G-estimation and artificial censoring: problems, challenges, and applications

In principle, G-estimation is an attractive approach for dealing with confounding by variables affected by treatment. It has rarely been applied for estimation of the effects of treatment on failure-time outcomes. Part of this is due to artificial censoring, an analytic device which considers some subjects who actually were observed to fail as if they were censored. Artificial censoring leads to a lack of smoothness in the estimating function, which can pose problems in variance estimation and in optimization. It also can lead to failure to have solutions to the usual estimating functions, which then raises questions about the appropriate criteria for optimization. To improve performance of the optimization procedures, we consider approaches for reducing the amount of artificial censoring, propose the substitution of smooth for indicator functions, and propose the use of estimating functions scaled to a measure of the information in the data; we evaluate performance of these approaches using simulation. We also consider appropriate optimization criteria in the presence of information loss due to artificial censoring. We motivate and illustrate our approaches using observational data on the effect of erythropoietin on mortality among subjects on hemodialysis.     

Insider Censoring: Distortion of Data with Nondetects

Environmental data often include low-level concentrations below reporting limits. These data may be reported as “< RL,” where RL is one of several types of reporting limits. Some values also may be reported as a single number, but flagged with a qualifier (J-values) to indicate a difference in precision as compared to values above the RL. A currently used method for reporting censored environmental data called “insider censoring” produces a strong upward bias, while also distorting the shape of the data distribution. This results in inaccurate estimates of summary statistics and regression coefficients, distorts evaluations of whether data follow a normal distribution, and introduces inaccuracies into risk assessments and models. Insider censoring occurs when data measured as below the detection limit (< DL) are reported as less than the higher quantitation limit (< QL), whereas values between the DL and QL are reported as individual numbers. Three unbiased alternatives to insider censoring are presented so that laboratories and their data users can recognize, and remedy, this problem.     

Nonparametric inference for median costs with censored data

Summary Increasingly, estimations of health care costs are used to evaluate competing treatments or to assess the expected expenditures associated with certain diseases. In health policy and economics, the primary focus of these estimations has been on the mean cost, because the total cost can be derived directly from the mean cost, and because information about total resources utilized is highly relevant for policymakers. Yet, the median cost also could be important, both as an intuitive measure of central tendency in cost distribution and as a subject of interest to payers and consumers. In many prospective studies, cost data collection is sometimes incomplete for some subjects due to right censoring, which typically is caused by loss to follow-up or by limited study duration. Censoring poses a unique challenge for cost data analysis because of so-called induced informative censoring, in that traditional methods suited for survival data generally are invalid in censored cost estimation. In this article, we propose methods for estimating the median cost and its confidence interval (CI) when data are subject to right censoring. We also consider the estimation of the ratio and difference of two median costs and their CIs. These methods can be extended to the estimation of other quantiles and other informatively censored data. We conduct simulation and real data analysis in order to examine the performance of the proposed methods.     

A Multiple Imputation Approach to Regression Analysis for Doubly Censored Data with Application to AIDS Studies

Sun, Liao, and Pagano (1999) proposed an interesting estimating equation approach to Cox regression with doubly censored data. Here we point out that a modification of their proposal leads to a multiple imputation approach, where the double censoring is reduced to single censoring by imputing for the censored initiating times. For each imputed data set one can take advantage of many existing techniques and software for singly censored data. Under the general framework of multiple imputation, the proposed method is simple to implement and can accommodate modeling issues such as model checking, which has not been adequately discussed previously in the literature for doubly censored data. Here we illustrate our method with an application to a formal goodness-of-fit test and a graphical check for the proportional hazards model for doubly censored data. We reanalyze a well-known AIDS data set.     

Cox regression model with randomly censored covariates

This paper deals with a Cox proportional hazards regression model, where some covariates of interest are randomly right‐censored. While methods for censored outcomes have become ubiquitous in the literature, methods for censored covariates have thus far received little attention and, for the most part, dealt with the issue of limit‐of‐detection. For randomly censored covariates, an often‐used method is the inefficient complete‐case analysis (CCA) which consists in deleting censored observations in the data analysis. When censoring is not completely independent, the CCA leads to biased and spurious results. Methods for missing covariate data, including type I and type II covariate censoring as well as limit‐of‐detection do not readily apply due to the fundamentally different nature of randomly censored covariates. We develop a novel method for censored covariates using a conditional mean imputation based on either Kaplan–Meier estimates or a Cox proportional hazards model to estimate the effects of these covariates on a time‐to‐event outcome. We evaluate the performance of the proposed method through simulation studies and show that it provides good bias reduction and statistical efficiency. Finally, we illustrate the method using data from the Framingham Heart Study to assess the relationship between offspring and parental age of onset of cardiovascular events.     

