Analyses of cumulative incidence functions via non-parametric multiple imputation

We describe a non-parametric multiple imputation method that recovers the missing potential censoring information from competing risks failure times for the analysis of cumulative incidence functions. The method can be applied in the settings of stratified analyses, time-varying covariates, weighted analysis of case-cohort samples and clustered survival data analysis, where no current available methods can be readily implemented. The method uses a Kaplan-Meier imputation method for the censoring times to form an imputed data set, so cumulative incidence can be analyzed using techniques and software developed for ordinary right censored survival data. We discuss the methodology and show from both simulations and real data examples that the method yields valid estimates and performs well. The method can be easily implemented via available software with a minor programming requirement (for the imputation step). It provides a practical, alternative analysis tool for otherwise complicated analyses of cumulative incidence of competing risks data.     

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.     

Dealing with missing outcome data in randomized trials and observational studies

Although missing outcome data are an important problem in randomized trials and observational studies, methods to address this issue can be difficult to apply. Using simulated data, the authors compared 3 methods to handle missing outcome data: 1) complete case analysis; 2) single imputation; and 3) multiple imputation (all 3 with and without covariate adjustment). Simulated scenarios focused on continuous or dichotomous missing outcome data from randomized trials or observational studies. When outcomes were missing at random, single and multiple imputations yielded unbiased estimates after covariate adjustment. Estimates obtained by complete case analysis with covariate adjustment were unbiased as well, with coverage close to 95%. When outcome data were missing not at random, all methods gave biased estimates, but handling missing outcome data by means of 1 of the 3 methods reduced bias compared with a complete case analysis without covariate adjustment. Complete case analysis with covariate adjustment and multiple imputation yield similar estimates in the event of missing outcome data, as long as the same predictors of missingness are included. Hence, complete case analysis with covariate adjustment can and should be used as the analysis of choice more often. Multiple imputation, in addition, can accommodate the missing-not-at-random scenario more flexibly, making it especially suited for sensitivity analyses.     

Analysis of accelerated failure time data with dependent censoring using auxiliary variables via nonparametric multiple imputation

We consider the situation of estimating the marginal survival distribution from censored data subject to dependent censoring using auxiliary variables. We had previously developed a nonparametric multiple imputation approach. The method used two working proportional hazards (PH) models, one for the event times and the other for the censoring times, to define a nearest neighbor imputing risk set. This risk set was then used to impute failure times for censored observations. Here, we adapt the method to the situation where the event and censoring times follow accelerated failure time models and propose to use the Buckley–James estimator as the two working models. Besides studying the performances of the proposed method, we also compare the proposed method with two popular methods for handling dependent censoring through the use of auxiliary variables, inverse probability of censoring weighted and parametric multiple imputation methods, to shed light on the use of them. In a simulation study with time‐independent auxiliary variables, we show that all approaches can reduce bias due to dependent censoring. The proposed method is robust to misspecification of either one of the two working models and their link function. This indicates that a working proportional hazards model is preferred because it is more cumbersome to fit an accelerated failure time model. In contrast, the inverse probability of censoring weighted method is not robust to misspecification of the link function of the censoring time model. The parametric imputation methods rely on the specification of the event time model. The approaches are applied to a prostate cancer dataset. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Sample size and robust marginal methods for cluster‐randomized trials with censored event times

In cluster‐randomized trials, intervention effects are often formulated by specifying marginal models, fitting them under a working independence assumption, and using robust variance estimates to address the association in the responses within clusters. We develop sample size criteria within this framework, with analyses based on semiparametric Cox regression models fitted with event times subject to right censoring. At the design stage, copula models are specified to enable derivation of the asymptotic variance of estimators from a marginal Cox regression model and to compute the number of clusters necessary to satisfy power requirements. Simulation studies demonstrate the validity of the sample size formula in finite samples for a range of cluster sizes, censoring rates, and degrees of within‐cluster association among event times. The power and relative efficiency implications of copula misspecification is studied, as well as the effect of within‐cluster dependence in the censoring times. Sample size criteria and other design issues are also addressed for the setting where the event status is only ascertained at periodic assessments and times are interval censored. Copyright © 2014 JohnWiley & Sons, Ltd.     

