Computationally efficient inference for center effects based on restricted mean survival time

Restricted mean survival time (RMST) has gained increased attention in biostatistical and clinical studies. Directly modeling RMST (as opposed to modeling then transforming the hazard function) is appealing computationally and in terms of interpreting covariate effects. We propose computationally convenient methods for evaluating center effects based on RMST. A multiplicative model for the RMST is assumed. Estimation proceeds through an algorithm analogous to stratification, which permits the evaluation of thousands of centers. We derive the asymptotic properties of the proposed estimators and evaluate finite sample performance through simulation. We demonstrate that considerable decreases in computational burden are achievable through the proposed methods, in terms of both storage requirements and run time. The methods are applied to evaluate more than 5000 US dialysis facilities using data from a national end-stage renal disease registry.     

Parametric accelerated failure time models with random effects and an application to kidney transplant survival

Accelerated failure time models with a shared random component are described, and are used to evaluate the effect of explanatory factors and different transplant centres on survival times following kidney transplantation. Different combinations of the distribution of the random effects and baseline hazard function are considered and the fit of such models to the transplant data is critically assessed. A mixture model that combines short- and long-term components of a hazard function is then developed, which provides a more flexible model for the hazard function. The model can incorporate different explanatory variables and random effects in each component. The model is straightforward to fit using standard statistical software, and is shown to be a good fit to the transplant data.     

Restricted mean survival time for interval-censored data

Restricted mean survival time (RMST) evaluates the mean event-free survival time up to a prespecified time point. It has been used as an alternative measure of treatment effect owing to its model-free structure and clinically meaningful interpretation of treatment benefit for right-censored data. In clinical trials, another type of censoring called interval censoring may occur if subjects are examined at several discrete time points and the survival time falls into an interval rather than being exactly observed. The missingness of exact observations under interval-censored cases makes the nonparametric measure of treatment effect more challenging. Employing the linear smoothing technique to overcome the ambiguity, we propose a new model-free measure for the interval-censored RMST. As an alternative to the commonly used log-rank test, we further construct a hypothesis testing procedure to assess the survival difference between two groups. Simulation studies show that the bias of our proposed interval-censored RMST estimator is negligible and the testing procedure delivers promising performance in detecting between-group difference with regard to size and power under various configurations of survival curves. The proposed method is illustrated by reanalyzing two real datasets containing interval-censored observations.     

限制平均生存时间回归模型在生存分析中的应用

目的 探讨限制平均生存时间(restricted mean survival time,RMST)回归模型在生存数据分析中的应用。 方法 运用伪值估计方法对医学数据进行限制平均生存时间回归模型实例分析,并与常见生存分析模型进行比较。 结果 RMST回归模型无特定模型假设,适用于不满足比例风险假定的生存数据;实例分析显示,RMST模型构建灵活,可通过设定多个τ值在多个时间段内进行估计;犯第一类错误的概率低于Cox比例风险模型,模型估算结果容易解释,能够提供在临床实践中更为实用的结论。 结论 在不满足比例风险假定且生存曲线有较大交叉的情形下,限制性平均生存时间模型能够提供稳定有效且易于解释的效应估计,在生存分析领域具有优良的适用性,可以作为Cox比例风险模型分析结果的补充。     

限制平均生存时间在临床随访研究中的应用

在临床随访研究中,风险比（HR）是衡量组间差异最常用的指标,但由于HR是两组间的风险率之比,含义较抽象,无法对患者的生存情况做出直观的解释,并且在使用过程中需满足比例风险假定。而此时,限制平均生存时间（RMST）可以作为一种相对有效的指标或统计量,本文介绍基于RMST的统计分析方法,包括RMST及其差值的估计、假设检验以及回归分析,并通过两个实例说明了RMST在数据分析中的应用。结果显示RMST可作为有效的分析工具,其结果易于解释,并且当比例风险假定失效时,在组间差异的比较上相对于HR更有效。因此,在疾病疗效评价和预后分析的过程中,建议RMST应随HR同时给出,两者相互配合、相互补充,共同反映数据的特征。     

Meta‐analysis of time‐to‐event outcomes from randomized trials using restricted mean survival time: application to individual participant data

Meta‐analysis of time‐to‐event outcomes using the hazard ratio as a treatment effect measure has an underlying assumption that hazards are proportional. The between‐arm difference in the restricted mean survival time is a measure that avoids this assumption and allows the treatment effect to vary with time. We describe and evaluate meta‐analysis based on the restricted mean survival time for dealing with non‐proportional hazards and present a diagnostic method for the overall proportional hazards assumption. The methods are illustrated with the application to two individual participant meta‐analyses in cancer. The examples were chosen because they differ in disease severity and the patterns of follow‐up, in order to understand the potential impacts on the hazards and the overall effect estimates. We further investigate the estimation methods for restricted mean survival time by a simulation study. Copyright © 2015 John Wiley & Sons, Ltd.     

