Joint modeling longitudinal semi‐continuous data and survival,with application to longitudinal medical cost data

It has been increasingly common to analyze simultaneously repeated measures and time to failure data. In this paper we propose a joint model when the repeated measures are semi‐continuous, characterized by the presence of a large portion of zero values, as well as right skewness of non zero (positive) values. Examples include monthly medical costs, car insurance annual claims, or annual number of hospitalization days. A random effects two‐part model is used to describe respectively the odds of being positive and the level of positive values. The random effects from the two‐part model are then incorporated in the hazard of the failure time to form the joint model. The estimation can be carried out by Gaussian quadrature techniques conveniently implemented in SAS Proc NLMIXED. Our model is applied to longitudinal (monthly) medical costs of 1455 chronic heart‐failure patients from the clinical data repository at the University of Virginia. Copyright © 2008 John Wiley & Sons, Ltd.     

A joint model of recurrent events and a terminal event with a nonparametric covariate function

We extend the shared frailty model of recurrent events and a dependent terminal event to allow for a nonparametric covariate function. We include a Gaussian random effect (frailty) in the intensity functions of both the recurrent and terminal events to capture correlation between the two processes. We employ the penalized cubic spline method to describe the nonparametric covariate function in the recurrent events model. We use Laplace approximation to evaluate the marginal penalized partial likelihood without a closed form. We also propose the variance estimates for regression coefficients. Numerical analysis results show that the proposed estimates perform well for both the nonparametric and parametric components. We apply this method to analyze the hospitalization rate of patients with heart failure in the presence of death.     

Regression analysis of failure time data with informative interval censoring

Interval censoring arises when a subject misses prescheduled visits at which the failure is to be assessed. Most existing approaches for analysing interval-censored failure time data assume that the censoring mechanism is independent of the true failure time. However, there are situations where this assumption may not hold. In this paper, we consider such a situation in which the dependence structure between the censoring variables and the failure time can be modelled through some latent variables and a method for regression analysis of failure time data is proposed. The method makes use of the proportional hazards frailty model and an EM algorithm is presented for estimation. Finite sample properties of the proposed estimators of regression parameters are examined through simulation studies and we illustrate the method with data from an AIDS study.     

Regression analysis of multivariate current status data with dependent censoring: application to ankylosing spondylitis data

Multivariate current‐status failure time data consist of several possibly related event times of interest, in which the status of each event is determined at a single examination time. If the examination time is intrinsically related to the event times, the examination is referred to as dependent censoring and needs to be taken into account. Such data often occur in clinical studies and animal carcinogenicity experiments. To accommodate for possible dependent censoring, this paper proposes a joint frailty model for event times and dependent censoring time. We develop a likelihood approach using Gaussian quadrature techniques for obtaining maximum likelihood estimates. We conduct extensive simulation studies for investigating finite‐sample properties of the proposed method. We illustrate the proposed method with an analysis of patients with ankylosing spondylitis, where the examination time may be dependent on the event times of interest. Copyright © 2013 John Wiley & Sons, Ltd.     

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。     

Covariate dimension reduction for survival data via the Gaussian process latent variable model

The analysis of high‐dimensional survival data is challenging, primarily owing to the problem of overfitting, which occurs when spurious relationships are inferred from data that subsequently fail to exist in test data. Here, we propose a novel method of extracting a low‐dimensional representation of covariates in survival data by combining the popular Gaussian process latent variable model with a Weibull proportional hazards model. The combined model offers a flexible non‐linear probabilistic method of detecting and extracting any intrinsic low‐dimensional structure from high‐dimensional data. By reducing the covariate dimension, we aim to diminish the risk of overfitting and increase the robustness and accuracy with which we infer relationships between covariates and survival outcomes. In addition, we can simultaneously combine information from multiple data sources by expressing multiple datasets in terms of the same low‐dimensional space. We present results from several simulation studies that illustrate a reduction in overfitting and an increase in predictive performance, as well as successful detection of intrinsic dimensionality. We provide evidence that it is advantageous to combine dimensionality reduction with survival outcomes rather than performing unsupervised dimensionality reduction on its own. Finally, we use our model to analyse experimental gene expression data and detect and extract a low‐dimensional representation that allows us to distinguish high‐risk and low‐risk groups with superior accuracy compared with doing regression on the original high‐dimensional data. Copyright © 2015 John Wiley & Sons, Ltd.     

