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Precision about the glmer model for Bernoulli variables

John Maindonald 
Apologies, the lower limit for the BB variance is pi(1-pi)rho.

Actually, with a suitable parameterization, the betabinomial allows
small negative correlations.  See
  Binary Regression Using an Extended Beta-Binomial Distribution, With Discussion of
 Correlation Induced by Covariate Measurement Errors
R. L. Prentice: Journal of the American Statistical Association
Vol. 81, No. 394 (Jun., 1986), pp. 321-327

A feature of the betabinomial (BB) , which ought to be better advertised than
it is in most accounts that I have seen, is that, for positive correlation rho,
it sets a strict lower limit of pi(1-pi)rho on the variance of the estimate
of the proportion pi.  This is in marked contrast to the variance
assumptions that define quasibinomial errors.

The gamlss package implements the double binomial (as well
as the logistic normal and BB), but as far as I can tell, not in a
multi-level model context.  The double binomial does allow a
negative correlation.

John Maindonald             email: john.maindonald at anu.edu.au<mailto:john.maindonald at anu.edu.au>
On 21/04/2020, at 19:18, David Duffy <David.Duffy at qimrberghofer.edu.au<mailto:David.Duffy at qimrberghofer.edu.au>> wrote:
You'd know the beta-binomial constrains the range of the correlation coefficient positive, but the logit-normal and probit-normal allow a negative correlation, with a lower bound constrained by the N, in the case of regular sized clusters. For your example of 2x2 tables, the correlation can go from -1 to 1. This is the usual genetics
type setup where you model the pairwise correlations for all pairs of observations in the dataset versus their
(pairwise) genetic relatedness.  In multi-trait models, the correlations between different phenotypes (say asthma and diabetes) in the same individual can be negative, so you can estimate negative variance components.

The pedigreemm package extends lme4 to allow glmms for pedigree data (see the mastitis example for a binary trait).

Cheers, David Duffy.

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Emmanuel Curis Precision about the glmer model for Bernoulli variables Apr 20 Vaida, Florin Precision about the glmer model for Bernoulli variables Apr 20 Emmanuel Curis Precision about the glmer model for Bernoulli variables Apr 20 Vaida, Florin Precision about the glmer model for Bernoulli variables Apr 20 David Duffy Precision about the glmer model for Bernoulli variables Apr 21 John Maindonald Precision about the glmer model for Bernoulli variables Apr 21 John Maindonald Precision about the glmer model for Bernoulli variables Apr 21 Emmanuel Curis Precision about the glmer model for Bernoulli variables Apr 21 John Maindonald Precision about the glmer model for Bernoulli variables Apr 21 Vaida, Florin Precision about the glmer model for Bernoulli variables Apr 21 Vaida, Florin Precision about the glmer model for Bernoulli variables Apr 21 Vaida, Florin Precision about the glmer model for Bernoulli variables Apr 21 Emmanuel Curis Precision about the glmer model for Bernoulli variables Apr 22 Emmanuel Curis Precision about the glmer model for Bernoulli variables Apr 22 David Duffy Precision about the glmer model for Bernoulli variables Apr 28 Dimitris Rizopoulos Precision about the glmer model for Bernoulli variables Apr 28