Analysis of linear transformation models with covariate measurement error and interval censoring

Among several semiparametric models, the Cox proportional hazard model is widely used to assess the association between covariates and the time-to-event when the observed time-to-event is interval-censored. Often, covariates are measured with error. To handle this covariate uncertainty in the Cox proportional hazard model with the interval-censored data, flexible approaches have been proposed. To fill a gap and broaden the scope of statistical applications to analyze time-to-event data with different models, in this paper, a general approach is proposed for fitting the semiparametric linear transformation model to interval-censored data when a covariate is measured with error. The semiparametric linear transformation model is a broad class of models that includes the proportional hazard model and the proportional odds model as special cases. The proposed method relies on a set of estimating equations to estimate the regression parameters and the infinite-dimensional parameter. For handling interval censoring and covariate measurement error, a flexible imputation technique is used. Finite sample performance of the proposed method is judged via simulation studies. Finally, the suggested method is applied to analyze a real data set from an AIDS clinical trial.     

A Bayesian mixture of semiparametric mixed‐effects joint models for skewed‐longitudinal and time‐to‐event data

In longitudinal studies, it is of interest to investigate how repeatedly measured markers in time are associated with a time to an event of interest, and in the mean time, the repeated measurements are often observed with the features of a heterogeneous population, non‐normality, and covariate measured with error because of longitudinal nature. Statistical analysis may complicate dramatically when one analyzes longitudinal–survival data with these features together. Recently, a mixture of skewed distributions has received increasing attention in the treatment of heterogeneous data involving asymmetric behaviors across subclasses, but there are relatively few studies accommodating heterogeneity, non‐normality, and measurement error in covariate simultaneously arose in longitudinal–survival data setting. Under the umbrella of Bayesian inference, this article explores a finite mixture of semiparametric mixed‐effects joint models with skewed distributions for longitudinal measures with an attempt to mediate homogeneous characteristics, adjust departures from normality, and tailor accuracy from measurement error in covariate as well as overcome shortages of confidence in specifying a time‐to‐event model. The Bayesian mixture of joint modeling offers an appropriate avenue to estimate not only all parameters of mixture joint models but also probabilities of class membership. Simulation studies are conducted to assess the performance of the proposed method, and a real example is analyzed to demonstrate the methodology. The results are reported by comparing potential models with various scenarios. Copyright © 2015 John Wiley & Sons, Ltd.     

Induced smoothing for rank‐based regression with recurrent gap time data

Various semiparametric regression models have recently been proposed for the analysis of gap times between consecutive recurrent events. Among them, the semiparametric accelerated failure time (AFT) model is especially appealing owing to its direct interpretation of covariate effects on the gap times. In general, estimation of the semiparametric AFT model is challenging because the rank‐based estimating function is a nonsmooth step function. As a result, solutions to the estimating equations do not necessarily exist. Moreover, the popular resampling‐based variance estimation for the AFT model requires solving rank‐based estimating equations repeatedly and hence can be computationally cumbersome and unstable. In this paper, we extend the induced smoothing approach to the AFT model for recurrent gap time data. Our proposed smooth estimating function permits the application of standard numerical methods for both the regression coefficients estimation and the standard error estimation. Large‐sample properties and an asymptotic variance estimator are provided for the proposed method. Simulation studies show that the proposed method outperforms the existing nonsmooth rank‐based estimating function methods in both point estimation and variance estimation. The proposed method is applied to the data analysis of repeated hospitalizations for patients in the Danish Psychiatric Center Register.     

