When a single gene influences more than one trait, known as pleiotropy, it is important to detect pleiotropy to improve the biological understanding of a gene. This can lead to improved screening, diagnosis, and treatment of diseases. Yet, most current multivariate methods to evaluate pleiotropy test the null hypothesis that none of the traits are associated with a variant; departures from the null could be driven by just one associated trait. A formal test of pleiotropy should assume a null hypothesis that one or fewer traits are associated with a genetic variant. We recently developed statistical methods to analyze pleiotropy for quantitative traits having a multivariate normal distribution. We now extend this approach to traits that can be modeled by generalized linear models, such as analysis of binary, ordinal, or quantitative traits, or a mixture of these types of traits. Based on methods from estimating equations, we developed a new test for pleiotropy. We then extended the testing framework to a sequential approach to test the null hypothesis that $k+1$ traits are associated, given that the null of $k$ associated traits was rejected. This provides a testing framework to determine the number of traits associated with a genetic variant, as well as which traits, while accounting for correlations among the traits. By simulations, we illustrate the Type-I error rate and power of our new methods, describe how they are influenced by sample size, the number of traits, and the trait correlations, and apply the new methods to a genome-wide association study of multivariate traits measuring symptoms of major depression. Our new approach provides a quantitative assessment of pleiotropy, enhancing current analytic practice.