Mathematical Methods for Small Sample Biostatistical Inference

小样本生物统计推断的数学方法

• 批准号: 0092659
• 负责人: John Kolassa
• 金额: $ 12.5万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-09-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0092659&HistoricalAwards=false
• 关键词: Mathematical; Methods; Small; Sample; Biostatistical

A wide variety of techniques exist for conditional inference on exponential families arising from discrete distributions. Normal theory methods, which rely on the approximate multivariate normality of the joint distribution of summary statistics from the data set, are often inaccurate for small data sets, and their quality can often be poor for summaries that indicate large parameter effects. They also ignore discreteness in the data. More sophisticated approximation techniques, known as saddlepoint techniques, are often used in cases when normal theory methods are inadequate. These techniques often do not account for discreteness in data, and hence are suboptimal in their unmodified forms. Exact inferential techniques are also available, but these techniques apply only to a limited number of models, require proprietary software, and fail when sample size reaches a moderate size. Extensions to this software that employ Monte Carlo techniques for larger sample sizes are not yet commercially available. These Monte Carlo techniques have the further disadvantage of delivering a variety of results for the same data set. The techniques proposed use saddlepoint approximations in a way that accounts for discreteness in the data while avoiding most of the computationally intractable aspects of exact calculations. Some of the projects proposed in this grant application involve new approximations, such as for approximating higher--dimensional distribution functions, and others involve modifications to existing approximations to avoid numerical instabilities. Other projects involve formulating confidence regions to make accurate calibration easy, and modifying the conditioning event to obtain a more powerful analysis, and performing diagnostics to ensure that the proper approximations are used. These methods will be general enough to apply to any canonical exponential family supported on a lattice, and hence to any generalized linear model with canonical link, observations supported on a lattice, and design matrix whose entries are confined to a lattice. Examples of models that will be accommodated are logistic regression, Poisson regression including log linear models for contingency tables, and multinomial models. Regression models with more exotic error structures, including positive Poisson and negative binomial distributions, will also be accommodated.This proposed research is intended to aid in statistical inference on multiple parameters, in the presence of other nuisance parameters that are not of direct interest, when the distribution modeled is discrete. For example, the probability that a cancer patient will stay in remission can be modeled as a function of a variety of factors. Some of these effects, like which treatment a patient received or whether the patient had other cancer--related pathologies, may generalize to other populations, and others, like the effect of a particular center where the patient was treated, may not generalize. Thus one might be interested in describing the possible values that the parameters of interested take on, without being required to simultaneously estimate the remaining parameters. Typically one treats information associated with nuisance parameters as held fixed, and performs inference conditionally on this information. That is, one assesses the the evidence concerning the parameter of interest by comparing experimental results to the population of possible results such that the information about nuisance parameters is held fixed. The research agenda proposed here presents methods for doing these calculations, which balance high computational costs of exact methods against potential inaccuracies of approximations, and introduces and combines new methods for bothexact and approximate calculations. These new methods will make the analysis of small discrete data sets, commonly occurring in applied sciences, quicker and more accurate.

存在各种各样的技术对离散分布产生的指数族进行条件推断。 正态理论方法依赖于来自数据集的汇总统计量的联合分布的近似多元正态性，对于小数据集通常是不准确的，并且对于指示大参数效应的汇总，其质量通常很差。 他们还忽略了数据的离散性。 更复杂的近似技术，称为鞍点技术，经常用于正常理论方法不充分的情况。 这些技术通常不考虑数据的离散性，因此在未修改的形式下是次优的。 精确的推理技术也是可用的，但这些技术只适用于有限数量的模型，需要专有软件，当样本量达到中等大小时失败。 这一软件的扩展，采用蒙特卡罗技术的更大的样本量尚未商业化。 这些蒙特卡罗技术具有为相同数据集提供各种结果的进一步缺点。 所提出的技术使用鞍点近似的方式，占离散的数据，同时避免了精确计算的计算上棘手的方面。 在此拨款申请中提出的一些项目涉及新的近似，如近似高维分布函数，和其他涉及修改现有的近似，以避免数值不稳定性。 其他项目涉及制定置信区域，使准确的校准容易，并修改条件事件，以获得更强大的分析，并执行诊断，以确保使用正确的近似。 这些方法将是一般性的，足以适用于任何典型的指数家庭支持的一个格，因此，任何广义线性模型与典型的链接，观察支持的一个格，和设计矩阵的条目被限制在一个格。 将被容纳的模型的例子是逻辑回归、泊松回归，包括列联表的对数线性模型和多项式模型。 此外，亦会考虑具有较奇异误差结构的回归模型，包括正泊松分布和负二项分布。这项研究旨在协助在多个参数的统计推断，在其他滋扰参数的存在下，不直接感兴趣，当建模的分布是离散的。 例如，癌症患者将保持缓解的概率可以被建模为各种因素的函数。 其中一些影响，如患者接受的治疗或患者是否患有其他癌症相关疾病，可能会推广到其他人群，而其他影响，如患者接受治疗的特定中心的影响，可能不会推广。 因此，人们可能对描述感兴趣的参数所取的可能值感兴趣，而不需要同时估计剩余的参数。 通常，人们将与滋扰参数相关联的信息视为保持固定，并且有条件地对该信息执行推断。 也就是说，通过将实验结果与可能结果的总体进行比较来评估关于感兴趣的参数的证据，使得关于干扰参数的信息保持固定。 这里提出的研究议程提出了做这些计算的方法，平衡高计算成本的精确方法对潜在的不准确的近似，并介绍和结合新的方法bothexact和近似计算。 这些新方法将使应用科学中常见的小型离散数据集的分析更快，更准确。