Flexible parametric proportional-hazards and proportional-odds models for censored survival data,with application to prognostic modelling and estimation of treatment effects

Modelling of censored survival data is almost always done by Cox proportional-hazards regression. However, use of parametric models for such data may have some advantages. For example, non-proportional hazards, a potential difficulty with Cox models, may sometimes be handled in a simple way, and visualization of the hazard function is much easier. Extensions of the Weibull and log-logistic models are proposed in which natural cubic splines are used to smooth the baseline log cumulative hazard and log cumulative odds of failure functions. Further extensions to allow non-proportional effects of some or all of the covariates are introduced. A hypothesis test of the appropriateness of the scale chosen for covariate effects (such as of treatment) is proposed. The new models are applied to two data sets in cancer. The results throw interesting light on the behaviour of both the hazard function and the hazard ratio over time. The tools described here may be a step towards providing greater insight into the natural history of the disease and into possible underlying causes of clinical events. We illustrate these aspects by using the two examples in cancer.     

A simulation study of predictive ability measures in a survival model II: explained randomness and predictive accuracy

Several R²‐type measures have been proposed to evaluate the predictive ability of a survival model. In Part I, we classified the measures into four categories and studied the measures in the explained variation category. In this paper, we study the remaining measures in a similar fashion, discussing their strengths and shortcomings. Simulation studies are used to examine the performance of the measures with respect to the criteria we set out in Part I. Our simulation studies showed that among the measures studied in this paper, the measures proposed by Kent and O'Quigley (and its approximation ) and Schemper and Kaider perform better with respect to our criteria. However, our investigations showed that is adversely affected by the distribution of covariate and the presence of influential observations. The results show that the other measures perform poorly, primarily because they are affected either by the degree of censoring or the follow‐up period. Copyright © 2012 John Wiley & Sons, Ltd.     

On robustness of marginal regression coefficient estimates and hazard functions in multivariate survival analysis of family data when the frailty distribution is mis-specified

The shared frailty model is an extension of the Cox model to correlated failure times and, essentially, a random effects model for failure time outcomes. In this model, the latent frailty shared by individual members in a cluster acts multiplicatively as a factor on the hazard function and is typically modelled parametrically. One commonly used distribution is gamma, where both shape and scale parameters are set to be the same to allow for unique identification of baseline hazard function. It is popular because it is a conjugate prior, and the posterior distribution possesses the same form as gamma. In addition, the parameter can be interpreted as a time-independent cross-ratio function, a natural extension of odds ratio to failure time outcomes. In this paper, we study the effect of frailty distribution mis-specification on the marginal regression estimates and hazard functions under assumed gamma distribution with an application to family studies. The simulation results show that the biases are generally 10% and lower, even when the true frailty distribution deviates substantially from the assumed gamma distribution. This suggests that the gamma frailty model can be a practical choice in real data analyses if the regression parameters and marginal hazard function are of primary interest and individual cluster members are exchangeable with respect to their dependencies.     

A simulation study of predictive ability measures in a survival model I: explained variation measures

Measures of predictive ability play an important role in quantifying the clinical significance of prognostic factors. Several measures have been proposed to evaluate the predictive ability of survival models in the last two decades, but no single measure is consistently used. The proposed measures can be classified into the following categories: explained variation, explained randomness, and predictive accuracy. The three categories are conceptually different and are based on different principles. Several new measures have been proposed since Schemper and Stare's study in 1996 on some of the existing measures. This paper is the first of two papers that study the proposed measures systematically by applying a set of criteria that a measure of predictive ability should possess in the context of survival analysis. The present paper focuses on the explained variation category, and part II studies the proposed measures in the other categories. Simulation studies are used to examine the performance of five explained variation measures with respect to these criteria, discussing their strengths and shortcomings. Our simulation studies show that the measures proposed by Kent and O'Quigley, R(PM)(2), and Royston and Sauerbrei, R(D)(2), appear to be the best overall at quantifying predictive ability. However, it should be noted that neither measure is perfect; R(PM)(2) is sensitive to outliers and R(D)(2) to (marked) non-normality of the distribution of the prognostic index. The results show that the other measures perform poorly, primarily because they are adversely affected by censoring.     

