Design of experiments for the precise estimation of dose-response parameters: the Hill equation

Optimal experimental designs were evaluated for the precise estimation of parameters of the Hill model. The optimally effective designs were obtained by using the criterion of D-optimization. For the Hill model, optimal designs replicate 3 sampling points. These points were shown to be quite sensitive to the behavior of the experimental error. Since an investigator is often uncertain about error conditions in biological studies, a practical approach would use the sampling scheme calculated for an intermediate error condition. Thus, if the behavior of error variances is not known, precise parameters of the Hill model are obtained by choosing concentrations which yield fractional responses (responses divided by their asymptotic, maximum value) of 0.086, 0.581 and 1.0. When experimental constraints limit the maximum attainable concentration and response, all design points are lowered. Appropriate designs can be constructed based on the design which is optimal when constraints result in a maximum attainable fractional response of 0.5. The optimal designs were found to be robust when the parameter values assumed by the investigator did not equal their true values. The estimating efficiencies obtained by using two frequently applied designs were assessed. Uniformly spaced concentrations yielded imprecise parameters. Six-point, geometrically spaced designs gave generally good results. However, their estimating efficiency was generally exceeded by the recommended sampling schemes even in the presence of uncertainty about error conditions. The method exemplified in this paper can be used for other models.