A Dirichlet process mixture model for survival outcome data: assessing nationwide kidney transplant centers

Mortality rates are probably the most important indicator for the performance of kidney transplant centers. Motivated by the national evaluation of mortality rates at kidney transplant centers in the USA, we seek to categorize the transplant centers based on the mortality outcome. We describe a Dirichlet process model and a Dirichlet process mixture model with a half‐cauchy prior for the estimation of the risk‐adjusted effects of the transplant centers, with strategies for improving the model performance, interpretability, and classification ability. We derive statistical measures and create graphical tools to rate transplant centers and identify outlying groups of centers with exceptionally good or poor performance. The proposed method was evaluated through simulation and then applied to assess kidney transplant centers from a national organ failure registry. Copyright © 2015 John Wiley & Sons, Ltd.     

Modelling discrete time survival data with random slopes: evaluating haemodialysis centres

We propose a hierarchical discrete time survival model to analyse registry data on haemodialysis patients in Rio de Janeiro, Brazil, collected at different dialysis centres. Our aim is to estimate differences in hazard ratios attributable to variation in dialysis centre performance, after adjusting for further observed covariates both at the individual and centre level. The proposed model allowed for the estimation of a residual calendar time trend varying across dialysis centres through the adoption of a random slope model. These calendar time trends turned out to have significant variation, after adjustment for important observed covariates. The technique can be easily adapted to other diseases as long as survival time is the measurement of interest.     

The use of restricted mean survival time to estimate the treatment effect in randomized clinical trials when the proportional hazards assumption is in doubt

In most randomized clinical trials (RCTs) with a right-censored time-to-event outcome, the hazard ratio is taken as an appropriate measure of the effectiveness of a new treatment compared with a standard-of-care or control treatment. However, it has long been known that the hazard ratio is valid only under the proportional hazards (PH) assumption. This assumption is formally checked only rarely. Some recent trials, particularly the IPASS trial in lung cancer and the ICON7 trial in ovarian cancer, have alerted researchers to the possibility of gross non-PH, raising the critical question of how such data should be analyzed. Here, we propose the use of the restricted mean survival time at a prespecified, fixed time point as a useful general measure to report the difference between two survival curves. We describe different methods of estimating it and we illustrate its application to three RCTs in cancer. The examples are graded from a trial in kidney cancer in which there is no evidence of non-PH, to IPASS, where the opposite is clearly the case. We propose a simple, general scheme for the analysis of data from such RCTs. Key elements of our approach are Andersen's method of 'pseudo-observations,' which is based on the Kaplan-Meier estimate of the survival function, and Royston and Parmar's class of flexible parametric survival models, which may be used for analyzing data in the presence or in the absence of PH of the treatment effect.     

Improving testing and description of treatment effect in clinical trials with survival outcomes

Cox model inference and the log-rank test have been the cornerstones for design and analysis of clinical trials with survival outcomes. In this article, we summarize some recently developed methods for analyzing survival data when the hazards may possibly be nonproportional and also propose some new estimators for summary measures of the treatment effect. These methods utilize the short-term and long-term hazard ratio model proposed in Yang and Prentice (2005), which contains the Cox model and also accommodates various nonproportional hazards scenarios. Without the proportional hazards assumption, these methods often improve the log-rank test and inference procedures based on the Cox model, as well as nonparametric procedures currently available in the literature. The proposed methods have sound theoretical justifications and can be computed quickly. R codes for implementing them are available. Detailed illustrations with 3 clinical trials are provided.     

