Abstract: Condom usage is reported as the fraction of protected sex acts out of the total number of sex acts in HIV behavioral intervention studies. The fraction of malformed fetuses out of the total number of fetuses is measured in teratology studies. Both fractions are examples of binomial data with random n's. Substance use is also measured in HIV behavioral interventions. Count measures for the number of times using a particular substance are often zero-inflated with more zeros than can be explained by a Poisson distribution. Binomial outcomes with random n's and zero-inflated Poisson (ZIP) data call for non-standard regression techniques. A binomial outcome is a count s of the number of successes out of the total number of independent trials n = s + f where f is a count of the failures. The n are random variables not fixed by design in many studies and provide potentially useful additional information about the probability π of success not directly incorporated in the logistic regression model. Observations where n = 0 are excluded from the binomial analysis yet may hold useful information about π. Correlation between s and f may exist and be of direct interest. I propose multivariate Poisson models for the bivariate response (s, f). Success and failure counts from the binomial outcome are modeled as bivariate Poisson counts, correlated through random effects. I extend my models to the analysis of longitudinal multivariate binomial outcomes. The ZIP regression model is a model for zero-inflated count data. A zero is allowed to come from two processes, with probability p one process, the zero state, has zeros as the only possibility and with probability 1 - p the other process has Poisson distributed counts. In longitudinal data, a subject may be in the zero-state at the subject level, across all time points, or at individual time points. I incorporate the two ways a subject may be in the zero state into hierarchical ZIP (HZIP) models for ZIP data and extend the HZIP models to multivariate ZIP data; outcomes are correlated through random effects.