Bayesian hierarchical model for multiple repeated measures and survival data: an application to Parkinson's disease

Multilevel item response theory models have been increasingly used to analyze the multivariate longitudinal data of mixed types (e.g., continuous and categorical) in clinical studies. To address the possible correlation between multivariate longitudinal measures and time to terminal events (e.g., death and dropout), joint models that consist of a multilevel item response theory submodel and a survival submodel have been previously developed. However, in multisite studies, multiple patients are recruited and treated by the same clinical site. There can be a significant site correlation because of common environmental and socioeconomic status, and similar quality of care within site. In this article, we develop and study several hierarchical joint models with the hazard of terminal events dependent on shared random effects from various levels. We conduct extensive simulation study to evaluate the performance of various models under different scenarios. The proposed hierarchical joint models are applied to the motivating deprenyl and tocopherol antioxidative therapy of Parkinsonism study to investigate the effect of tocopherol in slowing Parkinson's disease progression. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Interval‐censored parametric regression survival models and the analysis of longitudinal trials

This paper develops interval censoring likelihood methods in the context of parametric proportional hazard (PH) and non‐PH regression models in the longitudinal study setting to reanalyze the medical research council's randomized controlled trial of teletherapy in age‐related macular degeneration. We compare the performance of the interval censoring likelihood with proxy likelihoods that were used to analyze the original data. It is shown, analytically, that the use of such proxy likelihoods in selected PH models leads to biased estimators. Such estimators are artificially precise; further, the magnitude of their percentage bias is quantified in a data‐directed simulation study. For non‐PH models, we demonstrate that these results obtained from PH models do not hold uniformly and explain the implications of this finding for the reanalysis of proxy likelihood trial data. Our final analysis, of the age‐related macular degeneration trial data, based on fitting PH and non‐PH models, reassuringly confirms the published findings from the original trial. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Sensitivity analysis for the estimation of rates of change with non-ignorable drop-out: an application to a randomized clinical trial of the vitamin D3

The vitamin D(3) trial was a repeated measures randomized clinical trial for secondary hyperparathyroidism in haemodialysis patients where the efficacy of the vitamin D(3) infusions for suppressing the secretion of parathyroid hormone (PTH) was compared among four dose groups over 12 weeks. In this trial, patients terminated the study before the scheduled end of the study due to their elevated serum calcium (Ca) level, that is, the administration of the vitamin D(3) was expected to cause hypercalcaemia as an adverse event. In this setting of monotone missingness, there is a potential for bias in estimation of mean rates of decline in PTH for each treatment group using the standard methods such as the generalized estimating equations (GEE) which ignore the observed past Ca histories. We estimated the treatment-group-specific mean rates of decline in PTH by the inverse probability of censoring weighted (IPCW) methods which account for the observed past histories of time-dependent factors that are both a predictor of drop-out and are correlated with the outcomes. The IPCW estimator can be viewed as an extension of the GEE estimator that allows for the data to be MAR but not MCAR. With missing data, it is rarely appropriate to analyse the data solely under the assumption that the missing data process is ignorable, because the assumption of ignorable missingness cannot be guaranteed to hold and is untestable from the observed data. We proposed a sensitivity analysis that examines how inference about the IPCW estimates of the treatment-group-specific mean rates of decline in PTH changes as we vary the non-ignorable selection bias parameter over a range of plausible values.     

Joint modelling of bivariate longitudinal data with informative dropout and left-censoring, with application to the evolution of CD4+ cell count and HIV RNA viral load in response to treatment of HIV infection

Several methodological issues occur in the context of the longitudinal study of HIV markers evolution. Three of them are of particular importance: (i) correlation between CD4+ T lymphocytes (CD4+) and plasma HIV RNA; (ii) left-censoring of HIV RNA due to a lower quantification limit; (iii) and potential informative dropout. We propose a likelihood inference for a parametric joint model including a bivariate linear mixed model for the two markers and a lognormal survival model for the time to drop out. We apply the model to data from patients starting antiretroviral treatment in the CASCADE collaboration where all of the three issues needed to be addressed.     

Regression analysis of longitudinal data with informative observation times and application to medical cost data

Longitudinal data analysis is one of the most discussed and applied areas in statistics and a great deal of literature has been developed for it. However, most of the existing literature focus on the situation where observation times are fixed or can be treated as fixed constants. This paper considers the situation where these observation times may be random variables and more importantly, they may be related to the underlying longitudinal variable or process of interest. Furthermore, covariate effects may be time-varying. For the analysis, a joint modeling approach is proposed and in particular, for estimation of time-varying regression parameters, an estimating equation-based procedure is developed. Both asymptotic and finite sample properties of the proposed estimates are established. The methodology is applied to an acute myeloid leukemia trial that motivated this study.     

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.