Factor Levels in a Two-Way ANOVA

Question

I'm trying to figure out the explanations of the Effect Plots below. I believe I have figures out Figure 9a, but the explanation for Figures b, c, and d given in the textbook I do not comprehend.

Any further detail would be greatly appreciated.

Figure 9(a) shows that the means for Factor $A$ are equal (because the three dots are on the same horizontal line), but the response differs for the two levels of Factor $B$ (because the lines are one above the other). Thus, there is no effect of Factor $A$ on the response, but a Factor $B$ main effect is present. In Figure 9(b), the Factor $A$ means differ, but the Factor $B$ means are equal at each level of Factor $A$ (WHY?????). Here a Factor $A$ main effect is present, but no effect of Factor $B$ is present. Figure 9(c) and Figure 9(d) illustrate cases in which both factors affect the response. In Figure 9(c), the mean response between levels of Factor $A$ does not change for the three levels, so the effect of Factor $A$ on the response is independent of Factor $B$. That is, the two factors do not interact. In contrast, Figure 9(d) shows that the differences between mean distances between levels of Factor $B$ varies with the levels of Factor $A$. Thus, the effect of Factor $A$ on the response depends on the levels of Factor $B$, and the two factors do interact.

Answer 1

You have (a) correct! The key thing to understand here is that the effects of A and B will look totally different, because they are represented by different things in the diagrams. Moving a dot higher or lower means that that group mean is higher or lower. Factor A is split into three points along the x-axis, so a difference between the means across those will mean that the dots on a given line will be moved up or down relative to each other. I.e. if the line isn't flat: you have an effect of factor A.

Factor B on the other hand is represented by splitting the graph into two lines. So an effect of Factor B will be seen as moving the whole of one line higher or lower than the other.

It might make more sense to you if (b) had been drawn so that the two lines were directly on top of each other? That is the purest version of "no main effect of B" -- the two lines land in exactly the same place (neither shifted up or down relative to the other so that we don't really need two lines at all, because B has no effect). But in reality the lines never land exactly on top of each other (two sample means never come out the same: that's why we need the ANOVA in the first place) and also it wouldn't be easy to draw it. But the purest version of (b) would have both lines smack bang on top of each other. That is to say, factor B is having no effect and isn't able to shift the line up or down.

On the other hand in (c), one line is shifted up relative to the other. So factor B is having an effect (shifting the whole line up).

But notice that in all three graphs so far, the lines are parallel? Because whatever jump up or down we get as we move along one line, we get the same jump or down on the other. Those jumps up and down are the effects of factor A. So, the effect of factor A is the same in both levels of B (i.e. on both lines).

On the other hand, (d) is not even parallel. There are jumps up and down as we move across the graph to different levels of A, but those jumps up and down are different on the different lines, so not only is there an effect of A, but it differs across the levels of B. So, basically, we have something more complicated altogether.