The effect size measure for a two-way between-groups ANOVA is R², the same as for a one-way ANOVA.

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R² is a measure of the proportion of variance in the dependent variable that is explained by the independent variables. In a one-way ANOVA, there is only one independent variable, so R² measures the proportion of variance in the dependent variable that is explained by that variable. In a two-way between-groups ANOVA, there are two independent variables, so R² measures the proportion of variance in the dependent variable that is explained by both variables together.

The Effect Size Measure for a Two-Way Between-Groups ANOVA Is