A Bayesian proportional hazards regression model with non-ignorably missing time-varying covariates

Missing covariate data are common in observational studies of time to an event, especially when covariates are repeatedly measured over time. Failure to account for the missing data can lead to bias or loss of efficiency, especially when the data are non-ignorably missing. Previous work has focused on the case of fixed covariates rather than those that are repeatedly measured over the follow-up period, hence, here we present a selection model that allows for proportional hazards regression with time-varying covariates when some covariates may be non-ignorably missing. We develop a fully Bayesian model and obtain posterior estimates of the parameters via the Gibbs sampler in WinBUGS. We illustrate our model with an analysis of post-diagnosis weight change and survival after breast cancer diagnosis in the Long Island Breast Cancer Study Project follow-up study. Our results indicate that post-diagnosis weight gain is associated with lower all-cause and breast cancer-specific survival among women diagnosed with new primary breast cancer. Our sensitivity analysis showed only slight differences between models with different assumptions on the missing data mechanism yet the complete-case analysis yielded markedly different results.     

Visualizing covariates in proportional hazards model

We present a graphical method called the rank‐hazard plot that visualizes the relative importance of covariates in a proportional hazards model. The key idea is to rank the covariate values and plot the relative hazard as a function of ranks scaled to interval [0, 1]. The relative hazard is plotted with respect to the reference hazard, which can be, for example, the hazard related to the median of the covariate. Transformation to scaled ranks allows plotting of covariates measured in different units in the same graph, which helps in the interpretation of the epidemiological relevance of the covariates. Rank‐hazard plots show the difference of hazards between the extremes of the covariate values present in the data and can be used as a tool to check if the proportional hazards assumption leads to reasonable estimates for individuals with extreme covariate values. Alternative covariate definitions or different transformations applied to covariates can be also compared using rank‐hazard plots. We apply rank‐hazard plots to the data from the FINRISK study where population‐based cohorts have been followed up for events of cardiovascular diseases and compare the relative importance of the covariates cholesterol, smoking, blood pressure and body mass index. The data from the Study to Understand Prognoses Preferences Outcomes and Risks of Treatment (SUPPORT) are used to visualize nonlinear covariate effects. The proposed graphics work in other regression models with different interpretations of the y‐axis. Copyright © 2009 John Wiley & Sons, Ltd.     

Non-parametric estimation for baseline hazards function and covariate effects with time-dependent covariates

Often in many biomedical and epidemiologic studies, estimating hazards function is of interest. The Breslow's estimator is commonly used for estimating the integrated baseline hazard, but this estimator requires the functional form of covariate effects to be correctly specified. It is generally difficult to identify the true functional form of covariate effects in the presence of time-dependent covariates. To provide a complementary method to the traditional proportional hazard model, we propose a tree-type method which enables simultaneously estimating both baseline hazards function and the effects of time-dependent covariates. Our interest will be focused on exploring the potential data structures rather than formal hypothesis testing. The proposed method approximates the baseline hazards and covariate effects with step-functions. The jump points in time and in covariate space are searched via an algorithm based on the improvement of the full log-likelihood function. In contrast to most other estimating methods, the proposed method estimates the hazards function rather than integrated hazards. The method is applied to model the risk of withdrawal in a clinical trial that evaluates the anti-depression treatment in preventing the development of clinical depression. Finally, the performance of the method is evaluated by several simulation studies.     

Time‐varying coefficient proportional hazards model with missing covariates

Missing covariates often arise in biomedical studies with survival outcomes. Existing approaches for missing covariates generally assume proportional hazards. The proportionality assumption may not hold in practice, as illustrated by data from a mouse leukemia study with covariate effects changing over time. To tackle this restriction, we study the missing data problem under the varying‐coefficient proportional hazards model. On the basis of the local partial likelihood approach, we develop inverse selection probability weighted estimators. We consider reweighting and augmentation techniques for possible improvement of efficiency and robustness. The proposed estimators are assessed via simulation studies and illustrated by application to the mouse leukemia data. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiple imputation of missing covariates for the Cox proportional hazards cure model

We explore several approaches for imputing partially observed covariates when the outcome of interest is a censored event time and when there is an underlying subset of the population that will never experience the event of interest. We call these subjects ‘cured’, and we consider the case where the data are modeled using a Cox proportional hazards (CPH) mixture cure model. We study covariate imputation approaches using fully conditional specification. We derive the exact conditional distribution and suggest a sampling scheme for imputing partially observed covariates in the CPH cure model setting. We also propose several approximations to the exact distribution that are simpler and more convenient to use for imputation. A simulation study demonstrates that the proposed imputation approaches outperform existing imputation approaches for survival data without a cure fraction in terms of bias in estimating CPH cure model parameters. We apply our multiple imputation techniques to a study of patients with head and neck cancer. Copyright © 2016 John Wiley & Sons, Ltd.     