A mechanistic nonlinear model for censored and mismeasured covariates in longitudinal models,with application in AIDS studies

When modeling longitudinal data, the true values of time‐varying covariates may be unknown because of detection‐limit censoring or measurement error. A common approach in the literature is to empirically model the covariate process based on observed data and then predict the censored values or mismeasured values based on this empirical model. Such an empirical model can be misleading, especially for censored values since the (unobserved) censored values may behave very differently than observed values due to the underlying data‐generation mechanisms or disease status. In this paper, we propose a mechanistic nonlinear covariate model based on the underlying data‐generation mechanisms to address censored values and mismeasured values. Such a mechanistic model is based on solid scientific or biological arguments, so the predicted censored or mismeasured values are more reasonable. We use a Monte Carlo EM algorithm for likelihood inference and apply the methods to an AIDS dataset, where viral load is censored by a lower detection limit. Simulation results confirm that the proposed models and methods offer substantial advantages over existing empirical covariate models for censored and mismeasured covariates.     

Maximum likelihood estimation in generalized linear models with multiple covariates subject to detection limits

The analysis of data subject to detection limits is becoming increasingly necessary in many environmental and laboratory studies. Covariates subject to detection limits are often left censored because of a measurement device having a minimal lower limit of detection. In this paper, we propose a Monte Carlo version of the expectation–maximization algorithm to handle large number of covariates subject to detection limits in generalized linear models. We model the covariate distribution via a sequence of one‐dimensional conditional distributions, and sample the covariate values using an adaptive rejection metropolis algorithm. Parameter estimation is obtained by maximization via the Monte Carlo M‐step. This procedure is applied to a real dataset from the National Health and Nutrition Examination Survey, in which values of urinary heavy metals are subject to a limit of detection. Through simulation studies, we show that the proposed approach can lead to a significant reduction in variance for parameter estimates in these models, improving the power of such studies. Copyright © 2011 John Wiley & Sons, Ltd.     

Multiple imputation based on restricted mean model for censored data

Most multiple imputation (MI) methods for censored survival data either ignore patient characteristics when imputing a likely event time, or place quite restrictive modeling assumptions on the survival distributions used for imputation. In this research, we propose a robust MI approach that directly imputes restricted lifetimes over the study period based on a model of the mean restricted life as a linear function of covariates. This method has the advantages of retaining patient characteristics when making imputation choices through the restricted mean parameters and does not make assumptions on the shapes of hazards or survival functions. Simulation results show that our method outperforms its closest competitor for modeling restricted mean lifetimes in terms of bias and efficiency in both independent censoring and dependent censoring scenarios. Survival estimates of restricted lifetime model parameters and marginal survival estimates regain much of the precision lost due to censoring. The proposed method is also much less subject to dependent censoring bias captured by covariates in the restricted mean model. This particular feature is observed in a full statistical analysis conducted in the context of the International Breast Cancer Study Group Ludwig Trial V using the proposed methodology.     

Review: a gentle introduction to imputation of missing values

In most situations, simple techniques for handling missing data (such as complete case analysis, overall mean imputation, and the missing-indicator method) produce biased results, whereas imputation techniques yield valid results without complicating the analysis once the imputations are carried out. Imputation techniques are based on the idea that any subject in a study sample can be replaced by a new randomly chosen subject from the same source population. Imputation of missing data on a variable is replacing that missing by a value that is drawn from an estimate of the distribution of this variable. In single imputation, only one estimate is used. In multiple imputation, various estimates are used, reflecting the uncertainty in the estimation of this distribution. Under the general conditions of so-called missing at random and missing completely at random, both single and multiple imputations result in unbiased estimates of study associations. But single imputation results in too small estimated standard errors, whereas multiple imputation results in correctly estimated standard errors and confidence intervals. In this article we explain why all this is the case, and use a simple simulation study to demonstrate our explanations. We also explain and illustrate why two frequently used methods to handle missing data, i.e., overall mean imputation and the missing-indicator method, almost always result in biased estimates.     

Methods for comparing center‐specific survival outcomes using direct standardization

The evaluation of center‐specific outcomes is often through survival analysis methods. Such evaluations must account for differences in the distribution of patient characteristics across centers. In the context of censored event times, it is also important that the measure chosen to evaluate centers not be influenced by imbalances in the center‐specific censoring distributions. The practice of using center indicators in a hazard regression model is often invalid, inconvenient, or undesirable to carry out. We propose a semiparametric version of the standardized rate ratio (SRR) useful for the evaluation of centers with respect to a right‐censored event time. The SRR for center j can be interpreted as the ratio of the expected number of deaths in the total population (if the total population were in fact subject to the center j mortality hazard) to the observed number of events. The proposed measure is not affected by differences in center‐specific covariate or censoring distributions. Asymptotic properties of the proposed estimators are derived, with finite‐sample properties examined through simulation studies. The proposed methods are applied to national kidney transplant data. Copyright © 2014 John Wiley & Sons, Ltd.     

