Merlé Y, Mentré F
INSERM U194, CHU Pitié-Salpêtrière, Paris, France.
J Pharmacokinet Biopharm. 1995 Feb;23(1):101-25. doi: 10.1007/BF02353788.
In this paper 3 criteria to design experiments for Bayesian estimation of the parameters of nonlinear models with respect to their parameters, when a prior distribution is available, are presented: the determinant of the Bayesian information matrix, the determinant of the pre-posterior covariance matrix, and the expected information provided by an experiment. A procedure to simplify the computation of these criteria is proposed in the case of continuous prior distributions and is compared with the criterion obtained from a linearization of the model about the mean of the prior distribution for the parameters. This procedure is applied to two models commonly encountered in the area of pharmacokinetics and pharmacodynamics: the one-compartment open model with bolus intravenous single-dose injection and the Emax model. They both involve two parameters. Additive as well as multiplicative gaussian measurement errors are considered with normal prior distributions. Various combinations of the variances of the prior distribution and of the measurement error are studied. Our attention is restricted to designs with limited numbers of measurements (1 or 2 measurements). This situation often occurs in practice when Bayesian estimation is performed. The optimal Bayesian designs that result vary with the variances of the parameter distribution and with the measurement error. The two-point optimal designs sometimes differ from the D-optimal designs for the mean of the prior distribution and may consist of replicating measurements. For the studied cases, the determinant of the Bayesian information matrix and its linearized form lead to the same optimal designs. In some cases, the pre-posterior covariance matrix can be far from its lower bound, namely, the inverse of the Bayesian information matrix, especially for the Emax model and a multiplicative measurement error. The expected information provided by the experiment and the determinant of the pre-posterior covariance matrix generally lead to the same designs except for the Emax model and the multiplicative measurement error. Results show that these criteria can be easily computed and that they could be incorporated in modules for designing experiments.
本文提出了3条准则，用于在有先验分布的情况下，针对非线性模型的参数设计贝叶斯估计实验：贝叶斯信息矩阵的行列式、先验后验协方差矩阵的行列式以及实验提供的期望信息。针对连续先验分布的情况，提出了一种简化这些准则计算的方法，并将其与通过模型关于参数先验分布均值的线性化得到的准则进行了比较。该方法应用于药代动力学和药效学领域常见的两个模型：大剂量静脉单次注射单室开放模型和Emax模型。这两个模型都涉及两个参数。考虑了具有正态先验分布的加性和乘性高斯测量误差。研究了先验分布方差和测量误差的各种组合。我们的注意力仅限于测量次数有限（1次或2次测量）的设计。在实际进行贝叶斯估计时，这种情况经常发生。得到的最优贝叶斯设计随参数分布的方差和测量误差而变化。两点最优设计有时与参数先验分布均值的D最优设计不同，并且可能包括重复测量。对于所研究的情况，贝叶斯信息矩阵的行列式及其线性化形式导致相同的最优设计。在某些情况下，先验后验协方差矩阵可能远偏离其下限，即贝叶斯信息矩阵的逆，特别是对于Emax模型和乘性测量误差。除了Emax模型和乘性测量误差外，实验提供的期望信息和先验后验协方差矩阵的行列式通常导致相同的设计。结果表明，这些准则易于计算，并且可以纳入实验设计模块中。
