The proportional odds model for multivariate interval-censored failure time data

The proportional odds model is one of the most commonly used regression models in failure time data analysis and has been discussed by many authors (Appl. Stat. 1983; 32:165-171; J. Am. Stat. Assoc. 1999; 94:125-136; J. Am. Stat. Assoc. 1997; 92:960-967; Biometrics 2000; 56:511-518; J. Am. Stat. Assoc. 2001; 96:1446-1457). It specifies that covariates have multiplicative effects on the odds function and is often used when, for example, the covariate effect diminishes over time. Most of the existing methods for the model are for univariate failure time data. In this paper, we discuss how to fit the proportional odds model to multivariate interval-censored failure time data. For inference, the maximum likelihood approach is developed and evaluated by simulation studies, which suggest that the method works well for practical situations. The method is applied to a set of bivariate interval-censored data arising from an AIDS clinical trial.     

Efficient evaluation of treatment effects in the presence of missing covariate values

In clinical trials, treatment comparisons are often performed by models that incorporate important prognostic factors. Since these models require complete covariate information on all patients, statisticians frequently resort to complete case analysis or to omission of an important covariate. A probability imputation technique (PIT) is proposed that involves substituting conditional probabilities for missing covariate values when the covariate is qualitative. Simulation results are presented which demonstrate that the method neither violates the size of the treatment test nor introduces additional bias for the estimation of the treatment effect. It allows use of standard software. A clinical trial of breast cancer treatment, in which an important covariate was partly missing, was analysed by Cox's model. The use of PIT resulted in smaller observed error probability compared with case deletion, and sensitivity analysis supported these results.     

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.     

One‐stage individual participant data meta‐analysis models: estimation of treatment‐covariate interactions must avoid ecological bias by separating out within‐trial and across‐trial information

Stratified medicine utilizes individual‐level covariates that are associated with a differential treatment effect, also known as treatment‐covariate interactions. When multiple trials are available, meta‐analysis is used to help detect true treatment‐covariate interactions by combining their data. Meta‐regression of trial‐level information is prone to low power and ecological bias, and therefore, individual participant data (IPD) meta‐analyses are preferable to examine interactions utilizing individual‐level information. However, one‐stage IPD models are often wrongly specified, such that interactions are based on amalgamating within‐ and across‐trial information. We compare, through simulations and an applied example, fixed‐effect and random‐effects models for a one‐stage IPD meta‐analysis of time‐to‐event data where the goal is to estimate a treatment‐covariate interaction. We show that it is crucial to centre patient‐level covariates by their mean value in each trial, in order to separate out within‐trial and across‐trial information. Otherwise, bias and coverage of interaction estimates may be adversely affected, leading to potentially erroneous conclusions driven by ecological bias. We revisit an IPD meta‐analysis of five epilepsy trials and examine age as a treatment effect modifier. The interaction is ?0.011 (95% CI: ?0.019 to ?0.003; p = 0.004), and thus highly significant, when amalgamating within‐trial and across‐trial information. However, when separating within‐trial from across‐trial information, the interaction is ?0.007 (95% CI: ?0.019 to 0.005; p = 0.22), and thus its magnitude and statistical significance are greatly reduced. We recommend that meta‐analysts should only use within‐trial information to examine individual predictors of treatment effect and that one‐stage IPD models should separate within‐trial from across‐trial information to avoid ecological bias. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

A linear regression model for the analysis of life times

A linear model is suggested for the influence of covariates on the intensity function. This approach is less vulnerable than the Cox model to problems of inconsistency when covariates are deleted or the precision of covariate measurements is changed. A method of non-parametric estimation of regression functions is presented. This results in plots that may give information on the change over time in the influence of covariates. A test method and two goodness of fit plots are also given. The approach is illustrated by simulation as well as by data from a clinical trial of treatment of carcinoma of the oropharynx.     

Further results on the non-parametric linear regression model in survival analysis

This paper gives further developments of a non-parametric linear regression model in survival analysis. Three subjects are studied. First, martingale residuals, originally developed for the Cox model, are introduced for our linear model. Their theory is developed and they are shown to be useful for judging goodness of fit. The second focus of the paper is on the use of bootstrap replications to judge which features of the cumulative regression plots are likely to reflect real phenomena and not merely random variation. In particular, this is applied to judging whether the effect of a covariate disappears over time, a problem for which no formal test exists. The third subject is density type, or kernel, estimation of the regression functions themselves. This might give more direct information than the cumulative plots. The approaches are illustrated by data from a clinical trial of carcinoma of the oropharynx, and by survival times of grafts in renal patients.     