Multiple regression of cost data: use of generalised linear models

OBJECTIVE: Choosing an appropriate method for regression analyses of cost data is problematic because it must focus on population means while taking into account the typically skewed distribution of the data. In this paper we illustrate the use of generalised linear models for regression analysis of cost data. METHODS: We consider generalised linear models with either an identity link function (providing additive covariate effects) or log link function (providing multiplicative effects), and with gaussian (normal), overdispersed poisson, gamma, or inverse gaussian distributions. These are applied to estimate the treatment effects in two randomised trials adjusted for baseline covariates. Criteria for choosing an appropriate model are presented. RESULTS: In both examples considered, the gaussian model fits poorly and other distributions are to be preferred. When there are variables of prognostic importance in the model, using different distributions can materially affect the estimates obtained; it may also be possible to discriminate between additive and multiplicative covariate effects. CONCLUSIONS: Generalised linear models are attractive for the regression of cost data because they provide parametric methods of analysis where a variety of non-normal distributions can be specified and the way covariates act can be altered. Unlike the use of data transformation in ordinary least-squares regression, generalised linear models make inferences about the mean cost directly.     

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.     

Linear mixed models for replication data to efficiently allow for covariate measurement error

It is well known that measurement error in the covariates of regression models generally causes bias in parameter estimates. Correction for such biases requires information concerning the measurement error, which is often in the form of internal validation or replication data. Regression calibration (RC) is a popular approach to correct for covariate measurement error, which involves predicting the true covariate using error‐prone measurements. Likelihood methods have previously been proposed as an alternative approach to estimate the parameters in models affected by measurement error, but have been relatively infrequently employed in medical statistics and epidemiology, partly because of computational complexity and concerns regarding robustness to distributional assumptions. We show how a standard random‐intercepts model can be used to obtain maximum likelihood (ML) estimates when the outcome model is linear or logistic regression under certain normality assumptions, when internal error‐prone replicate measurements are available. Through simulations we show that for linear regression, ML gives more efficient estimates than RC, although the gain is typically small. Furthermore, we show that RC and ML estimates remain consistent even when the normality assumptions are violated. For logistic regression, our implementation of ML is consistent if the true covariate is conditionally normal given the outcome, in contrast to RC. In simulations, this ML estimator showed less bias in situations where RC gives non‐negligible biases. Our proposal makes the ML approach to dealing with covariate measurement error more accessible to researchers, which we hope will improve its viability as a useful alternative to methods such as RC. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian quantile regression‐based nonlinear mixed‐effects joint models for time‐to‐event and longitudinal data with multiple features

This article explores Bayesian joint models for a quantile of longitudinal response, mismeasured covariate and event time outcome with an attempt to (i) characterize the entire conditional distribution of the response variable based on quantile regression that may be more robust to outliers and misspecification of error distribution; (ii) tailor accuracy from measurement error, evaluate non‐ignorable missing observations, and adjust departures from normality in covariate; and (iii) overcome shortages of confidence in specifying a time‐to‐event model. When statistical inference is carried out for a longitudinal data set with non‐central location, non‐linearity, non‐normality, measurement error, and missing values as well as event time with being interval censored, it is important to account for the simultaneous treatment of these data features in order to obtain more reliable and robust inferential results. Toward this end, we develop Bayesian joint modeling approach to simultaneously estimating all parameters in the three models: quantile regression‐based nonlinear mixed‐effects model for response using asymmetric Laplace distribution, linear mixed‐effects model with skew‐t distribution for mismeasured covariate in the presence of informative missingness and accelerated failure time model with unspecified nonparametric distribution for event time. We apply the proposed modeling approach to analyzing an AIDS clinical data set and conduct simulation studies to assess the performance of the proposed joint models and method. Copyright © 2016 John Wiley & Sons, Ltd.     

Marginal regression models with time-varying coefficients for recurrent event data

Recurrent event data arise frequently from medical research. Examples include repeated infections, recurrence of tumors, relapse of leukemia, repeated hospitalizations, recurrence of symptoms of a disease, and so on. In the analysis of recurrent event data, the proportional rates model assumes that the regression coefficients are time invariant. In reality, however, these parameters may vary over time, and the temporal covariate effects on the event process are of great interest. In this article, we formulate a class of semiparametric marginal rates models, which incorporate a mixture of time-varying and time-independent parameters, to analyze recurrent event data. For statistical inference on model parameters, an estimation procedure is developed and asymptotic properties of the proposed estimators are established. In addition, we develop tests for investigating whether or not covariate effects vary with time. The finite-sample behaviors of the proposed methods are examined in simulation studies. An example of application of the proposed methodology is illustrated on a set of data from a clinic study on chronic granulomatous disease.     

