Empirical comparisons of proportional hazards and logistic regression models

We compare parameter estimates from the proportional hazards model, the cumulative logistic model and a new modified logistic model (referred to as the person-time logistic model), with the use of simulated data sets and with the following quantities varied: disease incidence, risk factor strength, length of follow-up, the proportion censored, non-proportional hazards, and sample size. Parameter estimates from the person-time logistic regression model closely approximated those from the Cox model when the survival time distribution was close to exponential, but could differ substantially in other situations. We found parameter estimates from the cumulative logistic model similar to those from the Cox and person-time logistic models when the disease was rare, the risk factor moderate, and censoring rates similar across the covariates. We also compare the models with analysis of a real data set that involves the relationship of age, race, sex, blood pressure, and smoking to subsequent mortality. In this example, the length of follow-up among survivors varied from 5 to 14 years and the Cox and person-time logistic approaches gave nearly identical results. The cumulative logistic results had somewhat larger p-values but were substantively similar for all but one coefficient (the age-race interaction). The latter difference reflects differential censoring rates by age, race and sex.     

队列研究资料分析中Cox比例风险回归模型的时间尺度选择和应用

Cox比例风险回归模型（Cox模型）是时间-事件数据分析中常用的多因素分析方法,拟合Cox模型时一个关键问题是如何选择合适的与结局事件发生相关的时间尺度。目前国内开展的队列研究在资料分析中较少关注Cox模型的时间尺度选择问题。本研究对文献报道中常见的几种时间尺度选择策略进行简要介绍和比较;并利用上海女性健康队列资料,以中心性肥胖与肝癌发病风险的关联为例,说明选择不同时间尺度的Cox模型对数据分析结果的影响;在此基础上提出几点Cox模型时间尺度选择上的建议,以期为队列研究资料的分析提供参考。     

实现logistic与Cox回归相乘相加交互作用的临床实践宏程序

病例对照研究常采用条件或非条件logistic分析,生存资料分析常采用Cox比例模型,但多数文献仅纳入主效应模型,然而广义线性模型不同于一般线性模型,其交互作用分为相乘交互与相加交互作用,前者只有统计学意义而后者更符合生物学意义。笔者以SAS 9.4软件编写宏,在计算logistic与Cox相乘交互项同时计算交互对比度、归因比、交互作用指数指标及利用Wald、Delta、PL(profile likelihood) 3种方法的可信区间评价相加交互作用,便于临床流行病学与遗传学大数据分析相乘相加交互作用时参考。     

A simple approach to power and sample size calculations in logistic regression and Cox regression models

For a given regression problem it is possible to identify a suitably defined equivalent two-sample problem such that the power or sample size obtained for the two-sample problem also applies to the regression problem. For a standard linear regression model the equivalent two-sample problem is easily identified, but for generalized linear models and for Cox regression models the situation is more complicated. An approximately equivalent two-sample problem may, however, also be identified here. In particular, we show that for logistic regression and Cox regression models the equivalent two-sample problem is obtained by selecting two equally sized samples for which the parameters differ by a value equal to the slope times twice the standard deviation of the independent variable and further requiring that the overall expected number of events is unchanged. In a simulation study we examine the validity of this approach to power calculations in logistic regression and Cox regression models. Several different covariate distributions are considered for selected values of the overall response probability and a range of alternatives. For the Cox regression model we consider both constant and non-constant hazard rates. The results show that in general the approach is remarkably accurate even in relatively small samples. Some discrepancies are, however, found in small samples with few events and a highly skewed covariate distribution. Comparison with results based on alternative methods for logistic regression models with a single continuous covariate indicates that the proposed method is at least as good as its competitors. The method is easy to implement and therefore provides a simple way to extend the range of problems that can be covered by the usual formulas for power and sample size determination.     