Quantile regression for doubly censored data

Double censoring often occurs in registry studies when left censoring is present in addition to right censoring. In this work, we propose a new analysis strategy for such doubly censored data by adopting a quantile regression model. We develop computationally simple estimation and inference procedures by appropriately using the embedded martingale structure. Asymptotic properties, including the uniform consistency and weak convergence, are established for the resulting estimators. Moreover, we propose conditional inference to address the special identifiability issues attached to the double censoring setting. We further show that the proposed method can be readily adapted to handle left truncation. Simulation studies demonstrate good finite-sample performance of the new inferential procedures. The practical utility of our method is illustrated by an analysis of the onset of the most commonly investigated respiratory infection, Pseudomonas aeruginosa, in children with cystic fibrosis through the use of the U.S. Cystic Fibrosis Registry.     

Median regression with censored cost data

Because of the skewness of the distribution of medical costs, we consider modeling the median as well as other quantiles when establishing regression relationships to covariates. In many applications, the medical cost data are also right censored. In this article, we propose semiparametric procedures for estimating the parameters in median regression models based on weighted estimating equations when censoring is present. Numerical studies are conducted to show that our estimators perform well with small samples and the resulting inference is reliable in circumstances of practical importance. The methods are applied to a dataset for medical costs of patients with colorectal cancer.     

Application of a rank-based genetic association test to age-at-onset data from the Collaborative Study on the Genetics of Alcoholism study

Association studies of quantitative traits have often relied on methods in which a normal distribution of the trait is assumed. However, quantitative phenotypes from complex human diseases are often censored, highly skewed, or contaminated with outlying values. We recently developed a rank-based association method that takes into account censoring and makes no distributional assumptions about the trait. In this study, we applied our new method to age-at-onset data on ALDX1 and ALDX2. Both traits are highly skewed (skewness > 1.9) and often censored. We performed a whole genome association study of age at onset of the ALDX1 trait using Illumina single-nucleotide polymorphisms. Only slightly more than 5% of markers were significant. However, we identified two regions on chromosomes 14 and 15, which each have at least four significant markers clustering together. These two regions may harbor genes that regulate age at onset of ALDX1 and ALDX2. Future fine mapping of these two regions with densely spaced markers is warranted.     

A tobit variance-component method for linkage analysis of censored trait data

Variance-component (VC) methods are flexible and powerful procedures for the mapping of genes that influence quantitative traits. However, traditional VC methods make the critical assumption that the quantitative-trait data within a family either follow or can be transformed to follow a multivariate normal distribution. Violation of the multivariate normality assumption can occur if trait data are censored at some threshold value. Trait censoring can arise in a variety of ways, including assay limitation or confounding due to medication. Valid linkage analyses of censored data require the development of a modified VC method that directly models the censoring event. Here, we present such a model, which we call the "tobit VC method." Using simulation studies, we compare and contrast the performance of the traditional and tobit VC methods for linkage analysis of censored trait data. For the simulation settings that we considered, our results suggest that (1) analyses of censored data by using the traditional VC method lead to severe bias in parameter estimates and a modest increase in false-positive linkage findings, (2) analyses with the tobit VC method lead to unbiased parameter estimates and type I error rates that reflect nominal levels, and (3) the tobit VC method has a modest increase in linkage power as compared with the traditional VC method. We also apply the tobit VC method to censored data from the Finland-United States Investigation of Non-Insulin-Dependent Diabetes Mellitus Genetics study and provide two examples in which the tobit VC method yields noticeably different results as compared with the traditional method.     

Interval mapping methods for detecting QTL affecting survival and time-to-event phenotypes

Quantitative trait loci (QTL) are usually searched for using classical interval mapping methods which assume that the trait of interest follows a normal distribution. However, these methods cannot take into account features of most survival data such as a non-normal distribution and the presence of censored data. We propose two new QTL detection approaches which allow the consideration of censored data. One interval mapping method uses a Weibull model (W), which is popular in parametrical modelling of survival traits, and the other uses a Cox model (C), which avoids making any assumption on the trait distribution. Data were simulated following the structure of a published experiment. Using simulated data, we compare W, C and a classical interval mapping method using a Gaussian model on uncensored data (G) or on all data (G'=censored data analysed as though records were uncensored). An adequate mathematical transformation was used for all parametric methods (G, G' and W). When data were not censored, the four methods gave similar results. However, when some data were censored, the power of QTL detection and accuracy of QTL location and of estimation of QTL effects for G decreased considerably with censoring, particularly when censoring was at a fixed date. This decrease with censoring was observed also with G', but it was less severe. Censoring had a negligible effect on results obtained with the W and C methods.     

