Nonparametric estimation of the cumulative incidence function under outcome misclassification using external validation data

Misclassification of outcomes or event types is common in health sciences research and can lead to serious bias when estimating the cumulative incidence functions in settings with competing risks. Recent work has shown how to estimate nonparametric cumulative incidence functions in the presence of nondifferential outcome misclassification when the misclassification probabilities are known. Here, we extend this approach to account for misclassification that is differential with respect to important predictors of the outcome using misclassification probabilities estimated from external validation data. Moreover, we propose a bootstrap approach in which the observations from both the main study data and the external validation study are resampled to allow the uncertainty in the misclassification probabilities to propagate through the analysis into the final confidence intervals, ensuring appropriate confidence interval coverage probabilities. The proposed estimator is shown to be uniformly consistent and simulation studies indicate that both the estimator and the standard error estimation approach perform well in finite samples. The methodology is applied to estimate the cumulative incidence of death and disengagement from HIV care in a large cohort of HIV infected individuals in sub-Saharan Africa, where a significant death underreporting issue leads to outcome misclassification. This analysis uses external validation data from a separate study conducted in the same country.     

Flexible parametric modelling of the cause‐specific cumulative incidence function

Competing risks arise with time‐to‐event data when individuals are at risk of more than one type of event and the occurrence of one event precludes the occurrence of all other events. A useful measure with competing risks is the cause‐specific cumulative incidence function (CIF), which gives the probability of experiencing a particular event as a function of follow‐up time, accounting for the fact that some individuals may have a competing event. When modelling the cause‐specific CIF, the most common model is a semi‐parametric proportional subhazards model. In this paper, we propose the use of flexible parametric survival models to directly model the cause‐specific CIF where the effect of follow‐up time is modelled using restricted cubic splines. The models provide smooth estimates of the cause‐specific CIF with the important advantage that the approach is easily extended to model time‐dependent effects. The models can be fitted using standard survival analysis tools by a combination of data expansion and introducing time‐dependent weights. Various link functions are available that allow modelling on different scales and have proportional subhazards, proportional odds and relative absolute risks as particular cases. We conduct a simulation study to evaluate how well the spline functions approximate subhazard functions with complex shapes. The methods are illustrated using data from the European Blood and Marrow Transplantation Registry showing excellent agreement between parametric estimates of the cause‐specific CIF and those obtained from a semi‐parametric model. We also fit models relaxing the proportional subhazards assumption using alternative link functions and/or including time‐dependent effects. Copyright © 2016 John Wiley & Sons, Ltd.     

Inference for cumulative incidence quantiles via parametric and nonparametric approaches

In survival analysis, a point estimate and confidence interval for median survival time have been frequently used to summarize the survival curve. However, such quantile analyses on competing risks data have not been widely investigated. In this paper, we propose parametric inferences for quantiles from the cumulative incidence function and develop parametric confidence intervals for quantiles. In addition, we study a simplified method of inference for the nonparametric approach. We compare the parametric and nonparametric inferences in empirical studies. Simulation studies show that the procedures perform well, with parametric analyses yielding smaller mean square error when the model is not too badly misspecified. We illustrate the methods with data from a breast cancer clinical trial.     

Multiple imputation methods for nonparametric inference on cumulative incidence with missing cause of failure

We propose a nonparametric approach for cumulative incidence estimation when causes of failure are unknown or missing for some subjects. Under the missing at random assumption, we estimate the cumulative incidence function using multiple imputation methods. We develop asymptotic theory for the cumulative incidence estimators obtained from multiple imputation methods. We also discuss how to construct confidence intervals for the cumulative incidence function and perform a test for comparing the cumulative incidence functions in two samples with missing cause of failure. Through simulation studies, we show that the proposed methods perform well. The methods are illustrated with data from a randomized clinical trial in early stage breast cancer. Copyright © 2014 John Wiley & Sons, Ltd.     

Direct likelihood inference on the cause‐specific cumulative incidence function: A flexible parametric regression modelling approach

In a competing risks analysis, interest lies in the cause‐specific cumulative incidence function (CIF) that can be calculated by either (1) transforming on the cause‐specific hazard or (2) through its direct relationship with the subdistribution hazard. We expand on current competing risks methodology from within the flexible parametric survival modelling framework (FPM) and focus on approach (2). This models all cause‐specific CIFs simultaneously and is more useful when we look to questions on prognosis. We also extend cure models using a similar approach described by Andersson et al for flexible parametric relative survival models. Using SEER public use colorectal data, we compare and contrast our approach with standard methods such as the Fine & Gray model and show that many useful out‐of‐sample predictions can be made after modelling the cause‐specific CIFs using an FPM approach. Alternative link functions may also be incorporated such as the logit link. Models can also be easily extended for time‐dependent effects.     

