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I have trouble interpreting interaction plots when there is an interaction between the two independent variables.

The following graphs are from this site:

Here, $A$ and $B$ are the independent variables and $DV$ is the dependent variable.

Question : There is interaction and main effect of $A$, but no main effect of $B$

enter image description here

I can see that the higher the value of $A$, the higher the value of $DV$, provided B is at $B_1$ otherwise, $DV$ is constant regardless of the value of $A$. Therefore, there is an interaction between $A$ and $B$ and main effect of $A$ (since higher $A$ leads to higher $DV$, holding $B$ constant at $B_1$).

Also, I can see that different levels of $B$ will lead to different levels of $DV$, holding $A$ constants. Therefore, there is main effect of B. But this apparently is not the case. So, this must mean I am wrongly interpreting the interaction plot. What am I doing wrong?

I am also wrongly interpreting plot 6-8. The logic I used to interpret them is the same as the one I used above so I if I know the error I am making above, I should be able to correctly interpret the rest. Otherwise, I will update this question.

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    $\begingroup$ How would you define "the main effect of B", knowing that there's an interaction between A & B? $\endgroup$ – Scortchi - Reinstate Monica Jun 3 '14 at 16:44
  • $\begingroup$ The logic you're using to interpret is implicit. If you're wrongly interpreting 6-8 perhaps add your incorrect interpretations into your question. BTW, your interpretation of the present graph isn't of the interaction per se but a description of the data through which you're inferring an interaction. Is the question really, "what about these graphs leads to the accompanying description?" (i.e. main effect and interaction) $\endgroup$ – John Jun 4 '14 at 17:23
  • $\begingroup$ @John Yes, the question I really wanted to ask was "what about these graphs leads to the accompanying description? (Plot 5 to Plot 8)" $\endgroup$ – mauna Jun 9 '14 at 17:25
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You're interpreting the individual points on the graph and calling that the interaction but it's not. Taking the example you provided, imagine how your description of the interaction would go if the main effect of A were much larger. Or perhaps if it was much smaller, or even 0. Your description would change but that main effect should be independent of the interaction. Therefore, your description is of the data but not the interaction per se.

You need to subtract out main effects to see just the interaction. Once you do that then ALL 2x2 interactions look like the last one on the page you reference, a symmetric "X". For example, in the linked document there is a data set

    A1 A2
B1   8 24
B2   4  6

There are clearly main effects in the rows and columns. If those are removed you can then see the interaction (think of the matrices below being operated on simultaneously).

8 24 -  10.5 10.5 -  5.5  5.5 -  -4.5 4.5 =  -3.5  3.5
4  6    10.5 10.5   -5.5 -5.5    -4.5 4.5     3.5 -3.5

(The subtracted matrices above can be calculated as the deviations from the grand mean expected based on the marginal means. The first matrix is the grand mean, 10.5. The second is based on the deviation of row means from the grand mean. The first row is 5.5 higher than the grand mean, etc.)

After the main effects are removed then the interaction can be described in effect scores from the grand mean or the reversing difference scores. An example of the latter for the example of above would be, "the interaction is that the effect of B at A1 is 7 and the effect of B at A2 is -7." This statement remains true regardless of the magnitudes of the main effects. It also highlights that the interaction is about the differences in effects rather than the effects themselves.

Now consider the various graphs at your link. Deep down, the interaction is the same shape as described above and in graph 8, a symmetric X. In that case the effect of B is in one direction at A1 and the other direction at A2 (note that your use of increasing A in your description suggests you know A isn't categorical). All that's happening when the main effects are added is that those shift around the final values. If you're just describing the interaction then the one for 8 is good for all of the ones where the interaction is present. However, if your plan is to describe the data then the best way is to just describe the effects and difference in effects. For example, for graph 7 it might be: "Both main effects increase from level 1 to 2, however the interaction causes a pattern of data where there is no effect of B at A1 and a positive effect at A2."

That's a concise accurate description of the data, data where an interaction is present, that contains no actual description of the interaction per se. It's a description of how the main effects are modified by the interaction. Which should be sufficient when no numbers are supplied.

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When an interaction effect exists between two factors, it no longer makes sense to talk about main effects. There is no main effect, for the sort of considerations that you mention in your post. You've got the point: you only know the effect of a level of B if you also know the level of A -- so, no main effects.