Corrected score estimation in the proportional hazards model with misclassified discrete covariates

We consider Cox proportional hazards regression when the covariate vector includes error-prone discrete covariates along with error-free covariates, which may be discrete or continuous. The misclassification in the discrete error-prone covariates is allowed to be of any specified form. Building on the work of Nakamura and his colleagues, we present a corrected score method for this setting. The method can handle all three major study designs (internal validation design, external validation design, and replicate measures design), both functional and structural error models, and time-dependent covariates satisfying a certain 'localized error' condition. We derive the asymptotic properties of the method and indicate how to adjust the covariance matrix of the regression coefficient estimates to account for estimation of the misclassification matrix. We present the results of a finite-sample simulation study under Weibull survival with a single binary covariate having known misclassification rates. The performance of the method described here was similar to that of related methods we have examined in previous works. Specifically, our new estimator performed as well as or, in a few cases, better than the full Weibull maximum likelihood estimator. We also present simulation results for our method for the case where the misclassification probabilities are estimated from an external replicate measures study. Our method generally performed well in these simulations. The new estimator has a broader range of applicability than many other estimators proposed in the literature, including those described in our own earlier work, in that it can handle time-dependent covariates with an arbitrary misclassification structure. We illustrate the method on data from a study of the relationship between dietary calcium intake and distal colon cancer.     

Time-dependent covariates in the Cox proportional hazards model. Theory and practice

In survival analysis, exposure that appears or changes during the follow-up of subjects must be taken into account as a time-dependent covariate in the Cox proportional hazards model. Two types of time-dependent covariates are defined: covariates with unique change, and covariates with multiple changes. The way of taking into account such changes in the exposure is presented in theory and illustrated from a small sample of 5 subjects enrolled in a French HIV cohort. The problems raised by missing data as well as alternative but more sophisticated solutions are also evocated. Annexes include programs for SAS software, and results from the "Log" and "Results" windows.     

An analysis for menstrual data with time-varying covariates

This paper concerns the analysis of menstrual data; in particular, methodology to identify variables that contribute to the variability of menstrual cycles both within and between women. The basis for the proposed methodology is a parameterization of the mean length of a menstrual cycle conditional upon the past cycles and covariates. This approach accommodates the length-bias and censoring commonly found in menstrual data. Data from a longitudinal study of menstrual patterns and other variables among Lese women of the Ituri Forest, Zaire, illustrate the methodology. A small simulation illustrates the bias caused by incorrectly deleting the censored cycles.     

A comparison of smoothing techniques for CD4 data measured with error in a time-dependent Cox proportional hazards model

The use of CD4+T-lymphocyte counts as a covariate presents some unique challenges in survivorship analyses due to the variability of this marker. If one does not account for the measurement error component of this variability in some manner, the estimate of the relative risk parameter in a time-dependent Cox model is biased towards zero, and coverage levels of confidence intervals may be seriously incorrect. We use a two-stage approach to reduce the variability in the observed CD4 counts in order to obtain a more accurate estimate of the relative risk parameter and more valid summary statistics. In the first stage, population based smoothing methods derived from a random-effects model plus a stochastic process or individual based smoothing methods are used to replace the observed longitudinal CD4 counts with less variable imputes at each failure time. In the second stage, we use the imputes in a time-dependent Cox model to estimate the risk parameter and its associated summary statistics. We compare the smoothing methods in simulation studies and find that the use of these smoothing methods results in a substantial reduction in bias for the true risk parameter estimate, better efficiency, and more accurate coverage rates in confidence intervals. We apply our two-stage smoothing methods to the marker CD4 in the ACTG-019 clinical trial part B. © 1998 John Wiley & Sons, Ltd.     

Bayesian proportional hazards model with time-varying regression coefficients: a penalized Poisson regression approach

One can fruitfully approach survival problems without covariates in an actuarial way. In narrow time bins, the number of people at risk is counted together with the number of events. The relationship between time and probability of an event can then be estimated with a parametric or semi-parametric model. The number of events observed in each bin is described using a Poisson distribution with the log mean specified using a flexible penalized B-splines model with a large number of equidistant knots. Regression on pertinent covariates can easily be performed using the same log-linear model, leading to the classical proportional hazard model. We propose to extend that model by allowing the regression coefficients to vary in a smooth way with time. Penalized B-splines models will be proposed for each of these coefficients. We show how the regression parameters and the penalty weights can be estimated efficiently using Bayesian inference tools based on the Metropolis-adjusted Langevin algorithm.     

