Mathematical Sciences: Model Building for Linear and Non- linear Mixed Effects Models

混合效应模型用于数据，其中对几个不同主题中的每个主题进行了几个观察，并且假设观察结果是由模型函数生成的。假设模型函数中的一些参数对所有受试者都是共同的（固定效应），而一些参数在受试者之间是不同的（随机效应）。分析的目的是估计固定效应和表征随机效应的分布。线性混合效应模型，其中所有参数在模型函数中线性出现，提供了明确的目标函数来定义最大似然或限制最大似然估计。我们将研究这些估计的性质，包括固定效应的渐近不相关和随机效应的方差-协方差。这项工作包括确定随机效应的方差-协方差的有效参数化。利用这一点，我们可以评估由Lindstrom和Bates提出的非线性混合效应模型的两阶段算法的有效性。统计学家称之为“混合效应”的数学模型已广泛应用于药物动力学，即研究人体吸收和消除药物的速率。这个统计工具允许药理学家建立一个药物在“普通人”体内的药代动力学的数学模型，然后用特定人群的一些浓度测量来更新该模型。通过这种方式，我们将有限的个体信息与总体信息相结合，以建立更准确的给药时间表。这既节省了时间又节省了金钱。我们将开发统计技术，使混合效应模型的开发更容易、更准确。

Mixed effects models are used with data where several observations are made on each of several different subjects and the observations are assumed to be generated by a model function. Some of the parameters in the model function are assumed to be common to all subjects (fixed effects) and some vary between subjects (random effects). The purpose of the analysis is to estimate the fixed effects and characterize the distribution of the random effects. Linear mixed effects models, in which all the parameters occur linearly in the model function, provide an explicit objective function to define maximum likelihood or restricted maximum likelihood estimates. We will investigate properties of these estimates including the asymptotic uncorrelation of the fixed effects and the variance-covariance of the random effects. This work includes determining effective parameterizations of the variance-covariance of the random effects. Using this we can assess the effectiveness of the two-stage algorithm for nonlinear mixed effects models proposed by Lindstrom and Bates. The type of mathematical modelling which statisticians call "mixed effects" has been widely applied in pharmacokinetics, the study of the rate at which the body absorbs and eliminates drugs. This statistical tool allows, a pharmacologist to build a mathematical model of the pharmacokinetics of a drug in the "average" person, then update that model with a few concentration measurements on a particular person. In this way, we augment limited information on the individual with general information about the population to establish more accurate dosage schedules. This saves both time and money. We will be developing statistical techniques to make the development of mixed effects models easier and more accurate.