Survival analysis using auxiliary variables via non-parametric multiple imputation

We develop an approach, based on multiple imputation, that estimates the marginal survival distribution in survival analysis using auxiliary variables to recover information for censored observations. To conduct the imputation, we use two working survival models to define a nearest neighbour imputing risk set. One model is for the event times and the other for the censoring times. Based on the imputing risk set, two non-parametric multiple imputation methods are considered: risk set imputation, and Kaplan-Meier imputation. For both methods a future event or censoring time is imputed for each censored observation. With a categorical auxiliary variable, we show that with a large number of imputes the estimates from the Kaplan-Meier imputation method correspond to the weighted Kaplan-Meier estimator. We also show that the Kaplan-Meier imputation method is robust to mis-specification of either one of the two working models. In a simulation study with time independent and time-dependent auxiliary variables, we compare the multiple imputation approaches with an inverse probability of censoring weighted method. We show that all approaches can reduce bias due to dependent censoring and improve the efficiency. We apply the approaches to AIDS clinical trial data comparing ZDV and placebo, in which CD4 count is the time-dependent auxiliary variable.     

A missing composite covariate in survival analysis: A case study of the Chinese Longitudinal Health and Longevity Survey

We estimate a Cox proportional hazards model where one of the covariates measures the level of a subject's cognitive functioning by grading the total score obtained by the subject on the items of a questionnaire. A case study is presented where the sample includes partial respondents, who did not answer some questionnaire items. The total score takes, hence, the form of an interval‐censored variable and, as a result, the level of cognitive functioning is missing on some subjects. We handle the partial respondents by taking a likelihood‐based approach where survival time is jointly modelled with the censored total score and the size of the censoring interval. Estimates are obtained by an E‐M‐type algorithm that reduces to the iterative maximization of three complete log‐likelihood functions derived from two augmented data sets with case weights, alternated with weights updating. This methodology is exploited to assess the Mini‐Mental State Examination index as a prognostic factor of survival in a sample of Chinese older adults. Copyright © 2009 John Wiley & Sons, Ltd.     

Subgroups from regression trees with adjustment for prognostic effects and postselection inference

Identification of subgroups with differential treatment effects in randomized trials is attracting much attention. Many methods use regression tree algorithms. This article addresses 2 important questions arising from the subgroups: how to ensure that treatment effects in subgroups are not confounded with effects of prognostic variables and how to determine the statistical significance of treatment effects in the subgroups. We address the first question by selectively including linear prognostic effects in the subgroups in a regression tree model. The second question is more difficult because it falls within the subject of postselection inference. We use a bootstrap technique to calibrate normal-theory t intervals so that their expected coverage probability, averaged over all the subgroups in a fitted model, approximates the desired confidence level. It can also provide simultaneous confidence intervals for all subgroups. The first solution is implemented in the GUIDE algorithm and is applicable to data with missing covariate values, 2 or more treatment arms, and outcomes subject to right censoring. Bootstrap calibration is applicable to any subgroup identification method; it is not restricted to regression tree models. Two real examples are used for illustration: a diabetes trial where the outcomes are completely observed but some covariate values are missing and a breast cancer trial where the outcome is right censored.     

Multiple augmentation for interval-censored data with measurement error

There has been substantial effort devoted to the analysis of censored failure time with covariates that are subject to measurement error. Previous studies have focused on right-censored survival data, but interval-censored survival data with covariate measurement error are yet to be investigated. Our study is partly motivated by analysis of the HIV clinical trial AIDS Clinical Trial Group (ACTG) 175 data, where the occurrence time of AIDS is interval censored and the covariate CD4 count is subject to measurement error. We assume that the data are realized from a proportional hazards model. A multiple augmentation approach is proposed to convert interval-censored data to right-censored data, and the conditional score approach is then employed to account for measurement error. The proposed approach is easy to implement and can be readily extended to other semiparametric models. Extensive simulations show that the proposed approach has satisfactory finite-sample performance. The ACTG 175 data are then analyzed.     

Combining multiple imputation and meta‐analysis with individual participant data

Multiple imputation is a strategy for the analysis of incomplete data such that the impact of the missingness on the power and bias of estimates is mitigated. When data from multiple studies are collated, we can propose both within‐study and multilevel imputation models to impute missing data on covariates. It is not clear how to choose between imputation models or how to combine imputation and inverse‐variance weighted meta‐analysis methods. This is especially important as often different studies measure data on different variables, meaning that we may need to impute data on a variable which is systematically missing in a particular study. In this paper, we consider a simulation analysis of sporadically missing data in a single covariate with a linear analysis model and discuss how the results would be applicable to the case of systematically missing data. We find in this context that ensuring the congeniality of the imputation and analysis models is important to give correct standard errors and confidence intervals. For example, if the analysis model allows between‐study heterogeneity of a parameter, then we should incorporate this heterogeneity into the imputation model to maintain the congeniality of the two models. In an inverse‐variance weighted meta‐analysis, we should impute missing data and apply Rubin's rules at the study level prior to meta‐analysis, rather than meta‐analyzing each of the multiple imputations and then combining the meta‐analysis estimates using Rubin's rules. We illustrate the results using data from the Emerging Risk Factors Collaboration. © 2013 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

