Linear mixed model: Adding an "irrelevant" continuous predictor cancels effect of interest

Question

I have a strange situation with a linear mixed model I can't make any sense of.

I have an LMM specified with one categorical predictor (cat_pred; with the three levels A, B, C) and one dependent variable (DV) where I specified random intercepts for my participants (par). The model looks like the following:

lmer(DV ~ cat_pred + (1|par), data = mydata)

My cat_pred shows clear effects on the DV

Fixed effects:
                Estimate Std. Error       df t value Pr(>|t|)    
(Intercept)       3.1250     0.1541 114.0570  20.278  < 2e-16 ***
cat_predB        -1.2500     0.2052  78.0000  -6.092 3.96e-08 ***
cat_predC        -1.4250     0.2052  78.0000  -6.945 1.00e-09 ***

I now add a continuous predictor (con_pred) and want to see if there is an interaction between my cat_pred and con_pred. So I add it to the model by specifying an interaction.

lmer(DV ~ cat_pred*con_pred + (1|par), data = mydata)

When I add it to the model, all my significant effects vanish.

Fixed effects:
                           Estimate Std. Error        df t value Pr(>|t|)    
(Intercept)                 2.50835    0.46787 110.76817   5.361 4.56e-07 ***
cat_predB                  -0.62591    0.62043  76.00000  -1.009    0.316    
cat_predC                  -0.63014    0.62043  76.00000  -1.016    0.313    
con_pred                    0.01060    0.00759 110.76817   1.397    0.165    
cat_predB:con_pred         -0.01073    0.01006  76.00000  -1.066    0.290    
cat_predC:con_pred         -0.01366    0.01006  76.00000  -1.357    0.179

When I now run the model just with the con_pred I see, that it is not at all related to my DV.

Fixed effects:
             Estimate Std. Error        df t value Pr(>|t|)    
(Intercept) 2.090e+00  3.207e-01 1.180e+02   6.517 1.85e-09 ***
con_pred    2.470e-03  5.202e-03 1.180e+02   0.475    0.636 

So this made me curious and I started to try out the effect of different other continuous predictors that have no association to my DV. Every time I add them to the model (it does not depend which one I use) my cat_pred loses it's significance. I even added random continuous variables just to check it.

Is this a power problem? My sample is rather small with N=40.

Thanks a lot for your input!!!

Answer 1

In DV ~ cat_pred*con_pred, you have an interaction, so the main effects of cat_pred are estimated at a baseline of con_pred == 0, and the interaction terms, cat_predB:con_pred and cat_predC:con_pred, indicate how this effects change as you move away from this baseline.

In contrast, in DV ~ cat_pred + con_pred or plain old DV ~ cat_pred, the main effects of cat_pred are estimated across all the data.

Try plotting DV ~ con_pred at each different value of cat_pred to get a better sense of what's going on.

Update: Also, yes, that's a very small sample for a model with 6 terms.