Flexible regression models for ROC and risk analysis,with or without a gold standard

A novel semiparametric regression model is developed for evaluating the covariate‐specific accuracy of a continuous medical test or biomarker. Ideally, studies designed to estimate or compare medical test accuracy will use a separate, flawless gold‐standard procedure to determine the true disease status of sampled individuals. We treat this as a special case of the more complicated and increasingly common scenario in which disease status is unknown because a gold‐standard procedure does not exist or is too costly or invasive for widespread use. To compensate for missing data on disease status, covariate information is used to discriminate between diseased and healthy units. We thus model the probability of disease as a function of ‘disease covariates’. In addition, we model test/biomarker outcome data to depend on ‘test covariates’, which provides researchers the opportunity to quantify the impact of covariates on the accuracy of a medical test. We further model the distributions of test outcomes using flexible semiparametric classes. An important new theoretical result demonstrating model identifiability under mild conditions is presented. The modeling framework can be used to obtain inferences about covariate‐specific test accuracy and the probability of disease based on subject‐specific disease and test covariate information. The value of the model is illustrated using multiple simulation studies and data on the age‐adjusted ability of soluble epidermal growth factor receptor – a ubiquitous serum protein – to serve as a biomarker of lung cancer in men. SAS code for fitting the model is provided. Copyright © 2015 John Wiley & Sons, Ltd.     

Time‐dependent tree‐structured survival analysis with unbiased variable selection through permutation tests

Incorporating time‐dependent covariates into tree‐structured survival analysis (TSSA) may result in more accurate prognostic models than if only baseline values are used. Available time‐dependent TSSA methods exhaustively test every binary split on every covariate; however, this approach may result in selection bias toward covariates with more observed values. We present a method that uses unbiased significance levels from newly proposed permutation tests to select the time‐dependent or baseline covariate with the strongest relationship with the survival outcome. The specific splitting value is identified using only the selected covariate. Simulation results show that the proposed time‐dependent TSSA method produces tree models of equal or greater accuracy as compared to baseline TSSA models, even with high censoring rates and large within‐subject variability in the time‐dependent covariate. To illustrate, the proposed method is applied to data from a cohort of bipolar youths to identify subgroups at risk for self‐injurious behavior. Copyright © 2014 John Wiley & Sons, Ltd.     

半参数治愈模型在长期生存者资料分析中的应用

目的 介绍长期生存者资料生存分析模型与方法 .方法 以SARS病人为例阐述半参数治愈模型原理与方法 ,并将长期生存者资料半参数治愈模型与Cox回归模型得到的结果 进行对比分析.结果 Cox比例风险回归模型得到四个协变量有统计学意义;半参数治愈模型比例风险回归部分得到一个有意义的协变量,logistic回归部分得到三个协变量有统计学意义.结论 在对长期生存者存在的资料分析时,半参数治愈模型比传统的Cox比例风险回归模型更具优势,不仅模型形式简明,参数估计解释合理,而且可从多角度提供更多有价值的信息,是一种适用范围更广,实用性更强的统计分析方法 .     

核平滑半参数回归模型在重复测量资料中的应用

目的探讨应用核平滑的半参数模型处理重复测量资料。方法利用核平滑和带约束的最小二乘法估计参数,并采用Bootstrap建立可信区间。结果用Matlab程序应用模拟数据得到相应结果。结论核平滑半参数模型可作为处理重复测量资料的一种方法。     

Nonlinear multiple imputation for continuous covariate within semiparametric Cox model: application to HIV data in Senegal