Missing data in a multi-item instrument were best handled by multiple imputation at the item score level

ObjectivesRegardless of the proportion of missing values, complete-case analysis is most frequently applied, although advanced techniques such as multiple imputation (MI) are available. The objective of this study was to explore the performance of simple and more advanced methods for handling missing data in cases when some, many, or all item scores are missing in a multi-item instrument.Study Design and SettingReal-life missing data situations were simulated in a multi-item variable used as a covariate in a linear regression model. Various missing data mechanisms were simulated with an increasing percentage of missing data. Subsequently, several techniques to handle missing data were applied to decide on the most optimal technique for each scenario. Fitted regression coefficients were compared using the bias and coverage as performance parameters.ResultsMean imputation caused biased estimates in every missing data scenario when data are missing for more than 10% of the subjects. Furthermore, when a large percentage of subjects had missing items (>25%), MI methods applied to the items outperformed methods applied to the total score.ConclusionWe recommend applying MI to the item scores to get the most accurate regression model estimates. Moreover, we advise not to use any form of mean imputation to handle missing data.     

Identification of significant host factors for HIV dynamics modelled by non-linear mixed-effects models

Non-linear mixed-effects models are powerful tools for modelling HIV viral dynamics. In AIDS clinical trials, the viral load measurements for each subject are often sparse. In such cases, linearization procedures are usually used for inferences. Under such linearization procedures, however, standard covariate selection methods based on the approximate likelihood, such as the likelihood ratio test, may not be reliable. In order to identify significant host factors for HIV dynamics, in this paper we consider two alternative approaches for covariate selection: one is based on individual non-linear least square estimates and the other is based on individual empirical Bayes estimates. Our simulation study shows that, if the within-individual data are sparse and the between-individual variation is large, the two alternative covariate selection methods are more reliable than the likelihood ratio test, and the more powerful method based on individual empirical Bayes estimates is especially preferable. We also consider the missing data in covariates. The commonly used missing data methods may lead to misleading results. We recommend a multiple imputation method to handle missing covariates. A real data set from an AIDS clinical trial is analysed based on various covariate selection methods and missing data methods.     

Properties of the full random-effect modeling approach with missing covariate data

During drug development, a key step is the identification of relevant covariates predicting between-subject variations in drug response. The full random effects model (FREM) is one of the full-covariate approaches used to identify relevant covariates in nonlinear mixed effects models. Here we explore the ability of FREM to handle missing (both missing completely at random (MCAR) and missing at random (MAR)) covariate data and compare it to the full fixed-effects model (FFEM) approach, applied either with complete case analysis or mean imputation. A global health dataset (20 421 children) was used to develop a FREM describing the changes of height for age Z-score (HAZ) over time. Simulated datasets (n = 1000) were generated with variable rates of missing (MCAR) covariate data (0%-90%) and different proportions of missing (MAR) data condition on either observed covariates or predicted HAZ. The three methods were used to re-estimate model and compared in terms of bias and precision which showed that FREM had only minor increases in bias and minor loss of precision at increasing percentages of missing (MCAR) covariate data and performed similarly in the MAR scenarios. Conversely, the FFEM approaches either collapsed at $$70% of missing (MCAR) covariate data (FFEM complete case analysis) or had large bias increases and loss of precision (FFEM with mean imputation). Our results suggest that FREM is an appropriate approach to covariate modeling for datasets with missing (MCAR and MAR) covariate data, such as in global health studies.     

Fitting additive hazards models for case‐cohort studies: a multiple imputation approach

In this paper, we consider fitting semiparametric additive hazards models for case‐cohort studies using a multiple imputation approach. In a case‐cohort study, main exposure variables are measured only on some selected subjects, but other covariates are often available for the whole cohort. We consider this as a special case of a missing covariate by design. We propose to employ a popular incomplete data method, multiple imputation, for estimation of the regression parameters in additive hazards models. For imputation models, an imputation modeling procedure based on a rejection sampling is developed. A simple imputation modeling that can naturally be applied to a general missing‐at‐random situation is also considered and compared with the rejection sampling method via extensive simulation studies. In addition, a misspecification aspect in imputation modeling is investigated. The proposed procedures are illustrated using a cancer data example. Copyright © 2015 John Wiley & Sons, Ltd.