Estimating survival gain for economic evaluations with survival time as principal endpoint: a cost-effectiveness analysis of adding early hormonal therapy to radiotherapy in patients with locally advanced prostate cancer

The problem of estimating expected outcomes for the economic evaluation of treatments for which the outcome of principal interest is (quality adjusted) survival time has so far not received sufficient attention in the literature. The best estimate of expected survival is mean survival time, but with censored survival data, the true survival time for all the subjects is not known, so the mean is not defined.A possible solution to this estimation problem is illustrated by a retrospective cost-effectiveness analysis of the addition of hormonal therapy to standard radiotherapy for patients with locally advanced prostate cancer. A recently proposed method is used to approach the problem caused by censored cost data, and the impact of uncertainty is assessed by bootstrap resampling techniques. Mean survival time is estimated by a restricted means analysis with the time point of restriction determined by statistical criteria. When average total costs and mean survival time is evaluated at this time point of restriction, the result is that the combined therapy (radiotherapy plus hormonal therapy) increases mean survival time by about 1 year, while reducing the costs per patient for the French health insurance system by 12 700 FF. The time point of restriction may also be determined by other criteria and mean survival time may be estimated by extrapolating the survival curves by means of various parametric survival distributions. We show that the exact results of the economic evaluation are decisively determined by the restriction time point chosen and the approach taken to estimate mean survival time.     

Ratio of geometric means to analyze continuous outcomes in meta-analysis: comparison to mean differences and ratio of arithmetic means using empiric data and simulation

Meta‐analyses pooling continuous outcomes can use mean differences (MD), standardized MD (MD in pooled standard deviation units, SMD), or ratio of arithmetic means (RoM). Recently, ratio of geometric means using ad hoc (RoGM _{ad hoc}) or Taylor series (RoGM _Taylor) methods for estimating variances have been proposed as alternative effect measures for skewed continuous data. Skewed data are suggested for summary measures of clinical parameters restricted to positive values which have large coefficients of variation (CV). Our objective was to compare performance characteristics of RoGM _{ad hoc} and RoGM _Taylor to MD, SMD, and RoM. We used empiric data from systematic reviews reporting continuous outcomes and selected from each the meta‐analysis with the most and at least 5 trials (Cochrane Database [2008, Issue 1]). We supplemented this with simulations conducted with representative parameters. Pooled results were calculated using each effect measure. Of the reviews, 232/5053 met the inclusion criteria. Empiric data and simulation showed that RoGM _{ad hoc} exhibits more extreme treatment effects and greater heterogeneity than all other effect measures. Compared with MD, SMD, and RoM, RoGM _Taylor exhibits similar treatment effects, more heterogeneity when CV ≤0.7, and less heterogeneity when CV > 0.7. In conclusion, RoGM _Taylor may be considered for pooling continuous outcomes in meta‐analysis when data are skewed, but RoGM _{ad hoc} should not be used. However, clinicians' lack of familiarity with geometric means combined with acceptable performance characteristics of RoM in most situations suggests that RoM may be the preferable ratio method for pooling continuous outcomes in meta‐analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of multilevel grouped survival data with time-varying regression coefficients

Correlated or multilevel grouped survival data are common in medical and dental research. Two common approaches to analyze such data are the marginal and the random-effects approaches. Models and methods in the literature generally assume that the treatment effect is constant over time. A researcher may be interested in studying whether the treatment effects in a clinical trial vary over time, say fade out gradually. This is of particular clinical value when studying the long-term effect of a treatment. This paper proposed to extend the random effects grouped proportional hazards models by incorporating the possibly time-varying covariate effects into the model in terms of a state-space formulation. The proposed model is very flexible and the estimation can be performed using the MCMC approach with non-informative priors in the Bayesian framework. The method is applied to a data set from a prospective clinical trial investigating the effectiveness of silver diamine fluoride (SDF) and sodium fluoride (NaF) varnish in arresting active dentin caries in the Chinese preschool children. It is shown that the treatment groups with caries removal prior to the topical fluoride applications are most effective in shortening the arrest times in the first 6-month interval, but their effects fade out rapidly since then. The effects of treatment groups without caries removal prior to topical fluoride application drop at a very slow rate and can be considered as more or less constant over time. The applications of SDF solution is found to be more effective than the applications of NaF vanish.     