Assessing cumulative incidence functions under the semiparametric additive risk model

In analyzing competing risks data, a quantity of considerable interest is the cumulative incidence function. Often, the effect of covariates on the cumulative incidence function is modeled via the proportional hazards model for the cause‐specific hazard function. As the proportionality assumption may be too restrictive in practice, we consider an alternative more flexible semiparametric additive hazards model of (Biometrika 1994; 81 :501–514) for the cause‐specific hazard. This model specifies the effect of covariates on the cause‐specific hazard to be additive as well as allows the effect of some covariates to be fixed and that of others to be time varying. We present an approach for constructing confidence intervals as well as confidence bands for the cause‐specific cumulative incidence function of subjects with given values of the covariates. Furthermore, we also present an approach for constructing confidence intervals and confidence bands for comparing two cumulative incidence functions given values of the covariates. The finite sample property of the proposed estimators is investigated through simulations. We conclude our paper with an analysis of the well‐known malignant melanoma data using our method. Published in 2009 by John Wiley & Sons, Ltd.     

Smooth semi‐nonparametric (SNP) estimation of the cumulative incidence function

This paper presents a novel approach to estimation of the cumulative incidence function in the presence of competing risks. The underlying statistical model is specified via a mixture factorization of the joint distribution of the event type and the time to the event. The time to event distributions conditional on the event type are modeled using smooth semi‐nonparametric densities. One strength of this approach is that it can handle arbitrary censoring and truncation while relying on mild parametric assumptions. A stepwise forward algorithm for model estimation and adaptive selection of smooth semi‐nonparametric polynomial degrees is presented, implemented in the statistical software R, evaluated in a sequence of simulation studies, and applied to data from a clinical trial in cryptococcal meningitis. The simulations demonstrate that the proposed method frequently outperforms both parametric and nonparametric alternatives. They also support the use of ‘ad hoc’ asymptotic inference to derive confidence intervals. An extension to regression modeling is also presented, and its potential and challenges are discussed. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Regression modeling of the cumulative incidence function with missing causes of failure using pseudo‐values

Competing risks arise when patients may fail from several causes. Strategies for modeling event‐specific quantities often assume that the cause of failure is known for all patients, but this is seldom the case. Several authors have addressed the problem of modeling the cause‐specific hazard rates with missing causes of failure. In contrast, direct modeling of the cumulative incidence function has received little attention. We provide a general framework for regression modeling of this function in the missing cause setting, encompassing key models such as the Fine and Gray and additive models, by considering two extensions of the Andersen–Klein pseudo‐value approach. The first extension is a novel inverse probability weighting method, whereas the second extension is based on a previously proposed multiple imputation procedure. We evaluated the gain in using these approaches with small samples in an extensive simulation study. We analyzed the data from an Eastern Cooperative Oncology Group breast cancer treatment clinical trial to illustrate the practical value and ease of implementation of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Semiparametric regression on cumulative incidence function with interval‐censored competing risks data

Many biomedical and clinical studies with time‐to‐event outcomes involve competing risks data. These data are frequently subject to interval censoring. This means that the failure time is not precisely observed but is only known to lie between two observation times such as clinical visits in a cohort study. Not taking into account the interval censoring may result in biased estimation of the cause‐specific cumulative incidence function, an important quantity in the competing risks framework, used for evaluating interventions in populations, for studying the prognosis of various diseases, and for prediction and implementation science purposes. In this work, we consider the class of semiparametric generalized odds rate transformation models in the context of sieve maximum likelihood estimation based on B‐splines. This large class of models includes both the proportional odds and the proportional subdistribution hazard models (i.e., the Fine–Gray model) as special cases. The estimator for the regression parameter is shown to be consistent, asymptotically normal and semiparametrically efficient. Simulation studies suggest that the method performs well even with small sample sizes. As an illustration, we use the proposed method to analyze data from HIV‐infected individuals obtained from a large cohort study in sub‐Saharan Africa. We also provide the R function ciregic that implements the proposed method and present an illustrative example. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

Estimation of the adjusted cause-specific cumulative probability using flexible regression models for the cause-specific hazards

In competing risks setting, we account for death according to a specific cause and the quantities of interest are usually the cause-specific hazards (CSHs) and the cause-specific cumulative probabilities. A cause-specific cumulative probability can be obtained with a combination of the CSHs or via the subdistribution hazard. Here, we modeled the CSH with flexible hazard-based regression models using B-splines for the baseline hazard and time-dependent (TD) effects. We derived the variance of the cause-specific cumulative probabilities at the population level using the multivariate delta method and showed how we could easily quantify the impact of a covariate on the cumulative probability scale using covariate-adjusted cause-specific cumulative probabilities and their difference. We conducted a simulation study to evaluate the performance of this approach in its ability to estimate the cumulative probabilities using different functions for the cause-specific log baseline hazard and with or without a TD effect. In the scenario with TD effect, we tested both well-specified and misspecified models. We showed that the flexible regression models perform nearly as well as the nonparametric method, if we allow enough flexibility for the baseline hazards. Moreover, neglecting the TD effect hardly affects the cumulative probabilities estimates of the whole population but impacts them in the various subgroups. We illustrated our approach using data from people diagnosed with monoclonal gammopathy of undetermined significance and provided the R-code to derive those quantities, as an extension of the R-package mexhaz .     

