Overview of the ANOVA Table
| Source | SS | df | MS | F | P value |
|---|---|---|---|---|---|
| Treatment A | SS_A | a − 1 | SS_A / df_A | MS_A / MS_Residual | From F distribution |
| Treatment B | SS_B | b − 1 | SS_B / df_B | MS_B / MS_Residual | From F distribution |
| Interaction A×B | SS_A×B | (a−1)(b−1) | SS_A×B / df_A×B | MS_A×B / MS_Residual | From F distribution |
| Residual (error) | SS_Error | ab(n−1) | SS_Error / df_Error | — | — |
| Total | SS_Total | abn − 1 | — | — | — |
a = number of levels of Factor A, b = number of levels of Factor B, n = number of replicates per cell.
Column-by-Column Explanation
SS (Sum of Squares)
SS quantifies the total variability attributable to each source. SS_Total is the sum of squared deviations of all observations from the grand mean. The ANOVA partitions this total into pieces: SS_A (attributable to Factor A), SS_B (attributable to Factor B), SS_A×B (the interaction), and SS_Residual (random error). All these pieces sum to SS_Total.
df (Degrees of Freedom)
Each SS is associated with a number of degrees of freedom. For Factor A with 'a' levels, df = a − 1. For the interaction, df = (a−1)(b−1). For the residual with n replicates per cell, df = ab(n−1). The df values also sum to the total df (abn − 1).
MS (Mean Square)
MS = SS / df. Dividing by df converts the sum of squares into a variance estimate. The MS Residual is the pooled within-cell variance and estimates the random error variance. MS for each factor estimates random error plus any systematic effect of that factor.
F ratio
F = MS_Factor / MS_Residual. If a factor has no effect, MS_Factor ≈ MS_Residual and F ≈ 1. A large F ratio indicates that the factor explains more variance than expected from random error alone. Separate F ratios are computed for Factor A, Factor B, and the interaction.
P value
The P value is the probability of observing an F ratio this large (or larger) if the null hypothesis is true — that the factor has no effect. Prism calculates P values from the F distribution using the numerator df (for the factor) and the denominator df (for the residual).
Repeated Measures Variations
In repeated measures two-way ANOVA, the ANOVA table gains a Subject (or matching) row. The within-subject variability is further partitioned into subject-by-treatment interactions. This reduces the error term and can increase statistical power when subjects are matched over time or within experiment.
Prism note: Prism clearly labels whether the analysis is ordinary two-way ANOVA, repeated measures on one factor, or repeated measures on both factors. The structure of the ANOVA table changes with each — check the source column labels carefully.