[R-sig-ME] bootstrapped CIs for fixed effects in binomial model with random slopes

[Christoph Huber-Huber]

Dear list,

I've made a puzzling observation concerning the bootstrap distribution 
of fixed effects in a logistic model after incorporating random slopes. 
I'm wondering whether that is actually to be expected or whether my 
model speficification is problematic in some way.

I have a datset with a binomial response variable, a couple of 
categorical predictors and one grouping variable.
I estimate the model with lme4 by:

m10 <- glmer(dv ~ cr * inter + (cr | vpnr),
                   family = binomial(link = "logit"),
                   data = dat,
                   control = glmerControl(optCtrl = list(maxfun = 
1e+07), optimizer = "bobyqa"))

The model converges and the output seems to be reasonable.

I'm specifically interested in whether random slope correlate. I use the 
"bootMer" function to get bootstrapped confidence intervals for the 
parameters of the variance-covariance matrix, and also for the fixed 
effects.
Now, the puzzling thing is that the bootstrapped distributions of the 
fixed effects are all centered on zero - for all fixed effects, the 
"bias" cancells out the the "original" value - although the fixed effect 
parameters all have a z-value between -6 and -11.

Bootstrap Statistics :   t1* to t12* are fixed effects
          original        bias    std. error
t1*   0.507682933 -5.065802e-01  0.05356729
t2*  -0.410385799  4.111768e-01  0.06641469
t3*  -0.632799315  6.329604e-01  0.07389701
...
t11*  0.046429240 -4.954121e-02  0.14428656
t12*  0.008759285 -7.260454e-03  0.05337620
t13*  1.000000000  0.000000e+00  0.00000000
t14*  0.276959335 -4.755610e-03  0.04368839
t15* -0.107974393  2.175404e-03  0.03900303
t16* -0.173489566  2.103498e-03  0.04720509
...
t34*  0.050002275 -1.520703e-03  0.01094700

When I remove the random slopes in the model, by replacing the 'cr' in 
the random term with '1':

m11 <- udpate(m10, dv ~ cr * inter + (1 | vpnr))

the corresponding bootstraped distributions for the fixed effects are 
now centered around the "original" parameter estimates, and not around 
zero anymore. The "bias" here is much smaller than the "original" 
parameter values.

Bootstrap Statistics :   t1* to t12* are fixed effects
         original        bias    std. error
t1*   0.48741049  1.603870e-04  0.05357392
t2*  -0.40698631 -6.327798e-04  0.05329463
t3*  -0.57912898  2.843670e-04  0.05492087
...
t11*  0.05052645 -1.667216e-03  0.12505046
t12*  0.01646606 -4.656007e-04  0.03281307
t13*  1.00000000  0.000000e+00  0.00000000
t14*  0.25614874 -4.534802e-03  0.04021043


I guess this change in the location/bias of the bootstrapped 
distribution of the fixed effect parameters has something to do with the 
somehow special residual variance term in a binomial model (sigma(m10) = 
1, sigma(m11) = 1) in connection with how bootMer works (sampling the 
residuals, etc.), but I'm far away from tracing down the exact reason.
Are both models m10 and m11 actually ok, or does this shift in the 
boostrapped distributions rather indicate any problems entailed by the 
random slopes? Should a different boostrap procedure (something else 
than bootMer) be used in this case?

Any advice is greatly appreciated.

Thank you very much,
Christoph



-- 
Christoph Huber-Huber, BA MSc
University of Vienna
Department of Basic Psychological Research and Research Methods
Liebiggasse 5
1010 Vienna
Austria

e-mail: christoph.huber-huber at univie.ac.at
phone: +43 1 4277 47147
www: http://psychologie.univie.ac.at/en/