Updating of covariates and choice of time origin in survival analysis: problems with vaguely defined disease states

This paper discusses survival analysis based on updated covariates with focus on proportional hazard regression in situations where some disease states may be vaguely defined. Analyses of a trial in liver cirrhosis are used to motivate the discussion. We use problems caused by inclusion of recordings from unscheduled follow-ups to illustrate the importance of appropriate coding of covariates and describe how such problems may be approached using appropriately 'lagged' covariates. The choice of time origin is discussed with emphasis on situations where disease initiation is difficult to define. Simulations are used to assess the effect of an erroneously specified time origin. It is argued that age or calendar time may frequently be sensible time variables.     

随机生存森林:基于机器学习算法的生存分析模型

传统的生存分析方法虽在生物医学领域已有广泛应用,但需满足一些前提假设。随机生存森林方法可克服这一弱点。本文以美国梅奥诊所的肝脏原发性胆汁肝硬化的数据为例,从随机生存森林的原理、建模步骤、实例演示和适用性讨论等方面进行阐述,以期为读者进行生存分析提供新的思路和方法。     

A special case of reduced rank models for identification and modelling of time varying effects in survival analysis

Flexible survival models are in need when modelling data from long term follow‐up studies. In many cases, the assumption of proportionality imposed by a Cox model will not be valid. Instead, a model that can identify time varying effects of fixed covariates can be used. Although there are several approaches that deal with this problem, it is not always straightforward how to choose which covariates should be modelled having time varying effects and which not. At the same time, it is up to the researcher to define appropriate time functions that describe the dynamic pattern of the effects. In this work, we suggest a model that can deal with both fixed and time varying effects and uses simple hypotheses tests to distinguish which covariates do have dynamic effects. The model is an extension of the parsimonious reduced rank model of rank 1. As such, the number of parameters is kept low, and thus, a flexible set of time functions, such as b‐splines, can be used. The basic theory is illustrated along with an efficient fitting algorithm. The proposed method is applied to a dataset of breast cancer patients and compared with a multivariate fractional polynomials approach for modelling time‐varying effects. Copyright © 2016 John Wiley & Sons, Ltd.     

Investigating heterogeneity in an individual patient data meta-analysis of time to event outcomes

Differences across studies in terms of design features and methodology, clinical procedures, and patient characteristics, are factors that can contribute to variability in the treatment effect between studies in a meta-analysis (statistical heterogeneity). Regression modelling can be used to examine relationships between treatment effect and covariates with the aim of explaining the variability in terms of clinical, methodological, or other factors. Such an investigation can be undertaken using aggregate data or individual patient data. An aggregate data approach can be problematic as sufficient data are rarely available and translating aggregate effects to individual patients can often be misleading. An individual patient data approach, although usually more resource demanding, allows a more thorough investigation of potential sources of heterogeneity and enables a fuller analysis of time to event outcomes in meta-analysis. Hierarchical Cox regression models are used to identify and explore the evidence for heterogeneity in meta-analysis and examine the relationship between covariates and censored failure time data in this context. Alternative formulations of the model are possible and illustrated using individual patient data from a meta-analysis of five randomized controlled trials which compare two drugs for the treatment of epilepsy. The models are further applied to simulated data examples in which the degree of heterogeneity and magnitude of treatment effect are varied. The behaviour of each model in each situation is explored and compared.     

Estimation of controlled direct effects in time‐varying treatments using structural nested mean models: application to a primary prevention trial for coronary events with pravastatin

For the estimation of controlled direct effects (i.e., direct effects controlling intermediates that are set at a fixed level for all members of the population) without bias, two fundamental assumptions must hold: the absence of unmeasured confounding factors for treatment and outcome and for intermediate variables and outcome. Even if these assumptions hold, one would nonetheless fail to estimate direct effects using standard methods, for example, stratification or regression modeling, when the treatment influences confounding factors. For such situations, the sequential g‐estimation method for structural nested mean models has been developed for estimating controlled direct effects in point‐treatment situations. In this study, we demonstrate that this method can be applied to longitudinal data with time‐varying treatments and repeatedly measured intermediate variables. We sequentially estimate the parameters in two structural nested mean models: one for a repeatedly measured intermediate and the other one for direct effects of a time‐varying treatment. The method was applied to data from a large primary prevention trial for coronary events, in which pravastatin was used to lower the cholesterol levels in patients with moderate hypercholesterolemia. Copyright © 2014 John Wiley & Sons, Ltd.