In the graph above, if there were main effects, but no interaction, your two lines would be parallel.

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    $\begingroup$ This is relative. Large main effects relative to the interaction, especially when the independent variables have genuinely limited scales (like sex variables) most definitely are meaningful even if there is an interaction. $\endgroup$ – John Jun 3 '14 at 20:11
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    $\begingroup$ My professor always emphasises that: once you've determined that the interaction effect is significative, you shouldn't interpret the main effect by itself anymore. I find it similar to running a model with a significative quadratic term, it's of no use interpreting the quadratic term by itself in the context of the problem (save for describing mathematical properties of the solution, say "the curve shifts downward because of the sign of the parameter attached to the quadratic term"). $\endgroup$ – mugen Jun 3 '14 at 21:32
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    $\begingroup$ Mugen, the magnitude of the main effect may be qualified by an interaction without ever qualifying the existence of said main effect. Placidia, I was simply qualifying your opening statement. It's not at all difficult for a main effect to be large enough with bounded variables that the interaction doesn't make the main effect ever go away and therefore nullify that first sentence. $\endgroup$ – John Jun 3 '14 at 23:14
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    $\begingroup$ @John it's not that the main effect "goes away". Rather, it is qualified in its impact. Consequently, I can't say that the main effect of A is, say, 42, unless I also know the level of B. Now if the interaction is small relative to the effects, the impact of A when B=0 might be $42+\epsilon$, and when B=1, it might be $42-\epsilon$, but I'm a mathematician and to me, an $\epsilon$ means something. $\endgroup$ – Placidia Jun 4 '14 at 2:27
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    $\begingroup$ Sure it does Placidia, but your comment does not support your opening sentence. It's a main effect who's magnitude varies, but still a main effect. $\endgroup$ – John Jun 4 '14 at 4:29
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If your model predicts a response $Y$ from predictors $x_1$ & $x_2$, the expected response is given by

$$\operatorname{E} Y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1 x_2$$

If the coefficients $\beta_1$ & $\beta_2$ are what you're calling "main effects" then note that, say, $\beta_1$ gives the change in $\operatorname{E} Y$ when $x_1$ changes by one (unit of whatever it's measured in) and when $x_2=0$. It's not always—indeed not often—the case that this quantity is of particular interest: if $x_2$ is temperature, the meaning of zero will depend on the arbitrary choice to measure it in either Celsius or Fahrenheit, if it's sex then the meaning of zero will depend on the arbitrary choice to use either male or female as the reference category; and therefore the "main effect" of $x_1$ depends on an arbitrary choice. Sometimes people code or translate predictors just in order for these parameters to have fairly sensible interpretations, which is fair enough, but this makes no substantive difference to the model—to its predictions or likelihood. @John's example corresponds to using -1 to code $A_1$ & $B_1$, & 1 to code $A_2$ & $B_2$: then $\beta_0$ is the grand mean over all four combinations of $A$ & $B$, $\beta_1$ the difference between the mean response for $A_2$ over both levels of $B$ and the grand mean, & so on.

I suspect that in the graph you show you're expected to assume, or have elsewhere been told, that a zero-value for $A$ lies mid-way between $A_1$ & $A_2$; at that precise point only moving from $B_1$ to $B_2$ makes no difference to the response.

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For the sake of intuitive simplicity, pretend this isn't a statistical problem, but just a mathematical problem. Say that the "data" include every single point exactly on those lines in your example, so that the task is to describe those lines entirely as functions of A and B. Arguably, this is in fact the case, and there's no pretending necessary, because your example provides no info about standard error or residuals. Then, assuming B1 bisects B2 perfectly, and that (B1,A2) is exactly as far above (B2,A2) as (B1,A1) is below (B2,A1), and ignoring the dashes (i.e., filling them in, basically)...

Half the points on B1 are above B2, and half are below, and their differences effectively cancel out. This means that DV(B1) = DV(B2) when averaging across all values of A. Yes, if you hold A constant at A1 or A2, B1 and B2 will differ, but since the differences are equal and opposite at opposite values of A, there is no main effect of B. Differences in DV(B) that depend on values of A are described entirely by the interaction effect. Similar logic can be applied to plots 6–8 to arrive at the intended conclusions.

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