Martin,

There is no residual variance in logistic regression (whether the regression includes mixed effects or not). One consequence of this is that the notion of intra-class correlation is far more tricky in a mixed logit model.

Goldstein et al. discuss some strategies for estimating ICC in mixed logit models here:
http://www.bristol.ac.uk/cmm/research/pvmm.pdf

I have not read that section of the paper very carefully, but the impression I get from skimming is that it is not straightforward. Wish I had better news for you.

Jake

> Date: Thu, 31 Oct 2013 21:30:45 -0400
> From: Martin.Leclerc2@USherbrooke.ca
> To: r-sig-mixed-models@r-project.org
> Subject: [R-sig-ME] ICC on random slopes in mixed logistic regression
> 
> Hi everyone,
> 
> I wanted to know if someone knows how to calculate repeatability on a  
> random slope of a mixed logistic regression with a nested intercept. I  
> want to evaluate the repeatability of the variable X. The function  
> glmer in the lme4 package doesn?t give me an estimate of the residual  
> variance and I am not sure if ICC is calculated the same way in mixed  
> linear and logistic regressions (and with nested intercept). Here is  
> what the model looks like:
> 
> reg = glmer ( Y ~ X + (X | ID / IDyear), family = binomial)
> 
> Where Y = dichotomous dependent variable, ID = factor of 34 levels,  
> IDyear = factor of 93 levels.
> 
> The output is the following:
> 
> Random effects:
> 
>   Groups            Name         Variance     Std.Dev.    Corr
>   IDyear : ID      (Intercept)   0.0056611    0.075241
>                     X            0.0898307    0.299718    0.670
> 
>   ID               (Intercept)   0.0331178    0.181983
>                     X            0.0699248    0.264433    0.564
> 
> Number of obs: 76740, groups: bearyear:bearID, 93; bearID, 34
> 
> Even though I am not familiar with other package, I have also tried  
> with the glmmPQL function. Here is the model (Is it normal that this  
> function is really much faster to converge? From 25 min. to 4 min.):
> 
> reg2 = glmmPQL ( Y ~ X, random = ~ X | ID / IDyear), family= binomial )
> 
> Here is the output:
> 
> Random effects:
> 
> Formula: ~ X | ID
>   Structure: General positive-definite, Log-Cholesky parametrization
> 
>                    StdDev      Corr
> (Intercept)       0.3067025    (Intr)
> X                 1.4141289    -0.943
> 
> 
> Formula: ~ X | IDyear  %in%  ID
>   Structure: General positive-definite, Log-Cholesky parametrization
> 
>                     StdDev         Corr
> (Intercept)        0.2676970    (Intr)
> X                  1.6009115    -0.965
> Residual           0.9964824
> 
> 
> Is there a way to calculate an ICC on each random slopes and intercept  
> or the ICC needs to be calculated at each level of my nested intercept  
> (i.e. one ICC for ID and one ICC for IDyear). The way I see it, is  
> that my random ID variance (intercept + slope) is my intergroup  
> variance and my IDyear variance (intercept + slope) is my intragroup  
> variance. Therefore, I would just evaluate the repeatability as:
> 
> random variance of ID / random variance of ID + random variance of IDyear
> 
> Thanks a lot for your comments and your help,
> 
> Martin
> -----------------------------------------------------
> Martin Leclerc, biol. M.Sc.,
> Candidat Ph.D.
> Labo de démographie évolutive et conservation
> 
> Département de biologie, Université de Sherbrooke
> Sherbrooke, Québec, Canada, J1K 2R1
> Tel: 1-(819)-821-8000 #63020
> Sans frais: 1-800-267-8337
> Fax: (819)-821-8049
> 
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