Testing for a Time Dependent Coefficient in Cox's Regression Model
本文讨论了一种检测非比例风险的常用检验方法，将其视为筛法假设检验的实例，证明了该检验对广泛的比例风险替代假设具有一致性，但代价是无法检测样本量平方根阶的局部替代假设。
This paper discusses a commonly used test to detect a nonproportional hazard. The test can be viewed as an example of the method of sieves approach to hypothesis testing. In this framework the test can be shown to be consistent against a wide class of alternatives to proportional hazards. However there is a price exacted for such consistency. This price is that the test can not detect local alternatives of order one over the square root of the sample size. In survial analysis, a frequently used method of associating covariates with the time to failure is via Cox's regression model (Cox, 1972; Andersen & Gill, 1982). If the covariates are time independent then this model can be called a proportional hazards model in that the hazard rates for different values of the covariates are proportional. Consider the simple example in which an invasive treatment (say surgery and medication) is compared to a less invasive treatment (only medication). In this situation, it is often expected that the hazard rate for the invasive treatment will be high for some length of time and then drop, possibly to or below the hazard rate for the noninvasive treatment. In the Cox regression model, the hazard at time u is given by eflo?x0(u), where X = 1 for the invasive treatment and 0 otherwise; that is, the hazard for the invasive treatment would be eflo%o(u) and 2O(u) would be the hazard rate for the noninvasive treatment. Obviously this does not allow the hazards to converge or