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Say I am plotting the following interaction: mpg ~ cyl*disp (from the mtcars dataset in R). If I am interested in the effect of disp would I make disp the moderator or the IV in the interaction plot to best show the effect of disp on mpg?

Would I then say:

Title: Conditional effect of cyl (IV) on mpg (DV) for values of disp (MODERATOR)? Or is it the other way around?

image of interaction

Short interpretation: The relationship of mpg to disp is influenced by cyl. At higher levels of disp the effect of cyl is increased?

R code:

data(mtcars)
lmres.model <- lmres(mpg~cyl*disp, data=mtcars)
S_slopes    <- simpleSlope(lmres.model, pred="cyl", mod1="disp")
PlotSlope(S_slopes) + ggtitle(paste("Conditional effect of cyl on mpg\nfor values of disp")) 

I am using APA style if that matters, I can't seem to find a rule there saying one way or the other though.

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The terms "moderation" and "interaction" are synonyms (cf., here). However, they indicate a subtle difference in how you are thinking about what you are modeling. Mathematically, neither variable included in an interaction has a different status, but often we are interested in the relationship between X and Y, and wonder if that relationship differs as a function of Z. In such cases, people sometimes refer to "moderation" instead of "interaction"; Z is the moderator here.

We usually try to visualize the relationship between two variables with a plot where we put the explanatory variable on the x-axis and the response on the y-axis. In this case, there are three variables, but you are still giving conceptual precedence to the relationship between X and Y. So it is most natural to put X on the x-axis, Y on the y-axis and then make distinct symbols or group them somehow to indicate Z. Referring to your concrete example, you are wondering about the relationship between engine displacement and fuel efficiency ("I am interested in the effect of disp"), but wonder if that relationship varies as a function of the number of cylinders the engine has. That suggests you put disp on the x-axis, mpg on the y-axis, and group the symbols by cyl.


I can demonstrate this using your example. (Note that I am taking this rather literally, which may or may not be relevant for whatever situation you really care about.) First of all, I would model the number of cylinders as a factor, not a continuous variable. Then I would use separate lines with distinct colors or types to indicate the different groupings. The existence of an interaction can be discerned by the lines not being parallel.

m          = lm(mpg~cyl*disp, mtcars)
mtcars$cyl = factor(mtcars$cyl, levels=c(4,6,8), labels=c(4,6,8))
mf         = lm(mpg~cyl*disp, mtcars)
AIC(m)   # [1] 159.1494
AIC(mf)  # [1] 153.4352

Despite the fact that using cyl as a factor consumes more degrees of freedom, the AIC is lower for the model that does so. At any rate, here is a plot:

seq.d4 = 70:150
seq.d6 = 140:260
seq.d8 = 270:475
windows()
  plot(seq.d4, predict(mf, newdata=data.frame(disp=seq.d4, cyl="4")), type="l", 
       xlim=c(70,475), ylim=c(5,35), xlab="disp", ylab="predicted MPG")
  lines(seq.d6, predict(mf, newdata=data.frame(disp=seq.d6, cyl="6")), col="blue")
  lines(seq.d8, predict(mf, newdata=data.frame(disp=seq.d8, cyl="8")), col="red")
  points(mpg~disp, mtcars, subset=mtcars$cyl=="4")
  points(mpg~disp, mtcars, subset=mtcars$cyl=="6", col="blue")
  points(mpg~disp, mtcars, subset=mtcars$cyl=="8", col="red")
  legend("topright", legend=c("4 cylinders", "6 cylinders", "8 cylinders"),
         pch=1, lty=1, col=c("black", "blue", "red"))

enter image description here

The interpretation would be: 'Fuel efficiency generally decreases as engine displacement increases, but the relationship varies as a function of the number of cylinders. Fuel efficiency decreases more rapidly with increasing displacement when the engine has 4 cylinders than for 6 or 8 cylinders.'

If you did want to model cyl as a continuous variable, you could use the mean of cyl, and the mean plus/minus one SD, or at the median and quartiles of cyl (see my answer here: How to visualize a fitted multiple regression model?).

This version adds confidence bands to the figure:

enter image description here


Update: It seems I misunderstood the research question the OP was trying to answer in the concrete example given.

