Global tests for multiple binary outcomes

The applied statistician often encounters the need to compare two or more groups with respect to more than one outcome or response. Several options are generally available, including reducing the dimension of the problem by averaging or summarizing the outcomes, using Bonferroni or other adjustments for multiple comparisons, or applying a global test based on a suitable multivariate model. For normally distributed data, it is well established that global tests tend to be significantly more sensitive than other procedures. While global tests have also been proposed for multiple binary outcomes, their properties have not been well studied nor have they been widely discussed in the context of clustered data. In this paper, we derive a class of quasi-likelihood score tests for multiple binary outcomes, and show that special cases of this class correspond to other tests that have been proposed. We discuss extensions to allow for clustered data, and compare the results to the simple approach of collapsing the data to a single binary outcome, indicating the presence or absence of at least one response. The asymptotic relative efficiencies of the tests are shown to depend not only on the correlation between the outcomes, but also on the response probabilities. Although global tests based on a multivariate model are generally recommended, our findings suggest that a test based on the collapsed data can maintain surprisingly high efficiency, especially when the outcomes of interest are rare. Data from several developmental toxicity studies illustrate our results.