The simultaneous analysis of discrete and continuous outcomes in a dose–response study: Using desirability functions

Multiple types of outcomes are sometimes measured on each animal in toxicology dose–response experiments. The potential false-positive rate from statistical tests on each endpoint may be inflated. We introduce a method of deriving a composite score that combines information from discrete and continuous outcomes through the use of desirability functions. These functions transform observed responses of any type to a 0-to-1 unitless scale. The geometric mean is used to combine the scores and then a statistical model is fit to the dose–response curve of the overall score. The overall desirability score is more sensitive to toxicity evident in only a few endpoints than other composite scores that are based on sums of components. We analyze the overall score using a nonlinear exponential model with a threshold parameter. In this example, the threshold parameter was statistically significant and its estimate was less than the lowest dose. Compared to the vehicle control, the lower overall scores at this dose group were due to lower levels of brain and blood cholinesterase (90% and 82% of control, respectively) whereas other endpoints were not altered, thus demonstrating the sensitivity of the desirability function to detect low levels of toxicity in a small number of outcomes.