Optimal Experimental Designs For Comparing Curves In Regression Models

The problem of the optimal design for comparing regression curves is one of the hot topics in recent years and has attracted the attentions of many researchers.The comparison of curves has been widely used in the fields of drug development,biology,industrial production and other fields.Two(or more)models used as the target of comparison describe the relation between a common response and the same covariates for two(or more)groups.The comparison of the curves is often used to determine whether one of the models has an advantage over the other models,or is relatively whether other models can be ignored,or whether these models are equivalent in some sense.This article investigates optimal designs for comparing curves in regression models.Based on the theories of the confidence bands and confidence tubes,we explore the optimal design problems for comparing curves in generalized linear models,several regression curves,multivariate linear models and by an L~2-point of view,respectively.The details of this article include the following aspects.Firstly,we study optimal designs for comparing curves in generalized linear models,and apply μp-optimality criterion to generalized linear models,and we give the equivalence theorems and the proofs.We also study the problem of optimal allocation.In addition,robust optimal design is constructed due to the model uncertainty,and the equivalence theorem is also presented.Secondly,we study optimal designs for comparing several regression curves,and propose μ_p~c-optimal criterion based on the confidence band of multiple comparison and give the criterion function.This article presents the equivalence theorems of μ_p~c-optimal criterion and the proofs.We also extend μ_p~c-optimal criterion to the generalized linear models and give the equivalence theorems.In addition,invariant property with respect to model reparameterization is also discussed.Thirdly,we study optimal designs for comparing multivariate regression curves,and the volume formula of the confidence tube is presented.We propose μ_D-optimal criterion based on the idea of D-optimal criterion,and the statistical significance of μ_D-optimal criterion is to minimize the volume of the confidence tube.We also study the equivalence theorem of μ_D-optimality criterion,and the proof of the theorem is given.Finally,we study optimal designs for comparing regression curves by an L~2-point of view and propose μL-optimal criterion.We give the equivalence theorem of μL-optimal criterion and the proof.We also explore the case when the sample size of each group is not fixed(can be freely chosen by the experimenters),and obtain the equivalence theorem.Moreover,the proof of the equivalence theorem is presented.