Khinkis Leonid A, Levasseur Laurence, Faessel Hélène, Greco William R
Department of Mathematics and Statistics, Canisius College, Buffalo, NY, U.S.A.;
Nonlinearity Biol Toxicol Med. 2003 Jul;1(3):363-77. doi: 10.1080/15401420390249925.
Many drug concentration-effect relationships are described by nonlinear sigmoid models. The 4-parameter Hill model, which belongs to this class, is commonly used. An experimental design is essential to accurately estimate the parameters of the model. In this report we investigate properties of D-optimal designs. D-optimal designs minimize the volume of the confidence region for the parameter estimates or, equivalently, minimize the determinant of the variance-covariance matrix of the estimated parameters. It is assumed that the variance of the random error is proportional to some power of the response. To generate D-optimal designs one needs to assume the values of the parameters. Even when these preliminary guesses about the parameter values are appreciably different from the true values of the parameters, the D-optimal designs produce satisfactory results. This property of D-optimal designs is called robustness. It can be quantified by using D-efficiency. A five-point design consisting of four D-optimal points and an extra fifth point is introduced with the goals to increase robustness and to better characterize the middle part of the Hill curve. Four-point D-optimal designs are then compared to five-point designs and to log-spread designs, both theoretically and practically with laboratory experiments.D-optimal designs proved themselves to be practical and useful when the true underlying model is known, when good prior knowledge of parameters is available, and when experimental units are dear. The goal of this report is to give the practitioner a better understanding for D-optimal designs as a useful tool for the routine planning of laboratory experiments.
许多药物浓度-效应关系由非线性S形模型描述。属于此类的四参数希尔模型是常用的。实验设计对于准确估计模型参数至关重要。在本报告中，我们研究D-最优设计的性质。D-最优设计可使参数估计的置信区域体积最小化，或者等效地，使估计参数的方差-协方差矩阵的行列式最小化。假定随机误差的方差与响应的某个幂次成正比。为生成D-最优设计，需要假定参数的值。即使这些关于参数值的初步猜测与参数的真实值有明显差异，D-最优设计仍能产生令人满意的结果。D-最优设计的这一特性称为稳健性。它可以用D-效率来量化。引入了一种由四个D-最优点和一个额外的第五个点组成的五点设计，目的是提高稳健性并更好地表征希尔曲线的中间部分。然后从理论和实验室实验实际操作两方面，将四点D-最优设计与五点设计以及对数分布设计进行比较。当真实的基础模型已知、有良好的参数先验知识且实验单元成本较高时，D-最优设计证明了自身的实用性和有效性。本报告的目的是让从业者更好地理解D-最优设计，将其作为实验室实验常规规划的一种有用工具。
