Perperoglou Aris, le Cessie Saskia, van Houwelingen Hans C
Department of Medical Statistics, Leiden University Medical Center, University of Leiden, P.O. Box 9604, 2300 RC, Leiden, The Netherlands.
Stat Med. 2006 Aug 30;25(16):2831-45. doi: 10.1002/sim.2360.
The Cox proportional hazards model is the most common method to analyse survival data. However, the proportional hazards assumption might not hold. The natural extension of the Cox model is to introduce time-varying effects of the covariates. For some covariates such as (surgical)treatment non-proportionality could be expected beforehand. For some other covariates the non-proportionality only becomes apparent if the follow-up is long enough. It is often observed that all covariates show similar decaying effects over time. Such behaviour could be explained by the popular (gamma-) frailty model. However, the (marginal) effects of covariates in frailty models are not easy to interpret. In this paper we propose the reduced-rank model for time-varying effects of covariates. Starting point is a Cox model with p covariates and time-varying effects modelled by q time functions (constant included), leading to a pxq structure matrix that contains the regression coefficients for all covariate by time function interactions. By reducing the rank of this structure matrix a whole range of models is introduced, from the very flexible full-rank model (identical to a Cox model with time-varying effects) to the very rigid rank one model that mimics the structure of a gamma-frailty model, but is easier to interpret. We illustrate these models with an application to ovarian cancer patients.
Cox比例风险模型是分析生存数据最常用的方法。然而，比例风险假设可能不成立。Cox模型的自然扩展是引入协变量的时变效应。对于某些协变量，如（手术）治疗，非比例性可以预先预期。对于其他一些协变量，只有随访时间足够长时，非比例性才会显现出来。人们经常观察到，所有协变量随时间都呈现出类似的衰减效应。这种行为可以用流行的（伽马）脆弱模型来解释。然而，脆弱模型中协变量的（边际）效应不易解释。在本文中，我们提出了协变量时变效应的降秩模型。出发点是一个具有p个协变量的Cox模型，其时变效应由q个时间函数（包括常数）建模，从而得到一个pxq结构矩阵，该矩阵包含所有协变量与时间函数相互作用的回归系数。通过降低这个结构矩阵的秩，引入了一系列模型，从非常灵活的满秩模型（等同于具有时变效应的Cox模型）到非常严格的秩一模型，该模型模仿伽马脆弱模型的结构，但更易于解释。我们通过应用于卵巢癌患者来说明这些模型。
