Three-Way ANOVA Calculator

A three-way ANOVA (analysis of variance) is a statistical test that examines the effects of three independent categorical variables (factors A, B, and C) on a continuous dependent variable. It evaluates seven effects simultaneously: three main effects (A, B, C), three two-way interactions (AB, AC, BC), and one three-way interaction (ABC).

A main effect is the independent influence of a single factor on the dependent variable, averaged across all levels of the other factors. An interaction effect occurs when the influence of one factor depends on the level of another factor. In a three-way ANOVA, a significant ABC interaction means the two-way interaction between any two factors changes depending on the level of the third.

The key assumptions are: (1) the dependent variable is continuous and approximately normally distributed within each cell, (2) observations are independent of each other, (3) there is homogeneity of variances across groups (homoscedasticity), and (4) the design is balanced (equal sample sizes per cell). Violations of normality are less critical with larger sample sizes.

Enter the dependent variable values as a comma-separated list. The order should vary Factor C fastest, then Factor B, then Factor A. For example, with 2 levels each and 2 replications: A1B1C1, A1B1C1 (rep2), A1B1C2, A1B1C2 (rep2), A1B2C1, ... continuing through all combinations. Specify the number of replications per cell so the calculator can correctly partition the values.

If the p-value for a source (e.g., Factor A or the AB interaction) is less than your chosen significance level α (commonly 0.05), you reject the null hypothesis for that source. This means there is statistically significant evidence that the factor or interaction has an effect on the dependent variable. A p-value above α suggests no significant effect was detected.

The full model is: Y_ijkl = μ + A_i + B_j + C_k + (AB)_ij + (AC)_ik + (BC)_jk + (ABC)_ijk + ε_ijkl, where μ is the grand mean, A_i, B_j, C_k are the main effects, the parenthesized terms are interaction effects, and ε_ijkl is the random error term.

A balanced design has the same number of observations (replications) in every cell (every combination of factor levels). This calculator assumes a balanced design. Balanced designs simplify computation and ensure that the sum-of-squares partitioning is orthogonal, meaning the effects are independent and the ANOVA results are straightforward to interpret.

The F-statistic for each effect is calculated by dividing its Mean Square (MS = SS / df) by the Mean Square Error (MSE) from within-cell variation. A large F-statistic indicates more variance is explained by the effect than by random error. The corresponding p-value is obtained from the F-distribution with the appropriate numerator and denominator degrees of freedom.