Mathematical Methods for Small--Sample Biostatistical Inference

为了扩大鞍点逼近技术和精确计数技术在统计推断中的应用范围，作者运用了各种数学方法。这些技术被应用于多维条件推理，主要是通过对充分统计向量的某些分量的以剩余分量为条件的多变量尾部概率开发新的近似来实现的，以及精确计算这些尾部概率的方法。特别关注其充分统计具有晶格分布的概率模型，因为标准的渐近技术在这方面经常失败。这些模型，包括Logistic模型、Poisson回归模型和列联表模型，在应用生物统计学工作中是非常常用的。具体地说，本研究包括多维鞍点近似在序限制假设中的应用。它将现有的适用于连续分布的近似推广到一般的非点阵分布。它为调整近似条件推理方法以获得更高的能力制定了指导方针。作为这项工作的一部分开发的计算算法是公开可用的。许多当前的统计技术依赖于数学近似；这些近似的精度从很好到不足。这项研究涉及已知的几乎总是高精度的近似，并将它们应用于科学家在不同学科中广泛使用的一些统计背景。这项研究允许研究人员从较小的数据集中得出有效的结论，特别是在重要的研究问题以一定数量的形式表述的情况下。这种情况发生在广泛的领域，从金融到政治学再到医学。例如，一种新的医学疗法可能会带来改善，可能会以多种方式衡量。研究人员希望证明，根据所有可能的措施，接受新疗法的患者至少与接受新疗法的患者一样好，至少在一项措施上比接受原始疗法的患者更好。标准的统计方法不能有效地处理这种情况；目前的研究代表着一项重大改进。

The investigator applies various mathematical methodsto extend the range of application of saddlepointapproximation techniques and exact enumeration techniquesin statistical inference. These techniques are applied tomulti-dimensional conditional inference, achieved primarilyby developing new approximations to multivariate tailprobabilities for some component of a vector of sufficientstatistics conditional on the remaining components, andmethods for calculating these tail probabilities exactly.Particular attention is paid to probability modelswhose sufficient statistics have lattice distributions,since standard asymptotic techniques frequently failin this context. These models, including logistic andPoisson regression and contingency table models, arevery frequently used in applied biostatistical work.Specifically, this research includes the applicationof multidimensional saddlepoint approximations toorder--restricted hypotheses. It extends existingApproximations applicable to continuous distributions togeneral non-lattice distributions. It develops guidelinesfor tuning approximate conditional inferential methods toobtain higher power. Computational algorithms developedas part of this work are publicly available.Many current statistical techniques rely on mathematical approximations;the accuracy of these approximations ranges from very good to inadequate.This research involves approximations known to be almost always of highaccuracy, and applies them in some statistical contexts that are widelyused by scientists in a variety of disciplines. This research allowsinvestigators to draw valid conclusions from smaller data sets,particularly in cases when important research questions are phrased in termsof a number of quantities that must be accounted for. This situation occurs in a wide range of areas, from finance to political science to medicine.For example, a new medical therapy might be expected to lead to improvements,potentially measured in a number of ways. The investigator wishes todemonstrate that individuals receiving the new therapy at least as well according to all of the potential measures, and better on at least onemeasure, than do patients on the original therapy. Standard statisticalmethods do not handle such situations in an efficient way; the currentresearch represents a significant improvement.