Applications and Computational Issues Involving Generalized Linear and Mixed Models

提出的研究涉及在广义线性和混合效应模型的背景下产生的理论和计算问题。提出了一种新的多变量模型作为协方差分析的结构，为平衡和非平衡设计设置下的处理方法调整提供了统一的框架。其次，对满足一定线性约束的列联表和0与1表的数目提出了一种新的近似。这是相关的，例如，在确定对数线性模型的精确条件分析的可行性。这种近似是用几何响应的广义线性模型对问题进行新的表述而产生的。这种近似比现有的竞争对手更普遍适用，而且似乎要准确得多。第三，提出了一类具有解析难处理似然函数的混合效应模型的实用拟合算法。该方法涉及蒙特卡罗EM算法的实现，该算法在e步使用随机球-径向积分规则。在测试用例中，使用这个积分规则将所需的蒙特卡罗样本大小减少了两个数量级。统计模型在现代研究的几乎所有领域都无处不在，包括农业、经济学、医学和社会学等不同领域。计算能力的进步使统计学家能够考虑模型并进行甚至在几年前还不可行的计算。本研究针对与广泛使用的统计模型相关的三个问题。第一个涉及在设计的实验中调整处理手段的技术，以解释与感兴趣的反应相关的观察协变量。这是一个经典问题，其根源在于农业田间试验。这项技术有着悠久的历史，可以追溯到20世纪中期。令人惊讶的是，即使在简单的平衡实验中，对正确的调整方法仍然存在分歧。一种解释是，当该方法最初被开发出来时，完整解决方案所需的数学工具和计算能力还不具备。第二个问题涉及当数据稀疏时精确统计检验的可行性，标准近似失效。例如，在医学研究中使用精确的方法检测与各种疾病有关的因素。最后，针对一类重要的统计模型，提出了一种新的拟合算法。测试用例表明，这些方法将显著地扩展模型的范围，使计算在实际中是可行的。

The proposed research concerns theoretical and computational issues arising in the context of generalized linear and mixed effects models. A new multivariate model is suggested as a construct for analysis of covariance which provides a unified framework for adjusting treatment means in balanced and unbalanced design settings. Second, a new approximation for the number of contingency tables and tables of zeros and ones, meeting certain linear constraints, is proposed. This is relevant, for example, in determining the feasibility of exact conditional analysis of log-linear models. The approximation arises from a novel formulation of the problem in terms of a generalized linear model for geometric responses. The approximation is much more generally applicable, and appears to be far more accurate, than exiting competitors. Third, a practical fitting algorithm for a broad class of mixed effects models with analytically intractable likelihood functions is proposed. The approach involves an implementation of the Monte Carlo EM algorithm that uses a randomized spherical-radial integration rule at the E-step. Use of this integration rule reduces the required Monte Carlo sample size by two orders of magnitude in test cases.Statistical models are ubiquitous in almost all areas of modern research, including such diverse fields as agriculture, economics, medicine, and sociology. Advances in computing power enable statisticians to consider models and do calculations that were not feasible even a few years ago. This research targets three problems related to widely-used statistical models. The first concerns a technique for adjusting treatment means in designed experiments to account for observed covariates related to the response of interest. This is a classical problem with its roots in agricultural field trials. The technique has a long history dating back to the mid-20th century. It is somewhat surprising then that there is still disagreement on the correct way to make the adjustments, even in simple balanced experiments. An explanation is that the mathematical tools and computing power necessary for a complete solution were not available when the method was first developed. The second problem relates to the feasibility of exact statistical tests when data is sparse, and the standard approximations break down. Exact methods are used, for example, in medical studies testing for factors associated with various diseases. Finally, a new fitting algorithm is proposed for an important class of statistical models. Test cases suggest that the methods will significantly extend the range of models for which the computations are practically feasible.