Covariate adjustment for two-sample treatment comparisons in randomized clinical trials: a principled yet flexible approach

There is considerable debate regarding whether and how covariate-adjusted analyses should be used in the comparison of treatments in randomized clinical trials. Substantial baseline covariate information is routinely collected in such trials, and one goal of adjustment is to exploit covariates associated with outcome to increase precision of estimation of the treatment effect. However, concerns are routinely raised over the potential for bias when the covariates used are selected post hoc and the potential for adjustment based on a model of the relationship between outcome, covariates, and treatment to invite a 'fishing expedition' for that leading to the most dramatic effect estimate. By appealing to the theory of semiparametrics, we are led naturally to a characterization of all treatment effect estimators and to principled, practically feasible methods for covariate adjustment that yield the desired gains in efficiency and that allow covariate relationships to be identified and exploited while circumventing the usual concerns. The methods and strategies for their implementation in practice are presented. Simulation studies and an application to data from an HIV clinical trial demonstrate the performance of the techniques relative to the existing methods. Copyright (c) 2007 John Wiley & Sons, Ltd.     

Joint models for efficient estimation in proportional hazards regression models

In survival studies, information lost through censoring can be partially recaptured through repeated measures data which are predictive of survival. In addition, such data may be useful in removing bias in survival estimates, due to censoring which depends upon the repeated measures. Here we investigate joint models for survival T and repeated measurements Y, given a vector of covariates Z. Mixture models indexed as f (T/Z) f (Y/T,Z) are well suited for assessing covariate effects on survival time. Our objective is efficiency gains, using non-parametric models for Y in order to avoid introducing bias by misspecification of the distribution for Y. We model (T/Z) as a piecewise exponential distribution with proportional hazards covariate effect. The component (Y/T,Z) has a multinomial model. The joint likelihood for survival and longitudinal data is maximized, using the EM algorithm. The estimate of covariate effect is compared to the estimate based on the standard proportional hazards model and an alternative joint model based estimate. We demonstrate modest gains in efficiency when using the joint piecewise exponential joint model. In a simulation, the estimated efficiency gain over the standard proportional hazards model is 6.4 per cent. In clinical trial data, the estimated efficiency gain over the standard proportional hazards model is 10.2 per cent.     

带时依协变量的重复测量资料的混合线性模型分析及其MIXED过程实现

目的 探讨混合线性模型在带有时依协变量的重复测量资料分析中的应用.方法 以治疗轻、中度原发性高血压病临床试验资料为例,考虑到给药方案在各个时间点随病情而变化,利用SAS中的MIXED过程,选择合适的协方差结构来实现带有时依协变量的重复测量资料的统计分析.结果 时依协变量(给药方案)对治疗轻、中度原发性高血压病有统计学意义(P＜0.05);时间因素有统计学意义(P＜0.05);给药方案与时间因素之间有交互效应(P＜0.05)、给药方案与处理因素之间有交互效应(P＜0.05).结论 采用混合线性模型对带有时依协变量的临床试验重复测量资料进行统计分析,可以更客观地进行药物疗效评价.     

A sequential Cox approach for estimating the causal effect of treatment in the presence of time‐dependent confounding applied to data from the Swiss HIV Cohort Study

When estimating the effect of treatment on HIV using data from observational studies, standard methods may produce biased estimates due to the presence of time‐dependent confounders. Such confounding can be present when a covariate, affected by past exposure, is both a predictor of the future exposure and the outcome. One example is the CD4 cell count, being a marker for disease progression for HIV patients, but also a marker for treatment initiation and influenced by treatment. Fitting a marginal structural model (MSM) using inverse probability weights is one way to give appropriate adjustment for this type of confounding. In this paper we study a simple and intuitive approach to estimate similar treatment effects, using observational data to mimic several randomized controlled trials. Each ‘trial’ is constructed based on individuals starting treatment in a certain time interval. An overall effect estimate for all such trials is found using composite likelihood inference. The method offers an alternative to the use of inverse probability of treatment weights, which is unstable in certain situations. The estimated parameter is not identical to the one of an MSM, it is conditioned on covariate values at the start of each mimicked trial. This allows the study of questions that are not that easily addressed fitting an MSM. The analysis can be performed as a stratified weighted Cox analysis on the joint data set of all the constructed trials, where each trial is one stratum. The model is applied to data from the Swiss HIV cohort study. Copyright © 2010 John Wiley & Sons, Ltd.     