In general, it is most natural to put the variable whose relation to the response you are interested in on the x-axis, so I have done that below. However, it is also true that the axes are better for continuous variables and grouping (e.g., by lines) is more consonant with categorical variables. Moreover, because the number of cylinders is highly correlated with the displacement, there are no 4 cylinder engines with large displacements in the dataset and the model's predicted value there is impossible. In addition, it is harder to combine the data with the predicted values coherently in this arrangement. For those reasons, the figures above might still be preferable. Nonetheless, I produce the alternate figure below. To do so, I computed the predicted values for the cylinder numbers at the mean of displacement, and plus/minus 1 SD.

md  = mean(mtcars$disp);                   md   # [1] 230.7219
mdH = mean(mtcars$disp) + sd(mtcars$disp); mdH  # [1] 354.6606
mdL = mean(mtcars$disp) - sd(mtcars$disp); mdL  # [1] 106.7832
ci = predict(mf, newdata=expand.grid(cyl=c("4","6","8"),disp=c(md,mdH,mdL)), 
             interval="confidence", level=.68)[,-1]

windows()
  plot(c(4,6,8), predict(mf, newdata=data.frame(disp=md, cyl=c("4","6","8"))), type="l", 
       ylim=c(0,30), xlab="number of cylinders", ylab="predicted MPG", xaxp=c(4,8,2))
  lines(c(4,6,8), predict(mf, newdata=data.frame(disp=mdH, cyl=c("4","6","8"))), col="blue")
  lines(c(4,6,8), predict(mf, newdata=data.frame(disp=mdL, cyl=c("4","6","8"))), col="red")
  arrows(x0=c(4,6,8), y0=ci[1:3,1], y1=ci[1:3,2], code=3, length=.1, angle=90)
  arrows(x0=c(4,6,8), y0=ci[4:6,1], y1=ci[4:6,2], code=3, length=.1, angle=90, col="blue")
  arrows(x0=c(4,6,8), y0=ci[7:9,1], y1=ci[7:9,2], code=3, length=.1, angle=90, col="red")
  legend("bottomright", legend=c("mean disp", "+1 SD disp", "-1 SD disp"), lty=1, 
         col=c("black", "blue", "red"))

enter image description here

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  • $\begingroup$ Sorry for the invading your answer, but I can't put images here. Would you consider adding confidence intervals or use my recreation(s). Whether to collapse 6 and 8 may be an unrelated question, but I'd be interested if you have any guidance since it supports your answer's stress on improving AIC. (My software, JMP, only calculates AICc so I'm not certain of the AIC values.) $\endgroup$ – xan Oct 2 '16 at 14:48
  • $\begingroup$ And in case it doesn't go without saying, feel free to delete or edit my additions as you see fit. $\endgroup$ – xan Oct 2 '16 at 14:55
  • $\begingroup$ It's OK to add these @xan; the confidence bands make the plot a little more cluttered (IMO), but add real information. FWIW, I have no problem w/ the AICc; I wouldn't typically combine 6 & 8 if it weren't a-priori, though. $\endgroup$ – gung - Reinstate Monica Oct 2 '16 at 16:10
  • $\begingroup$ @gung, I agree with your post, but I would like to point out that, I intended "interested in the effect of disp" to mean: I am interested in the effect of disp on the relationship between cyl and mpg. So the relationship varies as a function of disp $\endgroup$ – Rilcon42 Oct 2 '16 at 18:47
  • $\begingroup$ I see, @Rilcon42. In that case you do want cyl on the x-axis. I'll update this answer later today. $\endgroup$ – gung - Reinstate Monica Oct 2 '16 at 18:49
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Based on some research and a lot of thought I realized that the graph axes are correct. The graph shows a relationship between mpg and cyl then shows the change due to disp as separate lines. This highlights the effect of disp on the model best.

The title should be changed to: Conditional effect of disp on the relationship between cyl and mpg

Interpretation: The relationship between mpg and cyl is influenced by disp. At higher values of cyl disp has a smaller effect. (This can be seen because the points at X=8 are closer together than the points at X=3).

Resulting in the graph:

updated graph

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  • $\begingroup$ I disagree w/ the graph axes. I'm writing up an answer for you. $\endgroup$ – gung - Reinstate Monica Oct 2 '16 at 1:23
  • $\begingroup$ Fantastic, glad to hear it! $\endgroup$ – Rilcon42 Oct 2 '16 at 1:34
  • $\begingroup$ I now agree with these axes. I had misunderstood the question you were hoping to answer with these data. However, the model as you have fit it, & the plot as it was constructed, reflect the pattern accurately without quite showing the data / effects as they really are. Eg, the model's predicted value for 4 cyl +1SD disp is negative. I remain a little uneasy w/ this plot. $\endgroup$ – gung - Reinstate Monica Oct 3 '16 at 13:19
  • $\begingroup$ Would you edit your answer to explain why you dislike the plot I created, and propose a better alternative? I like your answer significantly better than mine, and if I had been clearer about my question in the first place I would have accepted your answer already. $\endgroup$ – Rilcon42 Oct 3 '16 at 13:42

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