A generalized weighted quantile sum approach for analyzing correlated data in the presence of interactions

A weighted quantile sum (WQS) regression has been used to assess the associations between environmental exposures and health outcomes. However, the currently available WQS approach, which is based on additive effects, does not allow exploring for potential interactions of exposures with other covariates in relation to a health outcome. In addition, the current WQS cannot account for clustering, thus it may not be valid for analysis of clustered data. We propose a generalized WQS approach that can assess interactions by estimating stratum‐specific weights of exposures in a mixture, while accounting for potential clustering effect of matched pairs of cases and controls as well as censored exposure data due to being below the limits of detection. The performance of the proposed method in identifying interactions is evaluated through simulations based on various scenarios of correlation structures among the exposures and with an outcome. We also assess how well the proposed method performs in the presence of the varying levels of censoring in exposures. Our findings from the simulation study show that the proposed method outperforms the traditional WQS, as indicated by higher power of detecting interactions. We also find no strong evidence that the proposed method falsely identifies interactions when there are no true interactive effects. We demonstrate application of the proposed method to real data from the Epidemiological Research on Autism Spectrum Disorder (ASD) in Jamaica (ERAJ) by examining interactions between exposure to manganese and glutathione S‐transferase family gene, GSTP1 in relation to ASD.     

Predicting patient survival from microarray data by accelerated failure time modeling using partial least squares and LASSO

We consider the problem of predicting survival times of cancer patients from the gene expression profiles of their tumor samples via linear regression modeling of log-transformed failure times. The partial least squares (PLS) and least absolute shrinkage and selection operator (LASSO) methodologies are used for this purpose where we first modify the data to account for censoring. Three approaches of handling right censored data-reweighting, mean imputation, and multiple imputation-are considered. Their performances are examined in a detailed simulation study and compared with that of full data PLS and LASSO had there been no censoring. A major objective of this article is to investigate the performances of PLS and LASSO in the context of microarray data where the number of covariates is very large and there are extremely few samples. We demonstrate that LASSO outperforms PLS in terms of prediction error when the list of covariates includes a moderate to large percentage of useless or noise variables; otherwise, PLS may outperform LASSO. For a moderate sample size (100 with 10,000 covariates), LASSO performed better than a no covariate model (or noise-based prediction). The mean imputation method appears to best track the performance of the full data PLS or LASSO. The mean imputation scheme is used on an existing data set on lung cancer. This reanalysis using the mean imputed PLS and LASSO identifies a number of genes that were known to be related to cancer or tumor activities from previous studies.     

A partially parametric estimator of survival in the presence of randomly censored data

Many biological or medical experiments have as their goal to estimate the survival function of a specified population of subjects when the time to the specified event may be censored due to loss to follow-up, the occurrence of another event that precludes the occurrence of the event of interest, or the study being terminated before the event of interest occurs. This paper suggests an improvement of the Kaplan-Meier product-limit estimator when the censoring mechanism is random. The proposed estimator treats the uncensored observations nonparametrically and uses a parametric model only for the censored observations. One version of this proposed estimator always has a smaller bias and mean squared error than the product-limit estimator. An example estimating the survival function of patients enrolled in the Ohio State University Bone Marrow Transplant Program is presented.     

Semiparametric analysis of recurrent events data in the presence of dependent censoring

Dependent censoring occurs in longitudinal studies of recurrent events when the censoring time depends on the potentially unobserved recurrent event times. To perform regression analysis in this setting, we propose a semiparametric joint model that formulates the marginal distributions of the recurrent event process and dependent censoring time through scale-change models, while leaving the distributional form and dependence structure unspecified. We derive consistent and asymptotically normal estimators for the regression parameters. We also develop graphical and numerical methods for assessing the adequacy of the proposed model. The finite-sample behavior of the new inference procedures is evaluated through simulation studies. An application to recurrent hospitalization data taken from a study of intravenous drug users is provided.     