A method to estimate treatment efficacy among latent subgroups of a randomized clinical trial

Subgroup analysis arises in clinical trials research when we wish to estimate a treatment effect on a specific subgroup of the population distinguished by baseline characteristics. Many trial designs induce latent subgroups such that subgroup membership is observable in one arm of the trial and unidentified in the other. This occurs, for example, in oncology trials when a biopsy or dissection is performed only on subjects randomized to active treatment. We discuss a general framework to estimate a biological treatment effect on the latent subgroup of interest when the survival outcome is right-censored and can be appropriately modelled as a parametric function of covariate effects. Our framework builds on the application of instrumental variables methods to all-or-none treatment noncompliance. We derive a computational method to estimate model parameters via the EM algorithm and provide guidance on its implementation in standard software packages. The research is illustrated through an analysis of a seminal melanoma trial that proposed a new standard of care for the disease and involved a biopsy that is available only on patients in the treatment arm.     

Controlling ecological bias in evidence synthesis of trials reporting on collapsed and overlapping covariate categories

Meta‐analysis of randomized controlled trials based on aggregated data is vulnerable to ecological bias if trial results are pooled over covariates that influence the outcome variable, even when the covariate does not modify the treatment effect, or is not associated with the treatment. This paper shows how, when trial results are aggregated over different levels of covariates, the within‐study covariate distribution, and the effects of both covariates and treatments can be simultaneously estimated, and ecological bias reduced. Bayesian Markov chain Monte Carlo methods are used. The method is applied to a mixed treatment comparison evidence synthesis of six alternative approaches to post‐stroke inpatient care. Results are compared with a model using only the stratified covariate data available, where each stratum is treated as a separate trial, and a model using fully aggregated data, where no covariate data are used. Copyright © 2010 John Wiley & Sons, Ltd.     

Point pattern analysis with spatially varying covariate effects,applied to the study of cerebrovascular deaths

This article proposes a modeling approach for handling spatial heterogeneity present in the study of the geographical pattern of deaths due to cerebrovascular disease.The framework involvesa point pattern analysis with components exhibiting spatial variation. Preliminary studies indicate that mortality of this disease and the effect of relevant covariates do not exhibit uniform geographic distribution. Our model extends a previously proposed model in the literature that uses spatial and non‐spatial variables by allowing for spatial variation of the effect of non‐spatial covariates. A number of relative risk indicators are derived by comparing different covariate levels, different geographic locations, or both. The methodology is applied to the study of the geographical death pattern of cerebrovascular deaths in the city of Rio de Janeiro. The results compare well against existing alternatives, including fixed covariate effects. Our model is able to capture and highlight important data information that would not be noticed otherwise, providing information that is required for appropriate health decision‐making. Copyright © 2014 John Wiley & Sons, Ltd.     

A graphical method to assess treatment-covariate interactions using the Cox model on subsets of the data

We introduce the subpopulation treatment effect pattern plot (STEPP) method, designed to facilitate the interpretation of estimates of treatment effect derived from different but potentially overlapping subsets of clinical trial data. In particular, we consider sequences of subpopulations defined with respect to a covariate, and obtain confidence bands for the collection of treatment effects (here obtained from the Cox proportional hazards model) associated with the sequences. The method is aimed at determining whether the magnitude of the treatment effect changes as a function of the values of the covariate. We apply STEPP to a breast cancer clinical trial data set to evaluate the treatment effect as a function of the oestrogen receptor content of the primary tumour.     

Imputing gene-treatment interactions when the genotype distribution is unknown using case-only and putative placebo analyses--a new method for the Genetics of Hypertension Associated Treatment (GenHAT) study

There is a sizeable literature on methods for detecting gene-environment interaction in the framework of case-control studies, particularly with reference to the assumption of independence of genotype and exposure. In the context of a clinical trial, wherein gene-drug interactions with regard to outcomes are examined, these methods may be readily applied, as gene and drug are independent by randomization. In an active-controlled trial (experimental treatment vs standard) that has collected genotype information, gene-drug interactions can be estimated. In addition, the effect of the experimental treatment vs placebo can be imputed by using data from a historical placebo-controlled trial (standard vs placebo) if either (a) genotype information is available from the historical trial or (b) assumptions are made about the prevalence of genotype and the odds ratios of genotype and disease in the historical trial using information from other studies. Motivation for these procedures is provided by the Genetics of Hypertension Associated Treatment, a large pharmacogenetics, ancillary study of a hypertension clinical trial, and examples from published hypertension trials will be used to illustrate the methods.     

