C- And D_A-Optimal Design Of Two-Group Heteroscedasticity Hierarchical Linear Model

The optimal experimental design is to propose various optimal criteria based on the studied model,in order to construct various optimal designs.The optimal design of hierarchical linear models is increasingly receiving attention among them.This model is applied to data analysis in fields such as socio-economic and healthcare,and by constructing the optimal design,the efficiency of statistical analysis can be greatly improved.The optimal design method usually assumes that the random error term have the same variance in the optimal design research of hierarchical linear models,but in practice,it often occurs that the observation error is heteroscedasticity,so studying the optimal design problem of heteroscedasticity stratified linear models has certain practical significance.This article mainly studies the c-and D_A-optimal design of a hierarchical linear model based on two groups when the observation error is heteroscedasticity.The research content is mainly divided into two parts:the first part is to define c-,D_A-optimal design criteria for two groups of heteroscedastic hierarchical linear models,and establish the equivalence theorem of c-,D_A-optimal design by using directional derivative on the basis of proving the convexity of the criterion function;the second part selects different forms of random effects and three types of heteroscedasticity error structures to solve the optimal design of the specific model.Firstly,through loewner order proof,it is found that the optimal design of the specific model can be limited to the endpoints of the unit space,thereby simplifying the search of the design and obtaining analytical or numerical results for c-,D_A-optimal designs.Finally,the optimality of the design is verified through the equivalence theorem.In addition,a special case of D-optimal design is given in the D_A-optimal design of the model.The optimal design of a fractional order polynomial mixed model under three types of heteroscedasticity structures is given through the matlab optimization algorithm,and an image is drawn to illustrate the optimality of the design.In addition,this article also concludes that the c-and D_A-optimal designs of heteroscedasticity stratified linear models and homoscedasticity stratified linear models under special heteroscedasticity structures are the same.