Five end points—anogenital distance (AGD), nipple retention (NR), and three organ weights (weights of the ventral prostate, seminal vesicles, and levator ani/bulbocavernosus muscles [LABC])—are assessed. Owing to the nature of the end points and the timing of their measurement, most pups were evaluated either for AGD and NR or for organ weights. The composite score adjusts for either case. As indicated in this example, the end points combined in the composite score may be different (for example, binary or categorical, count, or continuous or interval variables). The approach used here is based on desirability functions. Desirability functions were first proposed by Harrington (1965) for use in optimizing the quality of a manufactured product that is measured by multiple end points. Harrington’s approach is used to find the levels of the factors that optimize the overall quality of the many end points (Derringer and Suich 1980; Derringer 1994). It has been widely adopted in manufacturing and among engineers involved in product optimization is the most popular method for simultaneously analyzing many outcomes (Wu 2005). The method has also been applied to the titration of multiple-drug regimens in medical research (Shih et al. 2003) and in dose-response modeling in toxicology studies (Coffey et al. 2007).

Once a composite scoring method is specified, each animal is represented in the data analysis by a single score regardless of the number of variables measured. Dose-response curves are estimated for each chemical, and an additivity model is estimated. In this study, a fixed-ratio mixture of the three chemicals was also experimentally evaluated. It is of interest to determine whether there is evidence of interaction in the region of the mixing ratio and, even if there is evidence of an interaction, how different the dose-response curve of the mixture is from that predicted by dose addition.


Experimental data. Data, generously provided by Ulla Hass, are as described in Hass et al. (2007) and Metzdorff et al. (2007). In short, male sexual differentiation was studied after in utero and postnatal exposures to one or a mixture of three antiandrogens (vinclozolin, flutamide, and procymidone). The mixing ratio of the mixture was based on individual potencies for “causing retention of six nipples in male offspring” (Hass et al. 2007). Test chemicals and mixtures were administered by gavage to time-mated nulliparous, young adult Wistar rats from gestation day 7 to the day before expected birth and on postnatal days 1-16. Changes in AGD and NR in male offspring rats were evaluated. The ventral prostate, seminal vesicles, and LABC of one male per litter were excised and weighed.

Composite score. A composite score was calculated based on the basis of the desirability-function method (see, for example, Harrington 1965; Coffey et al. 2007) for the five end points chosen for analysis. In short, for each variable, a

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