Multiple imputation is commonly used to impute missing covariate in Cox semiparametric regression setting. It is to fill each missing data with more plausible values, via a Gibbs sampling procedure, specifying an imputation model for each missing variable. This imputation method is implemented in several softwares that offer imputation models steered by the shape of the variable to be imputed, but all these imputation models make an assumption of linearity on covariates effect. However, this assumption is not often verified in practice as the covariates can have a nonlinear effect. Such a linear assumption can lead to a misleading conclusion because imputation model should be constructed to reflect the true distributional relationship between the missing values and the observed values. To estimate nonlinear effects of continuous time invariant covariates in imputation model, we propose a method based on B‐splines function. To assess the performance of this method, we conducted a simulation study, where we compared the multiple imputation method using Bayesian splines imputation model with multiple imputation using Bayesian linear imputation model in survival analysis setting. We evaluated the proposed method on the motivated data set collected in HIV‐infected patients enrolled in an observational cohort study in Senegal, which contains several incomplete variables. We found that our method performs well to estimate hazard ratio compared with the linear imputation methods, when data are missing completely at random, or missing at random. Copyright © 2013 John Wiley & Sons, Ltd.     

A semiparametric imputation approach for regression with censored covariate with application to an AMD progression study

This research is motivated by studying the progression of age‐related macular degeneration where both a covariate and the response variable are subject to censoring. We develop a general framework to handle regression with censored covariate where the response can be different types and the censoring can be random or subject to (constant) detection limits. Multiple imputation is a popular technique to handle missing data that requires compatibility between the imputation model and the substantive model to obtain valid estimates. With censored covariate, we propose a novel multiple imputation‐based approach, namely, the semiparametric two‐step importance sampling imputation (STISI) method, to impute the censored covariate. Specifically, STISI imputes the missing covariate from a semiparametric accelerated failure time model conditional on fully observed covariates (Step 1) with the acceptance probability derived from the substantive model (Step 2). The 2‐step procedure automatically ensures compatibility and takes full advantage of the relaxed semiparametric assumption in the imputation. Extensive simulations demonstrate that the STISI method yields valid estimates in all scenarios and outperforms some existing methods that are commonly used in practice. We apply STISI on data from the Age‐related Eye Disease Study, to investigate the association between the progression time of the less severe eye and that of the more severe eye. We also illustrate the method by analyzing the urine arsenic data for patients from National Health and Nutrition Examination Survey (2003‐2004) where the response is binary and 1 covariate is subject to detection limit.     

A Bayesian semiparametric latent variable approach to causal mediation

In assessing causal mediation effects in randomized studies, a challenge is that the direct and indirect effects can vary across participants due to different measured and unmeasured characteristics. In that case, the population effect estimated from standard approaches implicitly averages over and does not estimate the heterogeneous direct and indirect effects. We propose a Bayesian semiparametric method to estimate heterogeneous direct and indirect effects via clusters, where the clusters are formed by both individual covariate profiles and individual effects due to unmeasured characteristics. These cluster‐specific direct and indirect effects can be estimated through a set of regression models where specific coefficients are clustered by a stick‐breaking prior. To let clustering be appropriately informed by individual direct and indirect effects, we specify a data‐dependent prior. We conduct simulation studies to assess performance of the proposed method compared to other methods. We use this approach to estimate heterogeneous causal direct and indirect effects of an expressive writing intervention for patients with renal cell carcinoma.     

Spline‐based accelerated failure time model

The accelerated failure time (AFT) model has been suggested as an alternative to the Cox proportional hazards model. However, a parametric AFT model requires the specification of an appropriate distribution for the event time, which is often difficult to identify in real‐life studies and may limit applications. A semiparametric AFT model was developed by Komárek et al based on smoothed error distribution that does not require such specification. In this article, we develop a spline‐based AFT model that also does not require specification of the parametric family of event time distribution. The baseline hazard function is modeled by regression B‐splines, allowing for the estimation of a variety of smooth and flexible shapes. In comprehensive simulations, we validate the performance of our approach and compare with the results from parametric AFT models and the approach of Komárek. Both the proposed spline‐based AFT model and the approach of Komárek provided unbiased estimates of covariate effects and survival curves for a variety of scenarios in which the event time followed different distributions, including both simple and complex cases. Spline‐based estimates of the baseline hazard showed also a satisfactory numerical stability. As expected, the baseline hazard and survival probabilities estimated by the misspecified parametric AFT models deviated from the truth. We illustrated the application of the proposed model in a study of colon cancer.     

