Robustness of Design in Dose–Response Studies

Abstract

We construct experimental designs for dose-response studies. The designs are robust against possibly misspecified link functions; for this they minimize the maximum meansquared error of the estimated dose required to attain a response in 100p% of the target population. Here p might be one particular value-p D 0:5 corresponds to ED 50 -estimation-or it might range over an interval of values of interest. The maximum of the mean-squared error is evaluated over a Kolmogorov neighbourhood of the fitted link. Both the maximum and the minimum must be evaluated numerically; the former is carried out by quadratic programming and the latter by simulated annealing.

A robust treatment of a dose–response study

“…a linear function of the discrepancies d i;n D p n .F n .˛0 Cˇ0x i / F 0 .˛0 Cˇ0x i //. In Li and Wiens [2], we also establish a representation of the limit of p n O…”

“…This question of how the designs might change in response to uncertainty about the appropriate link was the subject of our presentation, and our observations are detailed here. The theoretical development and mathematical details may be found in Li and Wiens . A basic feature of our approach is that we entertain a class of possible link functions, forming a neighbourhood of that used by the experimenter to analyse the data.…”

Model-Robust Designs for Quantile Regression

“…This was extended in one direction by Huber (1975), who derived minimax designs for straight line fits; these minimize the maximum mean squared error of the fitted values, with the maximum taken over a full L 2 -neighbourhood of the experimenter's assumed response. This work, for which it was assumed that the regression estimates would be obtained by least squares, has in turn been extended in numerous directions - Li (1984) to finite design spaces, Wiens (1992) to multiple regression, Woods, Lewis, Eccleston and Russell (2006) to GLMs, Li and Wiens (2011) to dose-response studies, to list but a few.…”

On generalized multinomial models and joint percentile estimation

“…The situation is similar for percentile estimation methods in multinomial response models. Though a huge number of research papers (namely, Hamilton (1979); Carter et al (1986); Williams (1986); Huang (2001); Biedermann et al (2006); Li and Wiens (2011)) have been written on percentile estimation and effect of link misspecification on percentile estimation for binary data, almost no work has been done in the case of multinomial data. There are, however, many experimental situations where multinomial responses may be observed for each setting of a group of control variables.…”