手术类单病种费用影响因素的Cox回归分析

目的为了控制日益增长的医疗费用,通过Cox回归分析找出影响单病种费用的主要因素,为制定有针对性的费用控制政策提供参考依据。方法对3768例手术类单病种病人费用的影响因素进行Cox回归分析。结果提示手术类单病种发生高费用与病人性别、年龄、手术方式、麻醉方式、麻醉检测时间、药占比、病人付款方式密切相关。结论通过COX回归分析,明确了手术类单病种费用的主要影响因素,提出严格筛选病种、结合合理的临床路径和合理的花费开展费用控制工作。     

Power analysis for multivariable Cox regression models

In power analysis for multivariable Cox regression models, variance of the estimated log-hazard ratio for the treatment effect is usually approximated by inverting the expected null information matrix. Because, in many typical power analysis settings, assumed true values of the hazard ratios are not necessarily close to unity, the accuracy of this approximation is not theoretically guaranteed. To address this problem, the null variance expression in power calculations can be replaced with one of the alternative expressions derived under the assumed true value of the hazard ratio for the treatment effect. This approach is explored analytically and by simulations in the present paper. We consider several alternative variance expressions and compare their performance to that of the traditional null variance expression. Theoretical analysis and simulations demonstrate that, whereas the null variance expression performs well in many nonnull settings, it can also be very inaccurate, substantially underestimating, or overestimating the true variance in a wide range of realistic scenarios, particularly those where the numbers of treated and control subjects are very different and the true hazard ratio is not close to one. The alternative variance expressions have much better theoretical properties, confirmed in simulations. The most accurate of these expressions has a relatively simple form. It is the sum of inverse expected event counts under treatment and under control scaled up by a variance inflation factor.     

Generating survival times to simulate Cox proportional hazards models

Simulation studies present an important statistical tool to investigate the performance, properties and adequacy of statistical models in pre-specified situations. One of the most important statistical models in medical research is the proportional hazards model of Cox. In this paper, techniques to generate survival times for simulation studies regarding Cox proportional hazards models are presented. A general formula describing the relation between the hazard and the corresponding survival time of the Cox model is derived, which is useful in simulation studies. It is shown how the exponential, the Weibull and the Gompertz distribution can be applied to generate appropriate survival times for simulation studies. Additionally, the general relation between hazard and survival time can be used to develop own distributions for special situations and to handle flexibly parameterized proportional hazards models. The use of distributions other than the exponential distribution is indispensable to investigate the characteristics of the Cox proportional hazards model, especially in non-standard situations, where the partial likelihood depends on the baseline hazard. A simulation study investigating the effect of measurement errors in the German Uranium Miners Cohort Study is considered to illustrate the proposed simulation techniques and to emphasize the importance of a careful modelling of the baseline hazard in Cox models.     

Cox模型标准回归系数的探讨

鉴于Cox模型标准回归系数目前尚缺乏合理的定义,笔者提出以下定义和推论:以标准化变量作Cox模型分析所获得的回归系数称为标准回归系数β′(与线性回归类同);β′等于回归系数β乘以变量的标准差s,即β′=βs。     

On model specification and selection of the Cox proportional hazards model

Prognosis plays a pivotal role in patient management and trial design. A useful prognostic model should correctly identify important risk factors and estimate their effects. In this article, we discuss several challenges in selecting prognostic factors and estimating their effects using the Cox proportional hazards model. Although a flexible semiparametric form, the Cox's model is not entirely exempt from model misspecification. To minimize possible misspecification, instead of imposing traditional linear assumption, flexible modeling techniques have been proposed to accommodate the nonlinear effect. We first review several existing nonparametric estimation and selection procedures and then present a numerical study to compare the performance between parametric and nonparametric procedures. We demonstrate the impact of model misspecification on variable selection and model prediction using a simulation study and an example from a phase III trial in prostate cancer. Copyright © 2013 John Wiley & Sons, Ltd.     

Cox回归模型与对数线性回归模型在生存分析中应用的比较

运用Ｃｏｘ回归模型和对数线性回归模型对１６８９例肝癌病人生存时间的分析，发现Ｃｏｘ回归模型能够提供主要的预后影响因素，其结果与特定的参数回归模型相接近，使临床上能够快速地获得预后的影响因素。     

