Flexible parametric models for relative survival, with application in coronary heart disease

Relative survival is frequently used in population-based studies as a method for estimating disease-related mortality without the need for information on cause of death. We propose an extension to relative survival of a flexible parametric model proposed by Royston and Parmar for censored survival data. The model provides smooth estimates of the relative survival and excess mortality rates by using restricted cubic splines on the log cumulative excess hazard scale. The approach has several advantages over some of the more standard relative survival models, which adopt a piecewise approach, the main being the ability to model time on a continuous scale, the survival and hazard functions are obtained analytically and it does not use split-time data.     

A pattern-mixture model for the analysis of censored quality-of-life data

We propose a pattern-mixture model for describing the joint distribution of incomplete repeated measurements of quality of life (QoL) and right-censored survival times. The model assumes that the survival times follow a multinomial distribution and that the quality of life outcome follows a multivariate normal distribution conditional on the survival time. We estimate the model using a Bayesian approach by importance sampling. We then use simulated parameters to create multiple imputations of the censored QoL outcomes, which can then be used to calculate individual values of quality-adjusted life-years (QALYs). We apply the method to data from the Randomized Evaluation of Mechanical Assistance in the Treatment of Congestive Heart Failure (REMATCH) clinical trial.     

Path consistent model selection in additive risk model via Lasso

As a flexible alternative to the Cox model, the additive risk model assumes that the hazard function is the sum of the baseline hazard and a regression function of covariates. For right censored survival data when variable selection is needed along with model estimation, we propose a path consistent model selector using a modified Lasso approach, under the additive risk model assumption. We show that the proposed estimator possesses the oracle variable selection and estimation property. Applications of the proposed approach to three right censored survival data sets show that the proposed modified Lasso yields parsimonious models with satisfactory estimation and prediction results.     

A multiple imputation approach for clustered interval‐censored survival data

Multivariate interval‐censored failure time data arise commonly in many studies of epidemiology and biomedicine. Analysis of these type of data is more challenging than the right‐censored data. We propose a simple multiple imputation strategy to recover the order of occurrences based on the interval‐censored event times using a conditional predictive distribution function derived from a parametric gamma random effects model. By imputing the interval‐censored failure times, the estimation of the regression and dependence parameters in the context of a gamma frailty proportional hazards model using the well‐developed EM algorithm is made possible. A robust estimator for the covariance matrix is suggested to adjust for the possible misspecification of the parametric baseline hazard function. The finite sample properties of the proposed method are investigated via simulation. The performance of the proposed method is highly satisfactory, whereas the computation burden is minimal. The proposed method is also applied to the diabetic retinopathy study (DRS) data for illustration purpose and the estimates are compared with those based on other existing methods for bivariate grouped survival data. Copyright © 2010 John Wiley & Sons, Ltd.     

INTERVAL CENSORED SURVIVAL DATA: A GENERALIZED LINEAR MODELLING APPROACH

A method is described for weak parametric modelling of arbitrarily interval censored survival data using generalized linear models. The method makes use of an associated Bernoulli model, with standard errors based on the observed information matrix. Three types of models are discussed: additive and multiplicative hazard models with piecewise constant baseline hazard, and a proportional hazards model with discrete baseline survivor function. These models may be fitted in the statistical package GLIM.     

EXPLAINED VARIATION IN SURVIVAL ANALYSIS

Several measures of explained variation have been suggested for the Cox proportional hazards regression model. We have categorized these measures into three classes which correspond to three different definitions of multiple R² of the general linear model. In an empirical study we compared the performance of these measures and classified them by their adherence to a set of criteria which we think should be met by a measure of explained variation for survival data. We suggest that currently there is no uniformly superior measure, particularly as the concepts of either uncensored or censored populations may lead to different choices. For uncensored populations, a measure by Kent and O'Quigley and the squared rank correlation between survival time and the predictor from a Cox regression model appear recommendable choices. For the latter, censored survival times are terminated using a very recent data augmentation algorithm for multiple imputation under proportional hazards. With censored populations, Schemper's measure, V₂, could be considered. We give an introductory example, discuss aspects of application and stress the desirability of routinely evaluating explained variation in studies of survival.     

Simulating survival data with predefined censoring rates for proportional hazards models

The proportional hazard model is one of the most important statistical models used in medical research involving time‐to‐event data. Simulation studies are routinely used to evaluate the performance and properties of the model and other alternative statistical models for time‐to‐event outcomes under a variety of situations. Complex simulations that examine multiple situations with different censoring rates demand approaches that can accommodate this variety. In this paper, we propose a general framework for simulating right‐censored survival data for proportional hazards models by simultaneously incorporating a baseline hazard function from a known survival distribution, a known censoring time distribution, and a set of baseline covariates. Specifically, we present scenarios in which time to event is generated from exponential or Weibull distributions and censoring time has a uniform or Weibull distribution. The proposed framework incorporates any combination of covariate distributions. We describe the steps involved in nested numerical integration and using a root‐finding algorithm to choose the censoring parameter that achieves predefined censoring rates in simulated survival data. We conducted simulation studies to assess the performance of the proposed framework. We demonstrated the application of the new framework in a comprehensively designed simulation study. We investigated the effect of censoring rate on potential bias in estimating the conditional treatment effect using the proportional hazard model in the presence of unmeasured confounding variables. Copyright © 2016 John Wiley & Sons, Ltd.     

