How to add a covariate in the linear mixed model

Question

I'm creating my LMM including three factors (A,B,C) as fixed effect, and D as the random intercept. At the same time, I also want to include a covariate (E) to test whether the covariate influence the dependent variables or not. I tried two models (see here below), I don't know which one is correct? or none of them?

model1 <- lmer(RT ~ A * B * C + E + (1|Subject))
model2 <- lmer(RT ~ A * B * C * E + (1|Subject))

In the output Table for model1, there is one row regarding the covariate E, but no information is related to the interaction between dependent variables (A,B,C) and E.

In the output Table for model2, there are several rows showing whether the interaction between the E and other dependent variables (A,B,C) is significant or not. But I'm not sure this is a correct way to test, as it seems more like that the covariate E is regarded as the dependent variable as well.

Answer 1

The answer depends on whether you are solely interested in testing "...whether the covariate influence the dependent variables or not." If so, then model1 is the winner. It allows for interactions between A, B, and C, while merely adjusting for the effect of E on the outcome. This would be a useful model if E were simply a variable that was not of substantive interest, but was associated with the outcome and you wanted to account for that association.

If instead E is of substantive interest, and in particular, you want to know whether the effect of A, B, and C vary depending on the levels of E, then model2 is your choice. Put differently (but equivalently), E is still a predictor in model2, but its effect on the outcome is moderated by each of the other predictors.