Misspecification of Cox regression models with composite endpoints

Researchers routinely adopt composite endpoints in multicenter randomized trials designed to evaluate the effect of experimental interventions in cardiovascular disease, diabetes, and cancer. Despite their widespread use, relatively little attention has been paid to the statistical properties of estimators of treatment effect based on composite endpoints. We consider this here in the context of multivariate models for time to event data in which copula functions link marginal distributions with a proportional hazards structure. We then examine the asymptotic and empirical properties of the estimator of treatment effect arising from a Cox regression model for the time to the first event. We point out that even when the treatment effect is the same for the component events, the limiting value of the estimator based on the composite endpoint is usually inconsistent for this common value. We find that in this context the limiting value is determined by the degree of association between the events, the stochastic ordering of events, and the censoring distribution. Within the framework adopted, marginal methods for the analysis of multivariate failure time data yield consistent estimators of treatment effect and are therefore preferred. We illustrate the methods by application to a recent asthma study. Copyright © 2012 John Wiley & Sons, Ltd.     

Cox回归模型在产后哺乳与避孕相关关系分析中的应用

应用Cox伴随时间变化变量回归模型来准确的评价产后哺乳与避孕的关系。从天津市河西区随机抽取9个街,344名产后12～18个月的哺乳期妇女进行问卷调查。调查结果表明产后启用避孕措施的时间与哺乳类型无关,但已恢复月经的妇女比仍闭经的妇女更可能选用避孕措施。年轻妇女产后性生活恢复和启用避孕措施的时间较早。建议妇女产后恢复性生活之后,无论完全哺乳还是闭经,都应尽早选择一种适宜的避孕方法。     

金昌队列人群痛风发病危险因素的Cox回归分析

目的 了解金昌队列人群痛风的发病状况及影响因素,为制订痛风的防治措施提供科学依据。方法 采用前瞻性队列研究方法,以金昌队列基线资料中未患痛风的人群作为研究对象,于2013年1月24日至2015年11月24日完成随访问卷调查、体格检查和实验室检测。采用Cox 回归模型分析金昌队列人群中痛风发病的影响因素,并用对数线性模型分析各影响因素之间的交互作用。结果 33 153例随访人群中新发痛风病例277例,痛风发病率为0.8%,男性整体上高于女性,但在60岁以后男女性痛风发病率相当。多因素Cox 回归分析结果显示,年龄在40岁以上（40~59岁：HR=2.982, 95%CI：1.503~5.981; 60~91岁：HR=2.588, 95%CI：1.107~6.049）、大量酒精摄入（HR=2.234, 95%CI：1.128~4.427）、肥胖（HR=2.204, 95%CI：1.216~3.997）、糖尿病（HR=2.725, 95%CI：1.500~4.950）和高尿酸（HR=5.963, 95%CI：3.577~9.943）是痛风发病的危险因素,每周豆类摄入 ≥ 250 g（HR=0.528, 95%CI：0.345~0.808）和经常体育锻炼（HR=0.499, 95%CI：0.286~0.869）是痛风发病的保护性因素。对数线性模型交互作用分析结果显示,各影响因素之间存在二阶效应。结论 年龄、豆类、酒类、体育锻炼、肥胖、糖尿病和高尿酸是痛风发病的重要影响因素。培养良好的生活饮食习惯,定期进行体检,有利于预防和控制该职业人群痛风病的发生。     

The cost of checking proportional hazards

Confidence intervals (CIs) and the reported predictive ability of statistical models may be misleading if one ignores uncertainty in the model selection procedure. When analyzing time-to-event data using Cox regression, one typically checks the proportional hazards (PH) assumption and subsequently alters the model to address any violations. Such an examination and correction constitute a model selection procedure, and, if not accounted for, could result in misleading CI. With the bootstrap, I study the impact of checking the PH assumption using (1) data to predict AIDS-free survival among HIV-infected patients initiating antiretroviral therapy and (2) simulated data. In the HIV study, due to non-PH, a Cox model was stratified on age quintiles. Interestingly, bootstrap CIs that ignored the PH check (always stratified on age quintiles) were wider than those which accounted for the PH check (on each bootstrap replication PH was tested and corrected through stratification only if violated). Simulations demonstrated that such a phenomenon is not an anomaly, although on average CIs widen when accounting for the PH check. In most simulation scenarios, coverage probabilities adjusting and not adjusting for the PH check were similar. However, when data were generated under a minor PH violation, the 95 per cent bootstrap CI ignoring the PH check had a coverage of 0.77 as opposed to 0.95 for CI accounting for the PH check. The impact of checking the PH assumption is greatest when the p-value of the test for PH is close to the test's chosen Type I error probability.     

