rare binary outcome, MCMCglmm, and priors (related to separation)

Some colleagues have collected data from 184 females in dating relationships. 
Data were collected daily using PDAs; the outcome is a binary indicator of 
whether any physical aggression occurred (intimate partner violence, or IPV).

They are interested in 3 covariates:

-- alcohol use: yes/no
-- anger: rated on 1-5 scale
-- verbal aggression: sum of handful of items, with 0-15 scale

Their hypothesis is that the interaction of all 3 covariates will lead to the 
highest likelihood of IPV.  As you might expect, the outcome is very rare 
with 51 instances of IPV out of 8,269 days of data, and 158 women (out of 
184) reported no instances of IPV.

I have read a bit about the problems of separation in logistic regression and 
know that Gelman et al suggest Bayesian priors as one "solution".  Moreover, 
I see in Jarrod Hadfield's course notes that his multinomial example has a 
"structural" zero that he addresses via priors on pp. 96-97, though I confess 
I don't quite follow exactly what he has done (and why).

Hi. why are you using a mixed model here: dispersion, or are there 
multiple reports per individual?  Another approach for separated/sparse 
data implemented in R is the penalized likelihood approach in the brlr, 
logistf, brglm (and Design) packages:

brglm(formula = cbind(ipv.yes, ipv.no) ~ (ang.cut + prov.cut +
     alc.cut)^2, family = binomial(), data = ipv)

Coefficients: (1 not defined because of singularities)
                  Estimate Std. Error z value Pr(>|z|)
(Intercept)       -8.9666     1.4145  -6.339 2.31e-10 ***
ang.cut            2.8959     1.4775   1.960  0.05000 .
prov.cut           2.3740     0.4587   5.175 2.27e-07 ***
alc.cut            7.8680     2.7082   2.905  0.00367 **
ang.cut:prov.cut       NA         NA      NA       NA
ang.cut:alc.cut   -7.0703     2.8616  -2.471  0.01348 *
prov.cut:alc.cut  -0.4007     0.9962  -0.402  0.68747

Model 1: cbind(ipv.yes, ipv.no) ~ (ang.cut + prov.cut + alc.cut)
Model 2: cbind(ipv.yes, ipv.no) ~ (ang.cut + prov.cut + alc.cut)^2
   Resid. Df Resid. Dev Df Deviance P(>|Chi|)
1         2     1.0875
2         0     1.8387  2 -0.75117

Cheers, David Duffy.

| David Duffy (MBBS PhD)                                         ,-_|\
| email: davidD at qimr.edu.au  ph: INT+61+7+3362-0217 fax: -0101  /     *
| Epidemiology Unit, Queensland Institute of Medical Research   \_,-._/
| 300 Herston Rd, Brisbane, Queensland 4029, Australia  GPG 4D0B994A v