R lmerTest step() function returns significant random effect without equivalent significant fixed effect

Question

I do have a 2 level data set with 3 observations nested in one person. I am fitting a mixed model including 71 predictors and 28 random slopes in the following manner:

model = lmer(var1 ~ a + b + c + (1|PersID) + (0+a|PersID) + (0+b|PersID) + (0+c|PersID)

I am using the step() function of the lmerTest package to do backwards elimination of random and fixed effects of the model and get the following model:

Linear mixed model fit by REML 
t-tests use  Satterthwaite approximations to degrees of freedom ['merModLmerTest']
Formula: Fufaksc1 ~ MusZ + HauZ + ArbZ + SpoZ + AusZ + TraZ + AAYYZ +  
AMYZ + AMSZ + TNAZ + AuZ + MusM + HauM + EmoM + SpoM + TraM +  
AMNM + WSM + AuM + SAE + Ex + RE + So + UC + (1 | PersID) +  
(0 + HauZ | PersID) + (0 + SpoZ | PersID) + (0 + VaZ | PersID) + (0 + AuZ | PersID) Data: Sitsort2

REML criterion at convergence: 2703.1

Scaled residuals: 
Min      1Q  Median      3Q     Max 
-3.6274 -0.4721  0.0070  0.4884  4.1122 

Random effects:
 Groups   Name        Variance Std.Dev.
 PersID   (Intercept) 0.259072 0.50899 
 PersID.1 HauZ        0.088366 0.29726 
 PersID.2 SpoZ        0.285073 0.53392 
 PersID.3 VaZ         0.008581 0.09263 
 PersID.4 AuZ         0.008177 0.09043 
 Residual             0.209756 0.45799 
Number of obs: 1300, groups:  PersID, 555

Fixed effects:
             Estimate Std. Error        df t value Pr(>|t|)    
(Intercept)  -1.00698    0.19540 515.50000  -5.154 3.65e-07 ***
MusZ          0.55662    0.07930 627.30000   7.019 5.80e-12 ***
HauZ         -0.26976    0.06780 456.30000  -3.979 8.06e-05 ***
ArbZ          0.30543    0.07223 704.20000   4.229 2.66e-05 ***
SpoZ         -0.45474    0.09816 130.70000  -4.633 8.62e-06 ***
AusZ          0.17835    0.08102 653.20000   2.201 0.028067 *  
TraZ         -0.10944    0.05563 648.90000  -1.967 0.049574 *  
AAYYZ        -0.20878    0.05301 769.00000  -3.939 8.93e-05 ***
AMYZ          0.10063    0.04699 767.70000   2.141 0.032550 *  
AMSZ          0.80907    0.16299 501.90000   4.964 9.49e-07 ***
TNAZ         -0.09069    0.04457 719.90000  -2.035 0.042261 *  
AuZ           0.06479    0.01266 455.80000   5.118 4.57e-07 ***
MusM          0.49393    0.18577 522.40000   2.659 0.008082 ** 
HauM         -0.47557    0.14806 517.50000  -3.212 0.001401 ** 
EmoM         -0.63551    0.30854 490.60000  -2.060 0.039950 *  
SpoM         -0.88607    0.20915 519.20000  -4.236 2.69e-05 ***
TraM         -0.35308    0.14697 513.00000  -2.402 0.016645 *  
AMNM         -0.41692    0.20316 530.50000  -2.052 0.040644 *  
WSM           0.04994    0.01685 530.80000   2.964 0.003172 ** 
AuM           0.09425    0.02577 527.50000   3.657 0.000281 ***
SAE           0.15065    0.02430 515.10000   6.198 1.17e-09 ***
Ex           -0.05206    0.02576 519.20000  -2.021 0.043836 *  
RE            0.07022    0.02686 514.70000   2.614 0.009201 ** 
So            0.09297    0.02611 523.20000   3.561 0.000404 ***
UC           -0.11635    0.02710 510.80000  -4.293 2.11e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The model contains 5 significant random effects including the intercept. The only thing I do not understand is why there is a significant random effect for VaZ but no significant fixed effect for VaZ included in the model. As far as I understand this it would only make sense if the fixed effect would be exact 0 (which is realisticly impossible). From my point of view it does not make sense to include a random effect without an equivalent fixed effect. Can anyone explain this to me or is it a bug in the step() function?

Thanks in advance!

Answer 1

There is no reason, in theory, why you can't have a positive random effect and a 0 fixed effect. It would mean that the average slope was 0 (flat line) and the slopes for each person vary around this. It might make no physical sense in your use case, but that's another matter.

Furthermore, the non-significance of the test does not mean that the fixed effect is 0 -- only that you can't prove that it's not with this method and these data.

Your fixed effect might also not be significant because it is highly correlated with another fixed effect. You're basically trying to do model selection using Wald statistics, and that's a method known to produce ambiguous results.

I'm sure there isn't a bug in lmerTest, but it might not be the right tool for the job. It's an approximate test whose results, I feel, would be hard to interpret with a model as complex as this one.

If it were me, I would first get the children off the street. Average the response over each person and do a fixed-effects only model on those variables that are applied at the person level. I would then attempt to get rid of variables that had no impact -- possibly using a lasso, or a best subsets. Then I would model a random effects model that included the within-person variables and the between person variables that I had selected in the previous step. This approach is not guaranteed to produce the best model, but you're going to have to simplify something here if you don't want to get lost in all the variables.

Another problem here is that lmerTest basically relies on classical likelihood theory to work its magic. But if you read the fine print, you can see that this theory assumes the maximum is at an interior point of the parameter space. This assumption fails when you are testing whether a random effect is zero ... because if it is, the true solution is on a boundary of the parameter space. So you want to be sure in advance that the random effects included in the model are actually greater than zero.