Optimizing the Precision of Toxicity Threshold Estimation Using a Two-Stage Experimental Design
An important consideration for risk assessment is the existence of a threshold, i.e., the highest toxicant dose where the response is not distinguishable from background. We have developed methodology for finding an experimental design that optimizes the precision of threshold model parameter estimation for single-chemical threshold experiments. Being interested in precisely estimating the threshold parameter, we used the D-optimality and the Ds-optimality criteria. The D-optimal design results in parameter estimates as precise as possible in the sense that the likelihood-based confidence region has minimum volume, while the Ds-optimal threshold design results in parameter estimates as precise as possible in that the variance of the threshold parameter estimate is minimized. For nonlinear models, optimal designs are a function of the unknown parameters via the information models, optimal designs are a function of the unknown parameters via the information matrix. Therefore, estimates of the parameters must be obtained before the optimal design of the experiment can be found. For this reason, a two-stage D-Ds-optimal design is recommended where the D-optimality criterion is used in the first stage followed by the Ds-optimality criterion in the second stage. The first stage is used for range finding and to obtain good global estimates to supply to the second stage. The second stage results in precise parameter estimates with minimum variance for the threshold parameter estimate. We propose that the use of this two-stage D-Ds-optimal design will provide toxicologists with the experimental parameters necessary to accurately estimate thresholds for risk assessment purposes in a more cost-effective and timely manner.