Two-Way Factorial ANOVA Calculator (Main Effects & Interaction

Statistical Foundations of Two-Way Factorial ANOVA

Two-Way ANOVA evaluates the simultaneous effects of two categorical independent variables (factors) on a continuous dependent variable, testing three separate null hypotheses in a single unified model:

• Main Effect of Factor A: Does the dependent variable differ across levels of Factor A averaging over Factor B)?

• Main Effect of Factor B: Does the dependent variable differ across levels of Factor B averaging over Factor A)?

• Interaction Effect (A × B): Does the effect of Factor A depend upon the specific level of Factor B? When lines on an interaction plot cross or diverge non-parallelly, a significant interaction exists.

• Partial Eta-Squared (η_p^2): Calculates the proportion of variance explained by each individual effect after partialling out all other model effects: .

How to Use This Calculator

Enter your Cell A1 × B1 Data (Factor A Level 1, Factor B Level 1) (e.g. 12, 14, 13, 15), Cell A1 × B2 Data (Factor A Level 1, Factor B Level 2) (e.g. 18, 20, 19, 17), Cell A2 × B1 Data (Factor A Level 2, Factor B Level 1) (e.g. 14, 16, 15, 13), Cell A2 × B2 Data (Factor A Level 2, Factor B Level 2) (e.g. 26, 28, 27, 25), Factor A Label (Row Factor) (e.g. Gender, Drug Formulation), and Factor B Label (Column Factor) (e.g. Dosage, Training Regimen). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Two-Way Factorial ANOVA Partitioning of Variance formulas:

SST = SSA + SSB + SSAB + SSE

df_total = N - 1, df_A = a - 1, df_B = b - 1, df_AB = (a - 1)(b - 1), df_E = N - ab

MSA = (SSA) / (df_A), MSB = (SSB) / (df_B), MSAB = (SSAB) / (df_AB), MSE = (SSE) / (df_E)

F_A = (MSA) / (MSE), F_B = (MSB) / (MSE), F_AB = (MSAB) / (MSE)

Partial Eta-Squared: η_p^2 = SS_effectSS_effect + SSE

Worked Example

For example, evaluating a 2x2 experiment with \ replications per cell: Cell A1B1 [12, 14, 13, 15]), A1B2 [18, 20, 19, 17]), A2B1 [14, 16, 15, 13]), and A2B2 [26, 28, 27, 25]): With \ and Main Effect A yields , Main Effect B yields , and Interaction yields , confirming highly significant main effects and interaction (Factor B enhances the response significantly more in Level A2 than Level A1).

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.