Optimal Designs For Comparing Population Curves In Hierarchical Models

In fields such as pharmaceutical and biological research,the problem of comparing each experimental group with the control group when there are multiple experimental groups is often solved as a comparison between two models,which is often used to determine the non-superiority of one model over another or to check whether the differences between the two models can be ignored.Optimal design is an important experimental design method,mainly through the establishment of appropriate design criteria and solving the corresponding design,so as to ensure the more accurate estimator.So in order to ensure the accuracy of comparison between curves,this thesis considers the problem of optimal designs for curve comparison.Dette and Schorning investigated the optimal design of the comparison between two regression curves in order to investigate the similarity between the dose-response relationship of two groups.Considering the wide application of hierarchical models in various fields such as biological research,agronomy,and pharmacokinetics,this thesis discusses the optimal designs for comparing curves in hierarchical models based on the research of Dette and Schorning.The main research of this thesis can be divided into two parts:theory and practical application.Theoretically,we construct?_pdesign criterion,the equivalence theorem under the criterion and the theorem about the lower bound of the effectiveness of the design are derived.In this thesis,we choose a pair of designs,each design corresponds to a group,and the pair of optimal designs should minimize the parametrization of the asymptotic variance of the difference between the two population curves,and thus we derive the equivalence theorem under the design criterion according to the general equivalence theorem.In order to investigate the quality of non-optimal designs,the efficiency of design is proposed and the lower bound of the efficiency of design is derived based on the equivalence theorem under the design criterion in this thesis.The application includes the following three main components:1.The optimal design of the three specific models for both diagonal and non-diagonal matrix of variance of random effect is proved according to the equivalence theorem under the design criterion of this thesis.2.According to the theorems about the lower bound of the efficiency,the efficiency of the equidistant design,the lower bound of the efficiency,and the difference between the lower bound of the efficiency and the efficiency are calculated for three specific models.It also discusses the changes in the efficiency of the equidistant design,the lower bound of the efficiency,and the difference between them as the number of design points and the random-effects covariance matrix change for a given parameter.3.The matlab optimization algorithm was used to find the optimal design under the two types of models,the asymptotic variance of the difference between the two population curves under the optimal design and the equidistant design is compared,and the confidence bands of the difference between the two population curves under the optimal design and the equidistant design are simulated stochastically,and the mean square error of the parameter estimates under the optimal and equidistant designs was calculated.The results show that the asymptotic variance of the difference between the two population curves under?₁ optimal design is smaller than the asymptotic variance of the difference between the two population curves under the equidistant design at most points of the interval,and the maximum value of the asymptotic variance of the difference between the two population curves under the?_?optimal design is the smallest;the width of the confidence band of the difference between the two population curves under the?₁ optimal design is the narrowest,and the maximum width of the confidence band of the difference between the two population curves under the?_?optimal design is the narrowest.In addition,the mean squared error of the parameter estimates under the optimal design is smaller than that under the equidistant design.It can be concluded that the optimal design is more accurate than the common design for the comparison of population curves in hierarchical model.