Suppose we have two factors A and B. What does an A x B interaction mean in a two way ANOVA?

In a two-way ANOVA, an A x B interaction means that the effect of factor A on the dependent variable is not the same at all levels of factor B. In other words, the effect of A depends on the level of B.

Here are the steps to understand this:

1. Two-way ANOVA is a statistical test used to determine the effect of two nominal predictor variables (in this case, A and B) on a continuous outcome variable.

2. The A x B interaction (also known as the interaction effect) is the effect on the outcome variable that arises from the combination of factors A and B.

3. If there is a significant interaction effect, it means that the effect of factor A on the outcome variable differs depending on the level of factor B.

4. For example, let's say factor A is a type of treatment (with levels: treatment and no treatment) and factor B is gender (with levels: male and female). A significant interaction could mean that the treatment has a different effect in males than in females.

5. To determine if there is an interaction effect, you would look at the p-value for the interaction term in the ANOVA table. If the p-value is less than your chosen significance level (often 0.05), you would conclude that there is a significant interaction effect.

6. If there is a significant interaction, it is important to probe the interaction to understand how the levels of factor A affect the outcome variable at each level of factor B. This is often done through simple effects analysis or plotting the interaction.