Optimal designs for comparing curves

Abstract

We consider the optimal design problem for a comparison of two regression curves, which is used to establish the similarity between the dose response relationships of two groups. An optimal pair of designs minimizes the width of the confidence band for the difference between the two regression functions. Optimal design theory (equivalence theorems, efficiency bounds) is developed for this non standard design problem and for some commonly used dose response models optimal designs are found explicitly. The results are illustrated in several examples modeling dose response relationships. It is demonstrated that the optimal pair of designs for the comparison of the regression curves is not the pair of the optimal designs for the individual models. In particular it is shown that the use of the optimal designs proposed in this paper instead of commonly used “non-optimal” designs yields a reduction of the width of the confidence band by more than 50%.

Optimal designs for comparing regression models with correlated observations

“…In particular, we consider the two regression curves 1,10], and study separately the cases of a Brownian motion and an exponential covariance kernel of the form K(t, t ) = exp{−λ|t − t |} for both error processes ε 1 (t) and ε 2 (t). Following Dette and Schorning (2015), here we focus on the µ ∞ -optimality criterion defined in (2.4) since, as they point out, it is probably of most practical interest and unlike the µ p -criteria for p < ∞, it is not necessarily differentiable. Throughout this section, we denote byθ * n = (θ * 1,n 1 ,θ * 2,n 2 ) the best pair of linear unbiased estimators defined by (4.1), where for each of the combinations of models (5.1) and (5.2) the optimal (vector-) weights have been found by Theorem 4.1 and the optimal design points t * i,j are determined minimising the criterion (4.9).…”

“…On the other hand, the efficient planning of experiments for comparing curves has not been dealt with in the literature although this would substantially improve the accuracy of the conclusions drawn regarding non-superiority or equivalence. To the best of the authors knowledge only Dette and Schorning (2015) investigate such a design problem. They consider regression models with independent observations and search for designs that minimise the width of the simultaneous confidence band proposed by Gsteiger, Bretz and Liu (2011), for the difference of the two models.…”

“…More precisely, an optimal pair of designs minimises an L p -norm calculated in the common covariate region of interest, of the asymptotic variance of the difference between the two regression curves estimated via maximum likelihood. Dette and Schorning (2015) provide explicit solutions for some commonly used dose-response models and demonstrate that using an optimal pair of designs, such as the one they propose, instead of a "standard design" results in the width of the confidence band to be reduced by more than 50%. Although this improvement is impressive, the results of Dette and Schorning (2015) can not be used, for example, to improve the statistical inference for the comparison of dissolution profiles since in such studies measurements are usually taken at the same patient and therefore can not be considered as uncorrelated.…”

“…Dette and Schorning (2015) provide explicit solutions for some commonly used dose-response models and demonstrate that using an optimal pair of designs, such as the one they propose, instead of a "standard design" results in the width of the confidence band to be reduced by more than 50%. Although this improvement is impressive, the results of Dette and Schorning (2015) can not be used, for example, to improve the statistical inference for the comparison of dissolution profiles since in such studies measurements are usually taken at the same patient and therefore can not be considered as uncorrelated. The goal of the present paper is to develop efficient statistical tools (estimation and design) for the comparison of two regression curves estimated from two samples of correlated measurements.…”

Optimal experimental design that minimizes the width of simultaneous confidence bands

“…The optimal design problem focusing on the width of the simultaneous confidence bands can be formalized in a different way. When comparing two estimated regression curves, Dette and Schorning (2016) and Dette, et al (2017) proposed to minimize the L por L ∞ -norm of the variance function of the estimator of the difference between the two curves. They demonstrated that their proposal reduces the width substantially compared with the pair of optimized designs for individual regression models.…”