Comparison of hypertabastic survival model with other unimodal hazard rate functions using a goodness‐of‐fit test

We studied the problem of testing a hypothesized distribution in survival regression models when the data is right censored and survival times are influenced by covariates. A modified chi‐squared type test, known as Nikulin‐Rao‐Robson statistic, is applied for the comparison of accelerated failure time models. This statistic is used to test the goodness‐of‐fit for hypertabastic survival model and four other unimodal hazard rate functions. The results of simulation study showed that the hypertabastic distribution can be used as an alternative to log‐logistic and log‐normal distribution. In statistical modeling, because of its flexible shape of hazard functions, this distribution can also be used as a competitor of Birnbaum‐Saunders and inverse Gaussian distributions. The results for the real data application are shown. Copyright © 2017 John Wiley & Sons, Ltd.     

A multivariate cure model for left‐censored and right‐censored data with application to colorectal cancer screening patterns

We develop a multivariate cure survival model to estimate lifetime patterns of colorectal cancer screening. Screening data cover long periods of time, with sparse observations for each person. Some events may occur before the study begins or after the study ends, so the data are both left‐censored and right‐censored, and some individuals are never screened (the ‘cured’ population). We propose a multivariate parametric cure model that can be used with left‐censored and right‐censored data. Our model allows for the estimation of the time to screening as well as the average number of times individuals will be screened. We calculate likelihood functions based on the observations for each subject using a distribution that accounts for within‐subject correlation and estimate parameters using Markov chain Monte Carlo methods. We apply our methods to the estimation of lifetime colorectal cancer screening behavior in the SEER‐Medicare data set. Copyright © 2016 John Wiley & Sons, Ltd.     

Joint modelling of repeated measurements and time-to-event outcomes: the fourth Armitage lecture

In many longitudinal studies, the outcomes recorded on each subject include both a sequence of repeated measurements at pre-specified times and the time at which an event of particular interest occurs: for example, death, recurrence of symptoms or drop out from the study. The event time for each subject may be recorded exactly, interval censored or right censored. The term joint modelling refers to the statistical analysis of the resulting data while taking account of any association between the repeated measurement and time-to-event outcomes. In this paper, we first discuss different approaches to joint modelling and argue that the analysis strategy should depend on the scientific focus of the study. We then describe in detail a particularly simple, fully parametric approach. Finally, we use this approach to re-analyse data from a clinical trial of drug therapies for schizophrenic patients, in which the event time is an interval-censored or right-censored time to withdrawal from the study due to adverse side effects.     

Methods for comparing center‐specific survival outcomes using direct standardization

The evaluation of center‐specific outcomes is often through survival analysis methods. Such evaluations must account for differences in the distribution of patient characteristics across centers. In the context of censored event times, it is also important that the measure chosen to evaluate centers not be influenced by imbalances in the center‐specific censoring distributions. The practice of using center indicators in a hazard regression model is often invalid, inconvenient, or undesirable to carry out. We propose a semiparametric version of the standardized rate ratio (SRR) useful for the evaluation of centers with respect to a right‐censored event time. The SRR for center j can be interpreted as the ratio of the expected number of deaths in the total population (if the total population were in fact subject to the center j mortality hazard) to the observed number of events. The proposed measure is not affected by differences in center‐specific covariate or censoring distributions. Asymptotic properties of the proposed estimators are derived, with finite‐sample properties examined through simulation studies. The proposed methods are applied to national kidney transplant data. Copyright © 2014 John Wiley & Sons, Ltd.     

A local likelihood proportional hazards model for interval censored data.

We discuss the use of local likelihood methods to fit proportional hazards regression models to right and interval censored data. The assumed model allows for an arbitrary, smoothed baseline hazard on which a vector of covariates operates in a proportional manner, and thus produces an interpretable baseline hazard function along with estimates of global covariate effects. For estimation, we extend the modified EM algorithm suggested by Betensky, Lindsey, Ryan and Wand. We illustrate the method with data on times to deterioration of breast cosmeses and HIV-1 infection rates among haemophiliacs.     

Linear regression for bivariate censored data via multiple imputation.

Bivariate survival data arise, for example, in twin studies and studies of both eyes or ears of the same individual. Often it is of interest to regress the survival times on a set of predictors. In this paper we extend Wei and Tanner's multiple imputation approach for linear regression with univariate censored data to bivariate censored data. We formulate a class of censored bivariate linear regression methods by iterating between the following two steps: 1. the data is augmented by imputing survival times for censored observations; 2. a linear model is fit to the imputed complete data. We consider three different methods to implement these two steps. In particular, the marginal (independence) approach ignores the possible correlation between two survival times when estimating the regression coefficient. To improve the efficiency, we propose two methods that account for the correlation between the survival times. First, we improve the efficiency by using generalized least squares regression in step 2. Second, instead of generating data from an estimate of the marginal distribution we generate data from a bivariate log-spline density estimate in step 1. Through simulation studies we find that the performance of the two methods that take the dependence into account is close and that they are both more efficient than the marginal approach. The methods are applied to a data set from an otitis media clinical trial.     

