marginsplot with Simpson's paradox

[Fred Lee]

Accoding to three-way interaction, the moderator c should play a negative moderating role, namely strengthening the negative relationship of a#b; however, the marginsplot shows it weakens the negative relationship on a#b.
I find it due to the the unblanced sample of c, in other words, Simpson's paradox.
According to my reading, the coefficient of three-way interaction is reliable, so how to figure this?
Note: the resut is apperaed in my large sample, I cannot replicate it in sub sample, so I have to use the screenshot to illustrate.

[Clyde Schechter]

Accoding to three-way interaction, the moderator c should play a negative moderating role, namely strengthening the negative relationship of a#b; however, the marginsplot shows it weakens the negative relationship on a#b.

No, this is a misinterpretation of the way three-way interactions work.

When you go from c = 0 to c = 1, you not only bring the a#b#c term into play, you also bring the a#c and b#c terms into play, changing the effects of a and b. What you see in the plot is the net effect of all three of these effect modifications by c, not just that corresponding to the a#b#c coefficient. As it happens, in your case, the combined results of these three effect modifications induced by c turns out to be opposite in direction to that of just the a#b#c coefficient.

I don't think this is a Simpson's paradox effect.

[Fred Lee]

Thanks Clyde! However, I want to examine what c plays in the relationship of a#b on Y. Before the three-way interaction, I have examined two-way interaction a#b, and the coefficient is negative. Combining the coefficient of a#b#c, I want to argue that c strengthens the negative relationship of a#b on Y. What I learned to figure out what moderating role does c play is according to coefficient of a#b, and a#b#c. Do you mean that only considering a#b, and a#b#c, cannot predict the role c plays? I have to use figure to illustrate?
By saying Simpson's paradox effect, I found if I use split sample analysis, I found for c=1, the coefficient of a#b is less negative than c=0, visually contrasting with the coeffcient of a#b#c.

[Fred Lee]

Or did I misunderstand this moderating figure? I need to compare the differences between blue line vs red line (for b=0, from c=0 to 1, the slope increases significantly) and green line vs yellow line (for b=1, from c=0 to1, the slope increases slowly). Does this still mean the c strengthen the negative relationship of a#b on Y?

[Clyde Schechter]

The problem is that when c changes from 0 to 1, in addition to "directly" affecting the interaction of the a and b effects, the a and b effects themselves also change--and it is not possible to keep this from happening in the interaction model. So the coefficient of a#b#c is simply only one piece of a three-piece puzzle. As you put it in #3, you cannot say what the impact of c on the a#b interaction will be by only considering a#b and a#b#c.

Added: I also think you are misreading your graph. When c = 0, the a#b interaction is represented in your graph by the difference between the slopes of the b = 0 c = 0 line and the b = 1 c = 0 line. The former is very close to zero, perhaps barely negative, and the latter is appreciably negative. So the difference is appreciably negative minus roughly 0, hence appreciably negative, which is also reflectd in thenegative coefficient of a#b in your regression output. When we go to c = 1, the a#b interaction is now represented by the difference between the slopes of the b = 0 c = 1 line and the b = 1 c = 1 line. This time, the former is a pretty steep positive, and the latter is almost the same as that of the earlier b = 1 c = 0 line, appreciably negative, and perhaps this time a small bit more so. So how has this difference changed? It is now appreciably negative minus large positive. So the difference is now even more negative than it was before: it has decreased, which is completely consistent with the negative sign of the a#b#c coefficient.

[Fred Lee]

Thanks Clyde! I want to clarify that am I correct? Accodrding to the figure, can we say c strengthens the negative relationship of a#b on Y?

[Clyde Schechter]

I think #6 probably crossed with my "Added:" on #5.

So, yes, c going from 0 to 1 makes the negative interaction of and b even more negative. If you want to use the word "strengthen" to describe that, that would be fair. I would not speak of any relationship between a#b and Y, however. a#b represents how a modifies the relationship of b to y, or, equivalently, how b modifies the relationship of a to y. a#b itself does not have a relationship of its own to y.

[Fred Lee]

Thnaks for the clarification, Clyde! Now I understand the three-way interaction graph better!