Assessment of stratum-covariate interactions in Cox's proportional hazards regression model

We assess stratum (e.g. treatment) interactions with covariates and with the baseline hazard function in the proportional hazards (PH) regression model for lifetime data. We consider models incorporating stratum interactions both with and without stratification of the risk sets in the likelihood function, and describe likelihood ratio statistics for tests of the presence of these interactions. We also present step-down methods for building reduced models which include stratum-specific parameters corresponding to covariates which interact with treatment. We apply PH models with such interactions to a clinical trial of DES in the treatment of prostate cancer to determine optimal treatment conditional on each patient's covariates.     

Bayesian random threshold estimation in a Cox proportional hazards cure model

In this paper, we develop a Bayesian approach to estimate a Cox proportional hazards model that allows a threshold in the regression coefficient, when some fraction of subjects are not susceptible to the event of interest. A data augmentation scheme with latent binary cure indicators is adopted to simplify the Markov chain Monte Carlo implementation. Given the binary cure indicators, the Cox cure model reduces to a standard Cox model and a logistic regression model. Furthermore, the threshold detection problem reverts to a threshold problem in a regular Cox model. The baseline cumulative hazard for the Cox model is formulated non‐parametrically using counting processes with a gamma process prior. Simulation studies demonstrate that the method provides accurate point and interval estimates. Application to a data set of oropharynx cancer patients suggests a significant threshold in age at diagnosis such that the effect of gender on disease‐specific survival changes after the threshold. Copyright © 2013 John Wiley & Sons, Ltd.     

The use of proportional hazards regression in investigating dropout rates in a longitudinal study

One difficulty with the interpretation of data from longitudinal studies is the bias associated with those who do not complete the study. In a 12-month study on the occurrence of musculoskeletal problems in 966 runners (583 of whom completed the study), a proportional hazards model with time-dependent covariates was used to assess factors associated with dropout at the various stages of the study. This approach allowed examination of baseline factors as well as the effect of change in mileage, the occurrence of a musculoskeletal problem, or the occurrence of another health problem on the rate of dropout. Those most likely to drop out of the study were younger and heavier at baseline and, prior to drop out, were less likely to experience general health problems and more likely to show a 40% decline in weekly running mileage in the month before dropout. Examination of factors associated with dropout is important since factors influencing dropout may also affect the study outcome for the risk factor analysis (a musculoskeletal problem severe enough to be seen by a physician). The results of the dropout analysis can be used to guide in the choice of analytic methods and to aid in the interpretation of the risk factor analyses.     

核函数logistic回归模型在全基因组关联研究中的应用

[导读]探讨基于基因水平的核函数logistic回归模型及其在全基因组关联研究中的应用.以全基因组关联研究模拟数据为例,介绍核函数logistic回归模型在基因水平检测遗传变异与复杂性疾病之间关联的分析策略.模拟结果表明,在所有已知基因检验结果中致病位点所在基因假设检验的P值最小.结果提示基于基因水平的核函数logistic回归模型能够充分提取和综合基因中多个遗传突变位点信息,降低统计学检验的自由度,同时还能够控制多种协变量因素和交互作用,在检测致病基因与疾病关联时具有一定的效能.     

Inference for a family of survival models encompassing the proportional hazards and proportional odds models

For survival data regression, the Cox proportional hazards model is the most popular model, but in certain situations the Cox model is inappropriate. Various authors have proposed the proportional odds model as an alternative. Yang and Prentice recently presented a number of easily implemented estimators for the proportional odds model. Here we show how to extend the methods of Yang and Prentice to a family of survival models that includes the proportional hazards model and proportional odds model as special cases. The model is defined in terms of a Box-Cox transformation of the survival function, indexed by a transformation parameter rho. This model has been discussed by other authors, and is related to the Harrington-Fleming G(rho) family of tests and to frailty models. We discuss inference for the case where rho is known and the case where rho must be estimated. We present a simulation study of a pseudo-likelihood estimator and a martingale residual estimator. We find that the methods perform reasonably. We apply our model to a real data set.