Two-Way ANOVA

Run a two-way ANOVA on a balanced 2×2 design when all you have are the four cell means, the cell size and a pooled standard deviation, for example from a paper or a report.

How the calculation works

In a 2×2 design, two factors (A and B) each have two levels, giving four cells: A1·B1, A1·B2, A2·B1 and A2·B2. You enter each cell mean, the number of observations per cell n (equal for all cells) and the pooled within-cell standard deviation SD. The calculator finds the grand mean and the marginal means, then partitions the variability using Fisher’s classic decomposition.

Sums of squares: SS_A = 2n · Σ(level mean of A − grand mean)², SS_B likewise for factor B, and SS_AB = n · Σ(cell mean − row mean − column mean + grand mean)². The within-cell mean square equals SD², so each effect’s F = SS / SD² with 1 degree of freedom, and the error has 4(n − 1) degrees of freedom. The p-value comes from the F distribution.

The headline figure is the F statistic for factor A. Below it you get the p-value for A, F and p for factor B, F and p for the A×B interaction, and the degrees of freedom. An effect is conventionally called significant when p < 0.05.

Worked example

Defaults: cell means 10, 12, 14 and 20, n = 10 per cell, SD = 4. Grand mean 14; A level means 11 and 17; B level means 12 and 16. SS_A = 20 × (9 + 9) = 360, SS_B = 20 × (4 + 4) = 160, SS_AB = 10 × 4 × 1 = 40, SD² = 16. That gives F_A = 22.5 (p < 0.001), F_B = 10 (p ≈ 0.0032) and F_AB = 2.5 (p ≈ 0.12) with df = 1 and 36: both main effects are significant, the interaction is not.

Things to keep in mind

• SD must be the pooled within-cell standard deviation, not the spread of the means and not the overall SD of all data. With equal n it is the square root of the average of the four cell variances.
• The tool covers only 2×2 designs with equal cell sizes. Unbalanced data or more levels need statistical software with Type II or Type III sums of squares.
• Look at the interaction first. If it is significant, interpret the main effects with care, because the effect of A depends on the level of B.
• Check the assumptions: independent observations, roughly normal data within cells and similar variances (Levene’s test, for instance).
• Effect size by hand: partial η² = SS_effect / (SS_effect + SD² × error df). For factor A in the example that is 360 / 936 ≈ 0.38.