异方差分层线性模型的A-最优设计

A-optimal design for hierarchical linear models with heteroscedasticity

分层线性模型在统计领域和实际应用中受到广泛关注,但是异方差分层线性模型的设计问题尚需更多关注和深入研究。对异方差分层线性模型的A-最优近似设计问题进行研究。以个体参数的精准预测为目标定义了模型的A-最优准则,证明了准则函数的凸性,并借助方向导数建立等价性定理以刻画A-最优设计。在不同的权重函数下以随机斜率一次回归模型和随机截距一次回归模型为例求解A-最优设计的解析或者数值结果,并比较了最优设计和等权重设计。结果显示,对于随机截距模型,等权重设计的效率与最优设计的效率接近,对于随机斜率模型,等权重设计的效率低于最优设计。

Since hierarchical linear model has attracted much attention in statistical fields and its practical application, hierarchical linear model with heteroscedasticity needs much attention and further study. The optimal approximate designs were studied for hierarchical linear models with heteroscedasticity. A-optimal criterion was defined for the accurate prediction of individual parameters. The convexity of the criterion function was proved and directional derivative was used to establish an equivalence theorem to describe A-optimal designs. The analytical solutions or numerical solutions of A-optimal designs were solved by taking the straight-line regression with random intercept or random slope as examples under different weight functions. The optimal designs were compared with the equireplicated design. The results show that the equireplicated design of the straight-line regression with random intercept is close to the optimal design, and the equireplicated design of the straight-line regression with random slope is less efficient than that of the optimal design.