New Methods to Reduce Bias and Mean Square Error of Maximum Likelihood Estimators

描述（由申请人提供）：逻辑回归是二进制数据最常用的模型，在健康，行为和物理科学中具有广泛的适用性。1999年发表的研究论文中，有2000多篇论文的标题或关键词中含有“逻辑回归”。最大似然法是计算Logistic回归模型中回归系数估计值的一种几乎通用的方法。这些估计是可靠的大样本的问题，当响应的比例既不太小也不太大。然而，几年前就已经知道，对于小的、稀疏的或不平衡的数据集，最大似然估计可能具有高偏倚和均方误差，后者指的是答复和不答复的数量之间的相当大的差异。精确逻辑回归是D. R.考克斯在这种情况下经常是有用的。然而，精确逻辑回归是计算密集型的，并且在实践中在数据集的大小和协变量的数量方面受到限制，它可以在耗尽内存或花费过多的计算时间之前处理。D. Firth开发了一种方法来减少logistic回归以及其他计算要求不高的广义回归模型的偏差和均方误差。文献中的研究表明，该方法通常改进最大似然法。Firth的方法在今天的任何商业软件包中都不可用。我们建议将Firth的方法纳入LogXact，Cytel的回归包，以及到PROC LOGXACT，一个模块，无缝运行的SAS软件系统的一部分。除了将弗斯的方法进行逻辑回归，我们打算开发它适用于条件逻辑回归，有序和无序多分类回归，泊松回归和负二项回归。 Firth的方法在logistic回归中的中等规模样本中的模型参数的某些范围内表现不佳。在某些情况下，它比最大似然法更糟。我们已经创建了一个新的方法，概括了弗斯的方法，以克服这个缺点。我们建议在LogXact和PROC LOGXACT中实现此方法。 在某些不寻常的条件下，最大似然法和Firth方法对logistic回归的估计都很差。我们已经开发了一种诊断措施，确定这种情况下，我们将把这种方法作为我们的推广弗斯的方法的一部分。我们还将研究一个贝叶斯估计和Cabrera和Fernholz建议的目标估计，在这种情况下有希望表现良好。

DESCRIPTION (provided by applicant): Logistic regression is the most frequently used model for binary data and has widespread applicability in the health, behavioral, and physical sciences. Over two thousand research papers were published in 1999 in which "logistic regression" was in the title of the paper or among the keywords. Maximum likelihood is the nearly universal method for computing estimates of regression coefficients in logistic regression models. These estimates are reliable for problems with large samples and when the proportion of responses is neither too small nor too large. However, it has been known for several years that maximum likelihood estimates can have high bias and mean square error for small, sparse or unbalanced datasets, with the latter referring to a considerable difference between the number of responses and non-responses. Exact logistic regression is a method invented by D. R. Cox that is often useful in such situations. However, exact logistic regression is computationally intensive and is limited in practice in terms of the size of datasets and the number of covariates that it can handle before running out of memory or taking an inordinate amount of computing time. D. Firth has developed a method for reducing bias and mean square error for logistic regression as well as other generalized regression models that is not as computationally demanding. Studies in the literature have shown that the method often improves on maximum likelihood. Firth's method is not available in any commercial software package today. We propose to incorporate Firth's method into LogXact, Cytel's regression package, as well as into PROC LOGXACT, a module that runs seamlessly as a part of the SAS software system. In addition to incorporating Firth's method for logistic regression we intend to develop it to apply to conditional logistic regression, ordered and unordered polytomous regression, Poisson regression and Negative Binomial regression. Firth's method does not perform well over certain ranges of model parameters in moderate sized samples in logistic regression. There are instances when it is worse than maximum likelihood. We have created a novel method that generalizes Firth's method to overcome this shortcoming. We propose to implement this method into LogXact and PROC LOGXACT. Under certain unusual conditions both maximum likelihood and Firth's method produce poor estimates for logistic regression. We have developed a diagnostic measure that identifies this situation and we will incorporate this method as part of our generalization of Firth's method. We will also investigate a Bayesian estimator and the target estimator suggested by Cabrerra and Fernholz that have promise of performing well in this situation.