Practical problems in fitting a proportional hazards model to data with udated measurements of the covariates

We review and discuss the practical problems encountered when analysing the effect on survival of covariates which are measured repeatedly over time. Specific issues arise over and above those met with the standard proportional hazards model and concern all stages of data preparation, data analysis and interpretation of the results. Data from a randomized clinical trial of patients with primary biliary cirrhosis, on whom several measurements were taken at regular intervals after entry, are presented as an illustration.     

The impact of time-dependent bias in proportional hazards modelling

In the clinical literature, time-dependent exposure status has regularly been analysed as if known at time origin. Although statisticians agree that such an analysis yields biased results when analysing the effect on the time until some endpoint of interest, this paper is the first to study in detail the bias arising in a proportional hazards analysis. We show that the biased hazard ratio estimate will always be less than the unbiased one; this leads to either an inflated or a damped effect of exposure, depending on the sign of the correct log hazard ratio estimate. We find an explicit formula of the asymptotic bias based on generalized rank estimators, and we investigate the role of censoring, which may prevent an individual from being considered as being baseline exposed in the biased analysis. We illustrate our results with data on hospital infection status and different censoring patterns.     

Data generation for the Cox proportional hazards model with time‐dependent covariates: a method for medical researchers

The proliferation of longitudinal studies has increased the importance of statistical methods for time‐to‐event data that can incorporate time‐dependent covariates. The Cox proportional hazards model is one such method that is widely used. As more extensions of the Cox model with time‐dependent covariates are developed, simulations studies will grow in importance as well. An essential starting point for simulation studies of time‐to‐event models is the ability to produce simulated survival times from a known data generating process. This paper develops a method for the generation of survival times that follow a Cox proportional hazards model with time‐dependent covariates. The method presented relies on a simple transformation of random variables generated according to a truncated piecewise exponential distribution and allows practitioners great flexibility and control over both the number of time‐dependent covariates and the number of time periods in the duration of follow‐up measurement. Within this framework, an additional argument is suggested that allows researchers to generate time‐to‐event data in which covariates change at integer‐valued steps of the time scale. The purpose of this approach is to produce data for simulation experiments that mimic the types of data structures applied that researchers encounter when using longitudinal biomedical data. Validity is assessed in a set of simulation experiments, and results indicate that the proposed procedure performs well in producing data that conform to the assumptions of the Cox proportional hazards model. Copyright © 2013 John Wiley & Sons, Ltd.     

尘肺患者生存年限的COX比例风险模型分析

目的探讨影响尘肺病患者生存年限的因素。方法以尘肺患者的死亡风险函数为应变量，生存年限为暴露时间变量，发病工龄、工种、是否并发肺结核、Ⅱ期晋升为Ⅲ期的年限及尘肺种类为方程自变量，运用STATA统计分析软件进行尘肺病例生存年限的COX比例风险模型分析。结果COX比例风险模型方程为：λ（t）=λ0（ε）exp（-0．0249×fb＋0．0344×gz＋0．2059×tb-0．0215×nx2—0．0280×cf），矽肺相对于煤工尘肺的COX比例风险模型方程为：λ（t）=λ0（t）exp（0．5289×cf）。由COX模型方程可得出各自变量的相对危险度（RR）值分别为1．025、1．035、1．229、1．022、1．697，提示，发病工龄对患者死亡的影响呈递减的形式，发病工龄每减少1年，尘肺病例在某时刻死亡的RR增加1．025倍；井下采矿工尘肺病例在某时刻死亡的RR值是非井下采矿工尘肺病例的1．035倍；尘肺并发肺结核在某时刻死亡的RR值是单纯尘肺的1．229倍；Ⅱ期尘肺晋升为Ⅲ期的年限每减少1年，尘肺病例在某时刻死亡的RR值增加1．022倍；矽肺病例在某时刻死亡的RR值是煤工尘肺的1．697倍。结论影响尘肺病例生存年限的因素有发病工龄、工种、是否并发肺结核、Ⅱ期尘肺晋升为Ⅲ期的年限及尘肺种类。     