Assessment of covariate effects in Aalen's additive hazard model

We study the properties of test statistics for a covariate effect in Aalen's additive hazard model and propose several new test statistics. The proposed statistics are derived by using the weights from linear rank statistics for comparing two survival curves. We compare these statistics with the two statistics proposed by Aalen using Monte Carlo simulations. Several different survival configurations are considered in the simulation study: proportional hazards; crossing hazards; hazard differences early in time, and hazard differences for large survival times. Of the proposed test statistics, one is superior for detecting hazard differences for large survival times and another is superior for detecting early hazard differences and crossing hazards. © 1998 John Wiley & Sons, Ltd.     

Effects of mid-point imputation on the analysis of doubly censored data.

Doubly censored data arise in some cohort studies of the AIDS incubation period because the time of infection may be known only up to an interval defined by two successive screening tests for HIV antibody. A simple analytic approach is to impute the infection time by the mid-point of the interval and then apply standard survival techniques for right censored data. The objective of this paper is to investigate the statistical properties of such a mid-point imputation approach. We investigated the asymptotic bias of the Kaplan-Meier estimate, coverage probabilities of associated confidence intervals, bias in hazard ratio, and the size of the logrank test. We show that the statistical properties of mid-point imputation depend strongly on the underlying distributions of infection times and the incubation periods, and the width of the interval between screening tests. In the absence of treatment, the median incubation period of HIV infection is approximately 10 years, and we conclude that, for this situation, mid-point imputation is a reasonable procedure for interval widths of 2 years or less.     

生存分析方法在道路交通事故致颅脑损伤住院时间估计中的应用

目的探讨用生存分析方法对道路交通事故致颅脑损伤住院时间进行估计。方法采用回顾性队列研究设计,从住院病历中收集因道路交通事故致颅脑损伤的病例(1294例)信息,对未死亡病例(1162例)住院时间的分布进行描述,采用KM法和寿命表法对总体住院时间和出院的瞬时速度进行估计,用Cox比例风险模型对多个预测因子的作用大小进行分析,在对Cox模型比例风险假设进行评价的基础上给出了预测平均住院时间的方法。结果住院时间呈正偏态分布,截尾数据比例达61%。平均住院时间的估计值为29天。出院的"快慢"除与病情相关外,还与病人年龄和伤者在事故中的角色密切相关。结论住院时间的分析应该考虑住院时间的分布和截尾现象,采用生存分析的方法比较合理。     

Variation over time of the effects of prognostic factors in a population-based study of colon cancer: comparison of statistical models

The authors compare the performance of different regression models for censored survival data in modeling the impact of prognostic factors on all-cause mortality in colon cancer. The data were for 1,951 patients, who were diagnosed in 1977-1991, recorded by the Registry of Digestive Tumors of C?te d'Or, France, and followed for up to 15 years. Models include the Cox proportional hazards model and its three generalizations that allow for hazard ratio to change over time: 1) the piecewise model where hazard ratio is a step function; 2) the model with interaction between a predictor and a parametric function of time; and 3) the non-parametric regression spline model. Results illustrate the importance of accounting for non-proportionality of hazards, and some advantages of flexible non-parametric modeling of time-dependent effects. The authors provide empirical evidence for the dependence of the results of piecewise and parametric models on arbitrary a priori choices, regarding the number of time intervals and specific parametric function, which may lead to biased estimates and low statistical power. The authors demonstrate that a single, a priori selected spline model recovers a variety of patterns of changes in hazard ratio and fits better than other models, especially when the changes are non-monotonic, as in the case of cancer stages.     

Multiple imputation for simple estimation of the hazard function based on interval censored data

A data augmentation algorithm is presented for estimating the hazard function and pointwise variability intervals based on interval censored data. The algorithm extends that proposed by Tanner and Wong for grouped right censored data to interval censored data. It applies multiple imputation and local likelihood methods to obtain smooth non-parametric estimates for the hazard function. This approach considerably simplifies the problem of estimation for interval censored data as it transforms it into the more tractable problem of estimation for right censored data. The method is illustrated for two real data sets: times to breast cosmesis deterioration and times to HIV-1 infection for individuals with haemophilia. Simulations are presented to assess the effects of various parameters on the estimates and their variances.     

Fitting regression models to censored survival data

A short review of regression models for the analysis of censored survival data is given. These include multiplicative hazard rate models, log-linear models (accelerated failure time models), linear models and polynomial models. An application of some of these models to the analysis of a large retrospective study on carcinomas of the oral cavity is described. The results obtained by parametric and semiparametric analyses are compared.