# LMM: fixed effect significant in complex model, but not in reduced model

I constructed two models with lme4::lmer:

decomposition ~ trait1 + trait2 + trait3 + (1|pair)

all trait effects are highly significant

yet when I run this simplified model:

decomposition ~ trait1 + trait2 + (1|pair)

the effect of trait1 is not significant, trait2 is significant.

How is this possible and what can I conclude about the effect of trait1? I have 50 observations, if that matters. Can this be due to model assumptions that are not met? I visually checked for homoscedasticity and did a formal test for normality of residuals, which appear to be fine though.

• FWIW this question is not at all specific to mixed models. The phenomenon is most likely still there even if you drop the random effect (i.e. use lm() to do a regular linear model). I believe trait3 is a confounder, i.e. a variable that if not controlled for obscures the effect of trait1. I bet there are lots of questions on this site that deal with variants of this question, but I'm not quite sure how to search for them. Commented Aug 6, 2020 at 21:36
• @BenBolker is likely correct. Read up on multicollinearity. Commented Aug 6, 2020 at 23:52
• I believe that multicollinearity would typically work in the opposite direction (i.e. traits would be non-significant in the full model but become significant in the reduced model?) I may not have used the "confounder" terminology correctly ... Commented Aug 6, 2020 at 23:56

## 1 Answer

This looks like you have a problem with multicollinearity: trait1 and trait3 are correlated.

You can imagine creating such a scenario like so:

• Find a covariate (we'll call it trait.unseen) that is useful for predicting the response;
• Create a variable that is just noise (trait1 <- rnorm());
• Create a second variable that is the combination of these: trait3 <- trait1 + trait.unseen.

Then trait3 is a noisy estimate of a useful variable and trait1 is used to eliminate the noise in trait3. Without trait3, trait1 is not useful. Thus a model with trait1 and trait3 will show both are significant while a model with only trait1 will show it as not significant.

• Thanks. I did not know that trait3 could make trait1 appear significant in that way. It seems counterintuitive that just noise can eliminate noise in trait3. Commented Aug 7, 2020 at 9:04
• It has to be the same or highly correlated noise, but yes it can happen. Commented Aug 7, 2020 at 15:05