ALGEBRAIC METHODS IN STATISTICS

该项目开发了统计学的新方法，包括理论和 应用，使用高等代数的方法。 结果 在一般线性混合模型的系统化中获得， 在具有结构化相关模式的数据分析中。 对于在同一病例上使用重复测量的生物医学数据， 通常会发现一个或多个数据点缺失或 得到了 用于分析此类数据的经典方法需要这样的 情况（例如，被完全从分析中剔除，尽管 通常大量的数据已经获得了相同的 案子 为了满足标准的数学和统计 在分析的条件下，这种缺失通常需要一半或 更多的案件被删除。 这是对生物医学的低效利用， 这些数据往往很难获得，而且成本很高， 收集的数据减少可能导致虚假的发现。 另一方面，Dempster的期望最大化算法， Laird和Rubin [1977]已经广泛使用了一段时间， 成功解决了数据缺失的问题。 基本的，迭代的 算法是众所周知的，但也是众所周知的收敛 难以诊断和解决的问题。 利用Rubin和Szatrowski [1982]首先提出的一个想法，我们给出了 一个完整的解决这个问题上面使用先进的方法 代数（英语：Jordan algebras） 现在又来了 众所周知的统计方法被证明是正确的，因为 一个隐含的使用约旦代数，所以是特殊情况下，我们的 结果 我们的算法在实验中找到总变化的估计值， 即使当已知该变化受任何一组 线性限制 结合严格的大样本统计 近似，研究人员可以更系统地探索 随时间进行的测量的影响（例如，真实变化与噪声） 而不必删除案例。 因此，在生物医学数据的背景下（经常有许多缺失的数据）， 数据点），新方法适用于增长曲线模型，方差 成分分析、遗传连锁分析、时间序列数据，以及 纵向数据通常在临床试验中获得，或 流行病学病例对照研究。

This project develops new methods in statistics, both theoretical and applied, using methods of advanced algebra. Results have been obtained in the systematization of the general linear mixed model and in the analysis of data having a structured pattern of correlation. For biomedical data using repeated measurements on the same case, it is often found that one or more data points are missing or were not obtained. Classical methods for analyzing such data require that such cases (e.g., subjects) be completely dropped from the analysis, despite the usually large amount of data that had been obtained on the same case. In order to satisfy the standard mathematical and statistical conditions for the analysis, such deletions often require that half or more of all cases be deleted. This is an inefficient use of biomedical data that is often difficult and costly to obtain, and using just the reduced data that was collected can lead to spurious findings. On the other hand, the Expectation-Maximization algorithm of Dempster, Laird, and Rubin [1977] has been in use for some time as a broadly successful antidote to this problem of missing data. The basic, iterative algorithm is well-known, but is also well-known to have convergence problems that are hard to diagnose and get around. Using an idea first proposed by Rubin and Szatrowski [1982], we give a complete solution to this problem above using methods of advanced algebra (technically: Jordan algebras). And now some other well-known statistical methods are shown to work precisely because of an implicit use of Jordan algebras, and so are special cases of our results. Our algorithm finds estimates for the total variation in an experiment, even when this variation is known to be constrained by any set of linear restrictions. Combined with rigorous, large-sample statistical approximations, the researchers can more systematically probe for effects in measurements taken over time (e.g, true variation vs. noise,) without having to delete cases. Thus, in the context of biomedical data (frequently having many missing data points), the new methods apply to growth curve models, variance components analysis, genetic linkage analysis, time series data, and to longitudinal data that is often acquired in clinical trials, or in epidemiological case-control studies.