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A two-way ANOVA compares three or more sample means categorized by two factors: a row factor and a column factor. Unlike a one-way ANOVA, it tests whether both factors independently affect the data and whether they interact simultaneously. This allows researchers to understand complex relationships between multiple variables in a single analysis.

An interaction effect is visualized by plotting line segments that connect mean values for each factor. If the line segments are parallel, no interaction exists between the factors. If they are not parallel, the two factors simultaneously affect the data values. Calculating the F statistic and P-value confirms whether this interaction is statistically significant.

Failing to reject the null hypothesis means the computed P-value exceeds the chosen significance level, such as 0.05. This indicates insufficient evidence that the factor or interaction effect significantly influences the data. For example, if age shows no substantial effect on mean height, the null hypothesis for age is not rejected.

After testing for interaction, the two-way ANOVA separately evaluates each factor by stating individual null hypotheses and calculating F statistics and P-values for the row factor and column factor independently. If a factor's P-value is lower than the significance level, that factor significantly affects the data values and the null hypothesis is rejected.

The row factor and column factor are the two independent variables organizing the data into categories. For example, when comparing height across age groups and gender, age is the row factor and gender is the column factor. Both factors are tested for their independent effects and their combined interaction effect on the dependent variable.

The P-value determines whether results are statistically significant by comparing it to a chosen significance level, typically 0.05. If the P-value is greater than this threshold, the null hypothesis is not rejected, suggesting no significant effect. If the P-value is lower, the null hypothesis is rejected, indicating a significant effect of the factor or interaction.

A two-way ANOVA is used when you need to simultaneously test two factors and their interaction effect on a dependent variable. Multiple comparison tests are post-hoc procedures applied after ANOVA to identify which specific groups differ. Use two-way ANOVA first to determine if factors significantly affect your data before conducting multiple comparison tests.