A}{MSE}, \quad F_B = \frac{MS_B}{MSE}, \quad F_{AB} = \frac{MS_{AB}}{MSE}$$

The interaction term captures whether the effect of one factor depends on the level of the other factor. A significant interaction means the factors do not act independently — you cannot interpret main effects in isolation when a strong interaction is present.

Degrees of freedom for a 2×2 design: df_A = 1, df_B = 1, df_AB = 1, df_error = 4(n−1). The MSE should be estimated from within-cell variability.

When interpreting two-way ANOVA results, follow this recommended order:

• Interaction (F_AB): Examine first. If significant, the effect of one factor depends on the level of the other. Plot cell means to visualize — non-parallel lines indicate interaction.
• Main Effect A (F_A): If no significant interaction, a significant F_A means the marginal means of Factor A levels differ.
• Main Effect B (F_B): Similarly, a significant F_B indicates different marginal means across Factor B levels.

Compare each F-statistic against critical F-values with appropriate df. For a 2×2 design: df_numerator = 1, df_denominator = 4(n−1).