Transient effects in the cox proportional hazards regression model

We consider a model for mortality rates that includes both the long and short term effects of switching from an initial to a second state, for example, when patients receive an initial treatment and then switch to a second treatment. We include transient effects associated with the switch in the model through the use of time-dependent covariates. One can choose the form of the time-dependent covariate to correspond with a variety of possible transition patterns. We use an exponential decay model to compare the survival experience of transplant versus dialysis treatment of end stage renal disease (ESRD) patients from the Michigan Kidney Registry (MKR). This model involves a hazard function that has an initial effect in mortality at the time of transplant, expected to be higher, followed by a smooth exponential decay to a long term effect, expected to be lower than the risk for those remaining on dialysis. Cox and Oakes used this model to analyse the Stanford Heart Transplant data. The model implicitly suggests there is a time at which the hazard curves (and survival curves) for the treatment groups cross. Those crossing times are useful in advising patients who have the option of receiving a transplant. We describe methods for obtaining estimates of the crossing times and their associated variances, and then apply them in analysing the MKR data.     

Mediation analysis with causally ordered mediators using Cox proportional hazards model

Causal mediation analysis aims to investigate the mechanism linking an exposure and an outcome. However, studies regarding mediation effects on survival outcomes are limited, particularly in multi-mediator settings. The existing multi-mediator analyses for survival outcomes are either performed under special model specifications such as probit models or additive hazard models, or they assume a rare outcome. Here, we propose a novel multi-mediation analysis based on the widely used Cox proportional hazards model without the rare outcome assumption. We develop a methodology under a counterfactual framework to identify path-specific effects (PSEs) of the exposure on the outcome through the mediator(s) and derive the closed-form formula for PSEs on a transformed survival time. Moreover, we show that the convolution of an extreme value and Gaussian random variables converges to another Gaussian, provided that the variance of the original Gaussian gets large. Based on that, we further derive closed-form expressions for PSEs on survival probabilities. Asymptotic properties are established for both estimators. Extensive simulation is conducted to evaluate the finite sample performance of our proposed estimators and to compare with existing methods. The utility of the proposed method is illustrated in a hepatitis study of liver cancer risk.     

The restricted cubic spline as baseline hazard in the proportional hazards model with step function time-dependent covariables

We incorporate a cubic spline function where the tails are linearly constrained, as the baseline hazard, into the proportional hazards model. We show estimation of covariable coefficients and survival probabilities with this model to be as efficient statistically as with the Cox proportional hazards model when covariables are fixed. Examples show that the inclusion of time-dependent covariables defined as step functions into the restricted cubic spline proportional hazards model reduces computation time by a factor of 213 over the Cox model. Advantages of the spline model also include flexibility of the hazard, smooth survival curves, and confidence limits for the survival and hazard estimates when there are time-dependent covariables present.     

Weighted estimating equations for longitudinal studies with death and non-monotone missing time-dependent covariates and outcomes

We propose a marginal modeling approach to estimate the association between a time-dependent covariate and an outcome in longitudinal studies where some study participants die during follow-up and both variables have non-monotone response patterns. The proposed method is an extension of weighted estimating equations that allows the outcome and covariate to have different missing-data patterns. We present methods for both random and non-random missing-data mechanisms. A study of functional recovery in a cohort of elderly female hip-fracture patients motivates the approach.     

The analysis of binary longitudinal data with time-dependent covariates

We consider longitudinal studies with binary outcomes that are measured repeatedly on subjects over time. The goal of our analysis was to fit a logistic model that relates the expected value of the outcomes with explanatory variables that are measured on each subject. However, additional care must be taken to adjust for the association between the repeated measurements on each subject. We propose a new maximum likelihood method for covariates that may be fixed or time varying. We also implement and make comparisons with two other approaches: generalized estimating equations, which may be more robust to misspecification of the true correlation structure, and alternating logistic regression, which models association via odds ratios that are subject to less restrictive constraints than are correlations. The proposed estimation procedure will yield consistent and asymptotically normal estimates of the regression and correlation parameters if the correlation on consecutive measurements on a subject is correctly specified. Simulations demonstrate that our approach can yield improved efficiency in estimation of the regression parameter; for equally spaced and complete data, the gains in efficiency were greatest for the parameter associated with a time-by-group interaction term and for stronger values of the correlation. For unequally spaced data and with dropout according to a missing-at-random mechanism, MARK1ML with correctly specified consecutive correlations yielded substantial improvements in terms of both bias and efficiency. We present an analysis to demonstrate application of the methods we consider. We also offer an R function for easy implementation of our approach.