I'm reading an online guide on two-way ANOVA and it says here we do not interpret the main effects if the interaction term is significant.
https://online.stat.psu.edu/stat502/lesson/4/4.1/4.1.1
Why is this the case? Would we not utilize the coefficients of the main effects in conjunction with the coefficient of the interaction term for final interpretation? For example, could we not say, given that factor 2 is _____, the deviation from the grand mean for factor 1 is ______. You get the final deviation from the grand mean by adding the two terms together.
Thanks!
Best Answer
Suppose that we have the following regression relationship:
$y=\beta_0 + \beta_1 X + \beta_2 Z + \beta_3 X \times Z + \varepsilon$.
If there is no the interaction term, i.e., $y=\beta_0 + \beta_1 X + \beta_2 Z + \varepsilon$, we can interpret the main effect as usual: "Keeping other variable, changing one unit in $X$ associates with $\beta_1$ units in $Y$".
That is not true if there is the interaction term. It is because the effect of $X$ depends on the value of $Z$ (through the interaction). Indeed, we can re-write the first formula as follows:
$y=\beta_0 + \beta_2 Z + (\beta_1 + \beta_3 Z) X + \varepsilon$.
Now we see that the coefficient of $X$ is $(\beta_1 + \beta_3 Z)$. After fixing $Z$ at a known value, we can interpret the effect of $X$ as usual. For example, with $Z=1$, the effect is represented by $\beta_1 + \beta_3$. Please note that significance of $\beta_1$ and $\beta_3$ does not guarantee a significant effect of $X$ (with $Z=1$). We need to test for the sum of those coefficients in this case.