A joint model for interval‐censored functional decline trajectories under informative observation

Multi‐state models are useful for modelling disease progression where the state space of the process is used to represent the discrete disease status of subjects. Often, the disease process is only observed at clinical visits, and the schedule of these visits can depend on the disease status of patients. In such situations, the frequency and timing of observations may depend on transition times that are themselves unobserved in an interval‐censored setting. There is a potential for bias if we model a disease process with informative observation times as a non‐informative observation scheme with pre‐specified examination times. In this paper, we develop a joint model for the disease and observation processes to ensure valid inference because the follow‐up process may itself contain information about the disease process. The transitions for each subject are modelled using a Markov process, where bivariate subject‐specific random effects are used to link the disease and observation models. Inference is based on a Bayesian framework, and we apply our joint model to the analysis of a large study examining functional decline trajectories of palliative care patients. Copyright © 2015 John Wiley & Sons, Ltd.     

POWER CONSIDERATIONS FOR CLINICAL TRIALS USING MULTIVARIATE TIME-TO-EVENT DATA

Many clinical trials involve the collection of data on the times to occurrence of different types of events, such as different fungal infections in AIDS research, or of recurrences of the same type, such as successive fits in epilepsy research. The multivariate proportional hazards model allows for analysis of this data and software for doing this is now widely available. In this paper, the approximate power of a clinical trial that aims to use such data for comparing two treatments is derived. Special attention is given to the bivariate case, both to show that the approximation works well and to illustrate how various design parameters affect the power of a trial. As with any multivariate data in clinical trials, there are many conceptual issues that should be considered during trial design; the paper closes with a discussion of some of these. © 1997 by John Wiley & Sons, Ltd. Stat. Med., Vol. 16, 865–882 (1997).     

Latent-variable models for longitudinal data with bivariate ordinal outcomes

We use the concept of latent variables to derive the joint distribution of bivariate ordinal outcomes, and then extend the model to allow for longitudinal data. Specifically, we relate the observed ordinal outcomes using threshold values to a bivariate latent variable, which is then modelled as a linear mixed model. Random effects terms are used to tie all together repeated observations from the same subject. The cross-sectional association between the two outcomes is modelled through the correlation coefficient of the bivariate latent variable, conditional on random effects. Assuming conditional independence given random effects, the marginal likelihood, under the missing data at random assumption, is approximated using an adaptive Gaussian quadrature for numerical integration. The model provides fixed effects parameters that are subject-specific, but retain the population-averaged interpretation when properly scaled. This is particularly well suited for the situation in which population comparisons and individual level contrasts are of equal importance. Data from a psychiatric trial, the Fluvoxamine (an antidepressant drug) study, are used to illustrate the methodology.     

鞅残差在Cox回归模型诊断中的应用

目的：探讨Cox模型中某给定连续变量对生存效应的最佳函数形式。方法：不含该变量的Cox模型下的鞅残差对该协变量作图可揭示该变量在全模型中的函数形式。结果：实例分析表明,Loess平滑鞅残差图可用于Cox模型中协变量函数形式的确定。结论：建议分析者把鞅残差图纳入Cox模型分析中,对协变量的生存效应作出正确估计。     

The choice of a genetic model in the meta-analysis of molecular association studies

BACKGROUND: To evaluate gene-disease associations, genetic epidemiologists collect information on the disease risk in subjects with different genotypes (for a bi-allelic polymorphism: gg, Gg, GG). Meta-analyses of such studies usually reduce the problem to a single comparison, either by performing two separate pairwise comparisons or by assuming a specific underlying genetic model (recessive, co-dominant, dominant). A biological justification for the choice of the genetic model is seldom available. METHODS: We present a genetic model-free approach, which does not assume that the underlying genetic model is known in advance but still makes use of the information available on all genotypes. The approach uses OR(GG), the odds ratio between the homozygous genotypes, to capture the magnitude of the genetic effect, and lambda, the heterozygote log odds ratio as a proportion of the homozygote log odds ratio, to capture the genetic mode of inheritance. The analysis assumes that the same unknown genetic model, i.e. the same lambda, applies in all studies, and this is investigated graphically. The approach is illustrated using five examples of published meta-analyses. RESULTS: Analyses based on specific genetic models can produce misleading estimates of the odds ratios when an inappropriate model is assumed. The genetic model-free approach gives appropriately wider confidence intervals than genetic model-based analyses because it allows for uncertainty about the genetic model. In terms of assessment of model fit, it performs at least as well as a bivariate pairwise analysis in our examples. CONCLUSIONS: The genetic model-free approach offers a unified approach that efficiently estimates the genetic effect and the underlying genetic model. A bivariate pairwise analysis should be used if the assumption of a common genetic model across studies is in doubt.