In numerous clinical/experimental studies, multiple endpoints are measured on each subject. It is often not clear which of these endpoints should be designated as of primary importance. The desirability function approach is a way of combining multiple responses into a single unitless composite score. The response variables may include multiple types of data: binary, ordinal, count, interval data. Each response variable is transformed to a 0 to1 unitless scale with zero representing a completely undesirable response and one representing the ideal value. In desirability function methodology, weights on individual components can be incorporated to allow different levels of importance to be assigned to different outcomes. The assignment of the weight values are subjective and based on individual or group expert opinion. In this dissertation, it is our goal to find the weights or response variable transformations that optimize an external empirical objective criterion. For example, we find the optimal weights/transformations that minimize the generalized variance of a prediction regression model relating the score and response of an external variable in pre-clinical and clinical data. For application of the weighting/transformation scheme, initial weighting or transformation values must be obtained then calculation of the corresponding value of the composite score follows. Based on the selected empirical model for the analyses, parameter estimates are found using the usual iterative algorithms (e.g., Gauss Newton). A direct search algorithm (e.g., the Nelder-Mead simplex algorithm) is then used for the minimization of a given objective criterion i.e. generalized variance. The finding of optimal weights/transformations can also be viewed as a model building process. Here relative importance levels are given to each variable in the score and less important variables are minimized and essentially eliminated.

In risk analysis, Benchmark dose (BMD)methodology is used to quantify the risk associated with exposure to stressors such as environmental chemicals. It consists of fitting a mathematical model to the exposure data and the BMD is the dose expected to result in a pre-specified response or benchmark response (BMR). Most available exposure data are from single chemical exposure, but living objects are exposed to multiple sources of hazards. Furthermore, in some studies, researchers may observe multiple endpoints on one subject. Statistical approaches to address multiple endpoints problem can be partitioned into a dimension reduction group and a dimension preservative group. Composite scores using desirability function is used, as a dimension reduction method, to evaluate neurotoxicity effects of a mixture of five organophosphate pesticides (OP) at a fixed mixing ratio ray, and five endpoints were observed. Then, a Bayesian hierarchical model approach, as a single unifying dimension preservative method is introduced to evaluate the risk associated with the exposure to mixtures chemicals. At a pre-specied vector of BMR of interest, the method estimates a tolerable area referred to as benchmark dose tolerable area (BMDTA) in multidimensional Euclidean plan. Endpoints defining the BMDTA are determined and model uncertainty and model selection problems are addressed by using the Bayesian Model Averaging (BMA) method.