D-optimal designs for mixed discrete and continuous outcomes analyzed using nonlinear models

Many dose-response experiments in toxicology and other biological sciences are designed to measure multiple outcomes. Unfortunately, most of these studies are powered or designed for a single response, and the inference on the under-powered endpoints is limited. As additional design challenges, the outcomes may have different regions and shapes of activity or have different response types. As a new application to the traditional D-optimality criterion, we have developed optimal designs for mixed discrete and continuous outcomes that are analyzed with nonlinear models. These designs use a numerical algorithm to choose the location of the dose groups and proportion of total sample size allocated to each group that minimize the generalized variance of a model-based covariance matrix that incorporates the correlation between outcomes. Using this methodology, we designed a dose-response experiment with binary, count, and continuous outcomes to evaluate neurotoxicity. In this example, the optimal designs placed dose groups at the predicted dose thresholds and throughout the active range. The designs were generally robust to different correlation structures. In addition, when the expected correlation was moderate or large, we observed a substantial gain in efficiency compared to optimal designs created for each outcome separately.