Regression models are among the most widespread statistical tools applied in the analysis of experimental and observational data. When the model is misspecified this can seriously affect the validity and efficiency of inference procedures. Unfortunately, investigators do not routinely check the adequacy of the specified model for their particular data analysis. Several authors have developed model checking techniques for regression models. In particular, for the linear regression model (Stute (1997); Stute et al. (1998)), the Cox model (Lin et al. (1993)) and generalized linear models (Su and Wei (1991); Lin et al. (2002); Stute and Zhu (2002)). The methodology of Lin et al. (1993) is based on martingale residuals and presently, robust estimation and model checking techniques for the censored linear regression model are not available.; In the first chapter we propose a new type of residual and an easily computed functional form test for the Cox proportional hazards model. The proposed test is a modification of the omnibus test for testing the overall fit of a parametric regression model, developed by Stute et al. (1998), and is based on what we call censoring consistent residuals. In addition, we develop residual plots that can be used to identify the correct functional forms of covariates. We compare our test with the functional form test of Lin et al. (1993) in a simulation study. The practical application of the proposed residuals and functional form test is illustrated using both a simulated data set and a real data set.; In the second chapter, we present robust inferences for certain covariate effects on the failure time in the presence of "nuisance" confounders under a semiparametric, partial linear regression setting. Specifically, the estimation procedures for the regression coefficients of interest are derived from a working linear model and are valid even when the function of the confounders in the model is not correctly specified. The new proposals are illustrated with two examples and their validity for cases with practical sample sizes is demonstrated via a simulation study.; The last chapter develops model checking techniques for assessing functional form specifications of covariates in censored linear regression models. These procedures are based on an censored data analog to taking cumulative sums of "robust" residuals over the space of the covariate under investigation. These cumulative sums are formed by integrating certain Kaplan-Meier estimators and may be viewed as "robust" censored data analogs to the processes considered by Lin et al. (2002). The null distributions of these stochastic processes can be approximated by the distributions of certain zero-mean Gaussian processes whose realizations can be generated by computer simulation. Each observed process can then be graphically compared with a few realizations from the Gaussian process. We also develop formal test statistics for numerical comparison. Such comparisons enable one to assess objectively whether an apparent trend seen in a residual plot reflects model misspecification or natural variation. We illustrate the methods with a well known dataset. In addition, we examine the finite sample performance of the proposed test statistics in simulation experiments. In our simulation experiments, the proposed test statistics have good power of detecting misspecification while at the same time controlling the size of the test.
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机译：回归模型是用于分析实验和观察数据的最广泛的统计工具之一。如果模型指定不正确，则会严重影响推理过程的有效性和效率。不幸的是，调查人员没有针对他们的特定数据分析常规检查指定模型的适当性。一些作者开发了用于回归模型的模型检查技术。特别是对于线性回归模型（Stute（1997）; Stute等人（1998）），Cox模型（Lin等人（1993））和广义线性模型（Su and Wei（1991）; Lin等人） （2002）； Stute and Zhu（2002）。林等人的方法。 （1993年）基于mar残差，目前尚无用于删失线性回归模型的鲁棒估计和模型检验技术。在第一章中，我们为Cox比例风险模型提出了一种新型的残差和易于计算的功能形式检验。拟议的测试是对综合测试的改进，该测试用于测试由Stute等人开发的参数回归模型的整体拟合。 （1998年），并基于我们所谓的审查一致残差。此外，我们开发了可用于识别协变量的正确函数形式的残差图。我们将我们的测试与Lin等人的功能形式测试进行了比较。 （1993）在模拟研究中。拟议的残差和功能形式测试的实际应用通过仿真数据集和真实数据集进行了说明。在第二章中，我们在半参数，部分线性回归设置下，在存在“烦人”混杂因素的情况下，针对故障时间的某些协变量效应给出了可靠的推断。具体来说，从工作线性模型中得出感兴趣的回归系数的估算程序，即使没有正确指定模型中混杂因素的功能，估算程序也有效。用两个例子说明了新建议，并通过模拟研究证明了它们对于具有实际样本量的案例的有效性。上一章开发了模型检查技术，以评估在审查的线性回归模型中协变量的函数形式规范。这些程序基于经过审查的数据类似物，以对调查的协变量空间上的“健壮”残差进行累积求和。这些累积的总和是通过整合某些Kaplan-Meier估计器而形成的，可以看作是Lin等人所考虑的过程的“稳健”的审查数据类似物。 （2002）。这些随机过程的零分布可以通过某些零均值高斯过程的分布来近似，其实现可以通过计算机仿真来生成。然后，可以将每个观察到的过程与高斯过程的一些实现进行图形比较。我们还开发了正式的测试统计数据以进行数值比较。这样的比较使人们能够客观地评估在残差图中看到的明显趋势是否反映了模型错误指定或自然变化。我们用一个众所周知的数据集来说明这些方法。此外，我们在仿真实验中检查了所提出的测试统计量的有限样本性能。在我们的模拟实验中，所提出的测试统计数据具有很好的检测错误规格的能力，同时可以控制测试的大小。
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