Can interaction be statistically significant while only one of two main effect are significant? I specifically want to know if I can report statistically significant interaction in a regression when one of the main effects hasn't reached the significance level
Plot the interaction, if you can, to attempt to understand it. From the plot, and by examining the coefficient of the interaction term, you might be able to tell if its statistical significance is there simply because you have a very large sample size, which causes almost anything to be statistically significant.
If this is the case, and if your research is exploratory, you may want to delete the interaction term from the model. Deleting it might cause the non-significant main effect to become statistically significant.
Either way, an examination of the interaction plots should help you decide whether the interaction is "impressive" or not.
Additionally, you do not need to center any variables unless centering the variable helps your interpretation. Since you can get a result (without centering), then centering the variables is a waste of time. See the following on why centering is irrelevant:
Consider the following example in R
dat = data.frame(fac = factor(rep(c(1,2), each = 10)), x = rep(seq(11,20), 2), y = c(2*seq(11,20) + 1, rev(2*seq(11,20)+1))) #our data looks like this png("example.png") plot(y ~ x, data = dat, col = fac, pch = 16) legend("top", c("1", "2"), col = 1:2, pch = 16, horiz = TRUE) dev.off()
It's clear there's no relationship between 'x' and 'y' if we don't take the factor 'fac' into account, so let's model that:
model = lm(y ~ x*fac, data = dat, contrasts = list(fac = c(-1,1))) #> coef(summary(model)) # Estimate Std. Error t value Pr(>|t|) #(Intercept) 3.20000e+01 1.091961e-14 2.930507e+15 2.850856e-239 #x -1.21003e-16 6.926982e-16 -1.746836e-01 8.635195e-01 #fac1 3.10000e+01 1.091961e-14 2.838928e+15 4.737922e-239 #x:fac1 -2.00000e+00 6.926982e-16 -2.887260e+15 3.616462e-239 #Warning message: #In summary.lm(model) : essentially perfect fit: summary may be unreliable
And so we uncovered the data generating process: no mean effect of 'x' alone, but an interaction with 'fac' instead.
Bottom line is the interaction is significant (we designed it to be so!) even if only a single main effect is significant.
We could even re-parametrize our problem in a way both 'x' and 'fac' are not significant, yet 'x:fac' is!
As I hinted before, we could make all other effects non-significant simply changing offsets in the data. Here's an example:
model = lm(scale(y) ~ scale(x)*fac, data = dat, contrasts = list(fac = c(-1,1))) # summary(model) # # Call: # lm(formula = scale(y) ~ scale(x) * fac, data = dat, contrasts = list(fac = # c(-1, 1))) # # Residuals: # Min 1Q Median 3Q Max # -5.703e-17 -2.652e-17 1.299e-18 2.777e-17 5.785e-17 # # Coefficients: # Estimate Std. Error t value Pr(>|t|) # (Intercept) 0.000e+00 8.815e-18 0.000e+00 1 # scale(x) 0.000e+00 9.044e-18 0.000e+00 1 # fac1 0.000e+00 8.815e-18 0.000e+00 1 # scale(x):fac1 -1.000e+00 9.044e-18 -1.106e+17 <2e-16 ***
Yes, main effects can be small or 0 while the interaction is large. Make sure in your tests of your main effects you either leave out the cross product or center the variables before multiplying. When a cross product is in the model, a "main effect" is actually a simple effect at the level of 0 for the other independent variable.
The short answer is: yes, you can include the significant interaction to the model even if one of the main effects is not significant. However, you should ask two questions before that:
- Is this interaction strong enough or it can be ignored?
- Does the interaction make sense, can you find a reasonable explanation for it?
The answers to those questions should help you decide what to do not just with the interaction, but all the terms under consideration.