# 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. – Ben Bolker Aug 6 at 21:36
• @BenBolker is likely correct. Read up on multicollinearity. – kurtosis Aug 6 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 ... – Ben Bolker Aug 6 at 23:56

• 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. – Sam Vanbergen Aug 7 at 9:04