# Binomial Mixed Effect Model Adjustments (and limitations thereof)

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?

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.