I have data on the number of hospitalizations over one year in patients with chronic obstructive pulmonary disease, and I need to estimate the difference in the number of hospitalization between two groups. I only have the number of hospitalizations, no information on hospitalizations before the study. About one third of participants had no events during the study. Each event increases the probability of having a further event, therefore a Poisson distribution does not seem to be appropriate. How would you suggest to model these data also taking into account age and sex as covariates?
Although it might seem unlikely that a Poisson generalized linear model will work, the first step is to try it anyway, incorporating your group membership and covariates. Then see if the variances of predicted counts are significantly different from corresponding predicted means; they should be equal if the Poisson model holds.* Several such tests are described on this page.
If the variance is greater than the mean (over-dispersion) a negative binomial model would be the next thing to try; it adds a single extra parameter to allow for variance to increase greater than the mean.
*A nonlinear relationship between age and predicted number of events might confuse this relationship; you might want to start by examining just sex and group membership as predictors within some stratified age groupings.