Equivalent to Dunnett's test for binomial endpoint? I have a trial comparing 2 treatments vs a control group.  Typically one would use Dunnett's test for this situation, but the outcome measure is binomial.  It seems strange to me but I've looked for the equivalent version of a Dunnett's test for binomial endpoints and not been successful in finding one.  Can anyone suggest which test I should use?  I need to ensure I control multiplicity.

This is easy, if you recognize the parallel construction of regressions when you have a categorical predictor and consider a continuous versus a binary outcome. Say you code treatment as a 3-level categorical predictor with "control" as the reference category.
Dunnett's test could be applied to a model like this in R:
lm(continuousOutcome ~ treatment)

which returns an intercept representing the estimated outcome in the control group and coefficients representing the difference of each of the treatment groups from control. Dunnett's test then effectively evaluates the differences of each of those treatment-group coefficients from 0, with correction for multiple comparisons.
With a binary outcome, the following logistic regression in R:
glm(binaryOutcome ~ treatment, family=binomial(link="logit"))

returns an intercept representing the log-odds of the outcome for the control group and coefficients representing the difference in log-odds for each of the treatment groups from control. So you similarly evaluate the treatment-group coefficients for differences from 0, with correction for multiple comparisons.
The continuous and binary-outcome models involve different assumptions about the distributions of the coefficient estimates, so Dunnett's test can't be used directly with the logistic regression. This is conveniently handled for a wide range of models, including logistic regression, by the "trt.vs.ctrl" contrasts in the R emmeans package, which provides "a close approximation to the Dunnett adjustment."
