Binomial Mixed Effect Model Adjustments (and limitations thereof)

Question

My question pertains to necessary caution/downfalls in adjustment of a binomial mixed effect model with a low number of "success" outcomes (p̂ = .05).

I have a data frame in the form:

data <- data.frame("SubjectID"=1:221, 
       "Group"=sample(gl(8, 28, labels = 280:288), size=221), 
       "Condition"=sample(relevel(gl(2, 130, labels = c("Intervention", "Control")), ref="Control"), size = 221),
       "Outcome"=rbinom(221, 1, prob=.05))

While in reality, the data has approximately 11 success outcomes (4 to intervention, 7 to control). This data stems from a pilot phase of a GRCT that's being modeled with a binomial mixed effect model (as below) to account for the random effect of group:

require(lme4)
binModel <- glmer(Outcome ~  Condition + (1 | Group), 
              data = data, family = binomial, 
              control = glmerControl(optimizer="bobyqa"), nAGQ = 50)

Further, there's a battery of psychometric and socio-demographic variables collected at baseline, tied to each of these these individuals I'd like to integrate a final model. Given the presence of this additional data, I'd like to build a model, inclusive of these variables, in order to better account for these outcomes. Is there any precautions/best practices to take in ensuring that adjustments are compatible with the nature/size of the data?

Answer 1

This is going to be tough.

Table 4.1 on p. 73 of Harrell's Regression Modeling Strategies (this Google Books link might work) states that the limiting sample size for binary data is $m=\textrm{min}(n_1,n_2)$ (where $n_1$, $n_2$ are the numbers of successes and failures; the accompanying text says that the number of parameters $p$ estimated should be less than about $m/15$. So if you want the model to be reliable you can't really include much more than the primary condition variable.