Critical assessment of the C-optimality design criteria for estimating the median effective dose in quantal dose-response curves

In this paper the properties of C-optimal designs constructed for estimating the median effective dose within the framework of two-parametric linear logistic models are critically assessed. It is well known that this design criterion which is based on the first-order variance approximation of the exact variance of the maximum likelihood estimate of the ED50 leads to a one-point design where the maximum likelihood theory breaks down. The single dose used in this design is identical with the true but unknown value of the ED50. It will be shown, that at this one-point design the asymptotic variance does not exist. A two-point design in the neighbourhood of the one-point design which is symmetrical about the ED50 and associated with a small dose-distance would be nearly optimal, but extremely nonrobust if the best guess of the ED50 differs from the true value. In this situation the asymptotic variance of the two-point design converging towards the one-point design tends to infinity. Moreover, taking in consideration, that for searching an optimal design the exact variance is of primary interest and the asymptotic variance serves only as an approximation of the exact variance, we calculate the exact Variance of the estimator from balanced, symmetric 2-point designs in the neighbourhood of the limiting 1-point design for various dose distances and initial best guesses of the ED50. We compare the true variance of the estimate of the ED50 with the asymptotic variance and show that the approximations generally do not represent suitable substitutes for the exact Variance even in case of unrealistically large sample sizes. KALISH (1990) proposed a criterion based on the second-order asymptotic variance of the maximum likelihood estimate of the ED50 to overcome the degenerated 1-point design as the solution of the optimization procedure. In fact, we are able to show that this variance approximation does not perform substantially better than the first-order variance. From these considerations it follows, that the C-optimality criterion is not useful in this estimation problem. Other criteria like the F-optimality should be used.