# Can there be three significant pairwise interactions without a significant three-way interaction in a linear model?

Given a response variable y and three predictors A, B, and C, can all predictors have significant pairwise interactive effects on y without having a three-way interactive effect on it? In other words, given the linear model (expressed in R notation):

model1 <- lm(y ~ A + B + C + A:B + A:C + B:C + A:B:C)

can A:B, A:C, and B:C have a significant effect on the variability of y without A:B:C having a significant effect on it?

Or, to phrase it once again differently: given the following alternative model without three-way interaction

model2 <- lm(y ~ A + B + C + A:B + A:C + B:C)

is it possible for model1 to not have a residual variability that is significantly less than that of model2 if A:B, A:C, and B:C all significantly contribute to reducing the residual variability of model2?

This question has been asked before (e.g. here, here) but, as far as I can tell, it has not been answered on this platform (although this is relevant). I am posing the question in statistical terms but I believe it's more of a logical question.

• Yes, of course. If the true coefficient of the product of A, B and C is 0, you would even hope to not get a significance here. Commented Jun 10, 2023 at 12:14

It's pretty simple to simulate data and models to show that it is possible:

set.seed(1234)
# predictor values:
x1 <- rnorm(1000)
x2 <- rnorm(1000)
x3 <- rnorm(1000)
# simulated y values for a model with three pairwise interactions
# and no three-way interaction (x1*x2*x3):
y <- x1 + x2 + x3 + x1*x2 + x1*x3 + x2*x3 + rnorm(1000, 0, 10)

# a linear model which doesn't include a three-way interaction:
m1 <- lm(y~x1 + x2 + x3 + x1*x2 + x1*x3 + x2*x3)
summary(m1) # all are significant
# a linear model which does include a three-way interaction:
m2 <- lm(y~x1 + x2 + x3 + x1*x2 + x1*x3 + x2*x3 + x1*x2*x3)
summary(m2) # all but the 3-way interaction are significant

• thanks! My intuition got in the way of my reasoning. I thought that, for some sort of transitive property, if x1*x2, x1*x3 and x2*x3 all had a significant effect on y, so should x1*x2*x3. I didn't think of running a simulation. Commented Jun 13, 2023 at 15:42
• You're welcome. I think that where you might have gotten this idea is from the reverse: If you include an interaction, many people (including me) think you have to include the main variables; if you include a 3 way interaction, you have to include all the 2 ways. David Rindskopf has written about some exceptions to this, but they are rare. Commented Jun 13, 2023 at 19:18