A conditional likelihood approach for regression analysis using biomarkers measured with batch‐specific error

Measurement error is common in epidemiological and biomedical studies. When biomarkers are measured in batches or groups, measurement error is potentially correlated within each batch or group. In regression analysis, most existing methods are not applicable in the presence of batch‐specific measurement error in predictors. We propose a robust conditional likelihood approach to account for batch‐specific error in predictors when batch effect is additive and the predominant source of error, which requires no assumptions on the distribution of measurement error. Although a regression model with batch as a categorical covariable yields the same parameter estimates as the proposed conditional likelihood approach for linear regression, this result does not hold in general for all generalized linear models, in particular, logistic regression. Our simulation studies show that the conditional likelihood approach achieves better finite sample performance than the regression calibration approach or a naive approach without adjustment for measurement error. In the case of logistic regression, our proposed approach is shown to also outperform the regression approach with batch as a categorical covariate. In addition, we also examine a ‘hybrid’ approach combining the conditional likelihood method and the regression calibration method, which is shown in simulations to achieve good performance in the presence of both batch‐specific and measurement‐specific errors. We illustrate our method by using data from a colorectal adenoma study. Copyright © 2012 John Wiley & Sons, Ltd.     

Considerations for analysis of time‐to‐event outcomes measured with error: Bias and correction with SIMEX

For time‐to‐event outcomes, a rich literature exists on the bias introduced by covariate measurement error in regression models, such as the Cox model, and methods of analysis to address this bias. By comparison, less attention has been given to understanding the impact or addressing errors in the failure time outcome. For many diseases, the timing of an event of interest (such as progression‐free survival or time to AIDS progression) can be difficult to assess or reliant on self‐report and therefore prone to measurement error. For linear models, it is well known that random errors in the outcome variable do not bias regression estimates. With nonlinear models, however, even random error or misclassification can introduce bias into estimated parameters. We compare the performance of 2 common regression models, the Cox and Weibull models, in the setting of measurement error in the failure time outcome. We introduce an extension of the SIMEX method to correct for bias in hazard ratio estimates from the Cox model and discuss other analysis options to address measurement error in the response. A formula to estimate the bias induced into the hazard ratio by classical measurement error in the event time for a log‐linear survival model is presented. Detailed numerical studies are presented to examine the performance of the proposed SIMEX method under varying levels and parametric forms of the error in the outcome. We further illustrate the method with observational data on HIV outcomes from the Vanderbilt Comprehensive Care Clinic.     

Bayesian measurement‐error‐driven hidden Markov regression model for calibrating the effect of covariates on multistate outcomes: Application to androgenetic alopecia

Multistate Markov regression models used for quantifying the effect size of state‐specific covariates pertaining to the dynamics of multistate outcomes have gained popularity. However, the measurements of multistate outcome are prone to the errors of classification, particularly when a population‐based survey/research is involved with proxy measurements of outcome due to cost consideration. Such a misclassification may affect the effect size of relevant covariates such as odds ratio used in the field of epidemiology. We proposed a Bayesian measurement‐error‐driven hidden Markov regression model for calibrating these biased estimates with and without a 2‐stage validation design. A simulation algorithm was developed to assess various scenarios of underestimation and overestimation given nondifferential misclassification (independent of covariates) and differential misclassification (dependent on covariates). We applied our proposed method to the community‐based survey of androgenetic alopecia and found that the effect size of the majority of covariate was inflated after calibration regardless of which type of misclassification. Our proposed Bayesian measurement‐error‐driven hidden Markov regression model is practicable and effective in calibrating the effects of covariates on multistate outcome, but the prior distribution on measurement errors accrued from 2‐stage validation design is strongly recommended.