To draw accurate conclusions from experiments involving multiple factors, you need to understand not just the main effects but also the interaction effects between those factors. Interaction effects occur when the effect of one factor depends on the level of another factor. A key tool for interpreting these relationships is the interaction plot.

An interaction plot typically displays the mean outcome (such as a response variable) on the y-axis, with one factor on the x-axis and different lines or colors representing levels of the second factor. By examining the shape and pattern of these lines, you can determine whether the effect of one factor changes depending on the other.

• If the lines are parallel, there is likely no interaction — the effect of one factor is consistent across the levels of the other;
• If the lines are not parallel, this suggests an interaction;
• The most striking case is when the lines cross or diverge, indicating that the direction or magnitude of one factor's effect reverses or shifts depending on the other factor's level.

This can reveal complex relationships that would be missed if you only looked at main effects in isolation.

To further illustrate the impact of interaction effects, consider the following table. It shows hypothetical outcomes for two factors, A and B, each with two levels (A1, A2; B1, B2). The first table represents a scenario without interaction, while the second includes a strong interaction. Notice how the conclusions about the effect of factor A change depending on the presence of interaction.

No interaction: the effect of moving from A1 to A2 is always +5, regardless of B.

With interaction: the effect of A changes depending on B. For B1, A2 is much higher than A1 (+15); for B2, A2 is lower than A1 (-10). This means the effect of A reverses depending on the level of B. Without considering interaction, you might miss this critical insight and make incorrect recommendations.