Abrahamowicz Michal, MacKenzie Todd A
Department of Epidemiology and Biostatistics, McGill University, Montreal, Que., Canada H3A 1A2.
Stat Med. 2007 Jan 30;26(2):392-408. doi: 10.1002/sim.2519.
In order to yield more flexible models, the Cox regression model, lambda(t;x) = lambda(0)(t)exp(betax), has been generalized using different non-parametric model estimation techniques. One generalization is the relaxation of log-linearity in x, lambda(t;x) = lambda(0)(t)exp[r(x)]. Another is the relaxation of the proportional hazards assumption, lambda(t;x) = lambda(0)(t)exp[beta(t)x]. These generalizations are typically considered independently of each other. We propose the product model, lambda(t;x) = lambda(0)(t)exp[beta(t)r(x)] which allows for joint estimation of both effects, and investigate its properties. The functions describing the time-dependent beta(t) and non-linear r(x) effects are modelled simultaneously using regression splines and estimated by maximum partial likelihood. Likelihood ratio tests are proposed to compare alternative models. Simulations indicate that both the recovery of the shapes of the two functions and the size of the tests are reasonably accurate provided they are based on the correct model. By contrast, type I error rates may be highly inflated, and the estimates considerably biased, if the model is misspecified. Applications in cancer epidemiology illustrate how the product model may yield new insights about the role of prognostic factors.
为了得到更灵活的模型，Cox回归模型lambda(t;x) = lambda(0)(t)exp(betax)已通过不同的非参数模型估计技术进行了推广。一种推广是放宽x中的对数线性关系，即lambda(t;x) = lambda(0)(t)exp[r(x)]。另一种是放宽比例风险假设，即lambda(t;x) = lambda(0)(t)exp[beta(t)x]。这些推广通常被认为是相互独立的。我们提出乘积模型lambda(t;x) = lambda(0)(t)exp[beta(t)r(x)]，它允许对两种效应进行联合估计，并研究其性质。使用回归样条同时对描述时间依赖的beta(t)和非线性r(x)效应的函数进行建模，并通过最大偏似然法进行估计。我们提出似然比检验来比较替代模型。模拟表明，只要基于正确的模型，两种函数形状的恢复和检验的规模都相当准确。相比之下，如果模型设定错误，I型错误率可能会大幅膨胀，估计也会有很大偏差。癌症流行病学中的应用说明了乘积模型如何能对预后因素的作用产生新的见解。