Dimension reduction in survival regressions with censored data via an imputed spline approach

Dimension reduction methods have been proposed for regression analysis with predictors of high dimension, but have not received much attention on the problems with censored data. In this article, we present an iterative imputed spline approach based on principal Hessian directions (PHD) for censored survival data in order to reduce the dimension of predictors without requiring a prespecified parametric model. Our proposal is to replace the right-censored survival time with its conditional expectation for adjusting the censoring effect by using the Kaplan-Meier estimator and an adaptive polynomial spline regression in the residual imputation. A sparse estimation strategy is incorporated in our approach to enhance the interpretation of variable selection. This approach can be implemented in not only PHD, but also other methods developed for estimating the central mean subspace. Simulation studies with right-censored data are conducted for the imputed spline approach to PHD (IS-PHD) in comparison with two methods of sliced inverse regression, minimum average variance estimation, and naive PHD in ignorance of censoring. The results demonstrate that the proposed IS-PHD method is particularly useful for survival time responses approximating symmetric or bending structures. Illustrative applications to two real data sets are also presented.     

Statistical estimation of gene expression using multiple laser scans of microarrays

We propose a statistical model for estimating gene expression using data from multiple laser scans at different settings of hybridized microarrays. A functional regression model is used, based on a non-linear relationship with both additive and multiplicative error terms. The function is derived as the expected value of a pixel, given that values are censored at 65 535, the maximum detectable intensity for double precision scanning software. Maximum likelihood estimation based on a Cauchy distribution is used to fit the model, which is able to estimate gene expressions taking account of outliers and the systematic bias caused by signal censoring of highly expressed genes. We have applied the method to experimental data. Simulation studies suggest that the model can estimate the true gene expression with negligible bias. AVAILABILITY: FORTRAN 90 code for implementing the method can be obtained from the authors.     

Flexible hazard regression modeling for medical cost data

The modeling of lifetime (i.e. cumulative) medical cost data in the presence of censored follow-up is complicated by induced informative censoring, rendering standard survival analysis tools invalid. With few exceptions, recently proposed nonparametric estimators for such data do not extend easily to handle covariate information. We propose to model the hazard function for lifetime cost endpoints using an adaptation of the HARE methodology (Kooperberg, Stone, and Truong, Journal of the American Statistical Association, 1995, 90, 78-94). Linear splines and their tensor products are used to adaptively build a model that incorporates covariates and covariate-by-cost interactions without restrictive parametric assumptions. The informative censoring problem is handled using inverse probability of censoring weighted estimating equations. The proposed method is illustrated using simulation and also with data on the cost of dialysis for patients with end-stage renal disease.     

Consistent estimation of the expected Brier score in general survival models with right-censored event times

In survival analysis with censored data the mean squared error of prediction can be estimated by weighted averages of time-dependent residuals. Graf et al. (1999) suggested a robust weighting scheme based on the assumption that the censoring mechanism is independent of the covariates. We show consistency of the estimator. Furthermore, we show that a modified version of this estimator is consistent even when censoring and event times are only conditionally independent given the covariates. The modified estimators are derived on the basis of regression models for the censoring distribution. A simulation study and a real data example illustrate the results.     

Analysis of longitudinal marginal structural models

In this article we construct and study estimators of the causal effect of a time-dependent treatment on survival in longitudinal studies. We employ a particular marginal structural model (MSM), proposed by Robins (2000), and follow a general methodology for constructing estimating functions in censored data models. The inverse probability of treatment weighted (IPTW) estimator of Robins et al. (2000) is used as an initial estimator and forms the basis for an improved, one-step estimator that is consistent and asymptotically linear when the treatment mechanism is consistently estimated. We extend these methods to handle informative censoring. The proposed methodology is employed to estimate the causal effect of exercise on mortality in a longitudinal study of seniors in Sonoma County. A simulation study demonstrates the bias of naive estimators in the presence of time-dependent confounders and also shows the efficiency gain of the IPTW estimator, even in the absence such confounding. The efficiency gain of the improved, one-step estimator is demonstrated through simulation.     

Population size estimation with interval censored counts and external information: Prevalence of multiple sclerosis in Rome

We discuss Bayesian log-linear models for incomplete contingency tables with both missing and interval censored cells, with the aim of obtaining reliable population size estimates. We also discuss use of external information on the censoring probability, which may substantially reduce uncertainty. We show in simulation that information on lower bounds and external information can each improve the mean squared error of population size estimates, even when the external information is not completely accurate. We conclude with an original example on estimation of prevalence of multiple sclerosis in the metropolitan area of Rome, where five out of six lists have interval censored counts. External information comes from mortality rates of multiple sclerosis patients.