Getting intraclass correlations from a binomial mixed model with logit link

Hi Ulf,

1. You can compute the following intraclass correlations. Let S (short for Subject) be the ID random intercept variance, I be the Item:Emotion random intercept variance, and E be the Emotion random intercept variance.


Correlation between responses with same Emotion, different Items, different IDs:
E / (S + I + E + pi^2/3)

Correlation between responses with same Emotion, same Items, different IDs:
(E + I) / (S + I + E + pi^2/3)

Correlation between responses with different Emotions, different Items, same IDs:
S / (S + I + E + pi^2/3)

Correlation between responses with same Emotions, different Items, same IDs:
(S + E) / (S + I + E + pi^2/3)

Correlation between responses with same Emotions, same Items, same IDs:
(S + I + E) / (S + I + E + pi^2/3)

2. The ICCs above are based on taking a latent variable view of the model. That is, we assume the responses arise from an underlying latent variable with a logistic distribution, and this logistic variable gets dichotomized around some threshold, so that we observe 0 below the threshold and 1 above the threshold. The ICCs above estimate various expected correlations in the value of this latent logistic variable. 

As hinted above, the intraclass correlation coefficient is, well, a bona fide correlation coefficient. So taking the inverse logit of a correlation doesn't really make sense.

The latent variable approach is nice because we don't have to specify a particular value of the predictor at which to assess the ICC. If you just want to talk about correlations involving the actually observed binary variable, with no latent variable baggage, you can do that, but you have to specify the expected value of Y that you're interested in (e.g., specify the values of all the predictors). That's because the variance of a binary variable depends on the mean, so accordingly the ICC is different for different expected Y values. In my opinion the notion of ICC loses its usefulness and intuitive appeal in this context. But if you want to compute it anyway, you can follow the simulation advice offered by Goldstein, Browne, & Rasbash, 2002, section 3.2.

http://www.bris.ac.uk/cmm/research/pvmm.pdf

Jake

Date: Wed, 20 Aug 2014 19:30:30 +0200
From: ukoether at uke.de
To: r-sig-mixed-models at r-project.org
Subject: [R-sig-ME] Getting intraclass correlations from a binomial mixed model with logit link

Dear list members,

I am asking you for help on interpretating the random effects from a
binomial model (strictly 0-1 responses) with a logit link:

Random effects:
 Groups                     Name        Variance Std.Dev.
 ID                         (Intercept) 0.1475   0.3840  
 Item:Emotion               (Intercept) 2.7546   1.6597  
 Emotion                    (Intercept) 0.6822   0.8259  
Number of obs: 4788, groups:  ID, 114; Item:Emotion, 42; Emotion, 7


I would like to get an ICC for each random intercept, but there are some
conceptional problems here I cannot solve yet:

1.) ID and Item are completely crossed random effects, but Items are
nested within Emotion, so I do not know if I just can get an ICC for
each variance component via

sigma-ID^2 / (sigma-ID^2 + sigma-Item:Emotion^2 + sigma-Emotion^2 +
(pi^2/3))

with pi^2/3 as the "residual variance"-equivalent term for a binomial
logit model. The variance parts for the other random intercepts would be
calculated accordingly.

I read the Papers from Goldstein (2002) and Browne (2005) about
partitioning the variance but did not find any concrete hints about such
a model which consists of crossed and nested random effects.

2.) As a side question, I would like to know if one can get these ICCs
(if it is possible to get in the first place) on the probability scale
and not on the logit scale as presented in the model output? I assume
that just applying the inverse link function on the ICC would be no good
idea, but this is just a feeling... Does anyone know, why that is wrong?

Thanks for your help,

kind regards, Ulf

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Dipl.-Psych. Ulf K?ther

PEPP-Team 
Klinik f?r Psychiatrie und Psychotherapie
Universit?tsklinikum Hamburg-Eppendorf
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20246 Hamburg

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Tel.: +49 (0) 40 7410 53248
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Pers?nlich:
Tel.: +49 (0) 40 7410 55851
Mobil: (9) 55851
ukoether at uke.de
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