Non-parametric inference of adverse events under informative censoring

In long-term treatments or in treatments associated with frequent severe adverse events (AEs) such as those for oncology, it is important to know the probability of occurrence of AEs over time and their severity. However, some patients discontinue treatment and drop out of the clinical trial. Assumption that the drop-outs are non-informative is not always true and are not validated by data.We propose a method of applying competing risk analysis by defining events of 'occurrence of AE' and 'drop-out prior to AE'. We focus on one AE at a time. We distinguish obvious non-informative censoring from other censorings that may not be non-informative. Therefore, our approach does not need an independent assumption for drop-outs. The cumulative incidence function estimator (CIFE) for the AE by severity can be obtained by treating the degree of severity as a competing risk within the AE. We also propose a non-parametric estimator of CIF for sequential occurrence of the same AE by forming a subset of subjects with prior occurrence(s) of the same AE and by applying Wang and Wells' estimator. We give a very simple formulation of the cumulative joint incidence function estimator (CJIFE) for subjects who drop out of the clinical trial after having suffered from the AE at least one time. We evaluate the performance of Pepe's variance estimator for CJIFE with small samples by simulations. We find that it works well with a sample size more than 100. A useful graphical presentation for CIFEs and CJIFE is shown.     

Small sample GEE estimation of regression parameters for longitudinal data

Longitudinal (clustered) response data arise in many bio‐statistical applications which, in general, cannot be assumed to be independent. Generalized estimating equation (GEE) is a widely used method to estimate marginal regression parameters for correlated responses. The advantage of the GEE is that the estimates of the regression parameters are asymptotically unbiased even if the correlation structure is misspecified, although their small sample properties are not known. In this paper, two bias adjusted GEE estimators of the regression parameters in longitudinal data are obtained when the number of subjects is small. One is based on a bias correction, and the other is based on a bias reduction. Simulations show that the performances of both the bias‐corrected methods are similar in terms of bias, efficiency, coverage probability, average coverage length, impact of misspecification of correlation structure, and impact of cluster size on bias correction. Both these methods show superior properties over the GEE estimates for small samples. Further, analysis of data involving a small number of subjects also shows improvement in bias, MSE, standard error, and length of the confidence interval of the estimates by the two bias adjusted methods over the GEE estimates. For small to moderate sample sizes (), either of the bias‐corrected methods GEEBc and GEEBr can be used. However, the method GEEBc should be preferred over GEEBr, as the former is computationally easier. For large sample sizes, the GEE method can be used. Copyright © 2014 John Wiley & Sons, Ltd.     

Sample-size calculations for studies with correlated ordinal outcomes

Correlated ordinal response data often arise in public health studies. Sample-size (power) calculations are a crucial step in designing such studies to ensure an adequate sample to detect a significant effect. Here we extend Rochon's method of sample-size estimation with a repeated binary response to the ordinal case. The proposed sample-size calculations are based on an analysis with generalized estimating equations (GEE) and inference with the Wald test. Simulation results demonstrate the merit of the proposed power calculations. Analysis of an arthritis clinical trial is used for illustration.     

Three methods for analysing correlated ROC curves: a comparison in real data sets from multi-reader,multi-case studies with a factorial design

This paper compares three published methods for analysing multiple correlated ROC curves: a method using generalized estimating equations with marginal non-proportional ordinal regression models; a method using jackknifed pseudovalues of summary statistics; a method using a corrected F-test from analysis of variance of summary statistics. Use of these methods is illustrated through six real data examples from studies with the common factorial design, that is, multiple readers interpreting images obtained with each test modality on each study subject. The issue of the difference between typical summary statistics and summary statistics from typical ROC curves is explored. The examples also address similarities and differences among the analytical methods. In particular, while point estimates of differences between test modalities are similar, the standard errors of these differences do not agree for all three methods. A simulation study supports the standard errors provided by the generalized estimating equations with marginal non-proportional ordinal regression models.     

A simulation-based comparison of techniques to correct for measurement error in matched case-control studies

The presence of measurement errors affecting the covariates in regression models is a relevant topic in many scientific areas, as, for example, in epidemiology. An example is given by an epidemiological population-based matched case-control study on the aetiology of childhood malignancies, which is currently under completion in Italy. This study was aimed at evaluating the effects of childhood exposure to extremely low electromagnetic fields on the risk of disease occurrence by taking into account the possibility of erroneous measures of the exposure. Within this framework, we focus on the application of likelihood methods to correct for measurement error. This approach, which has received less attention in literature with respect to alternatives, is compared with commonly used methods such as regression calibration and SIMEX. The comparison is performed by simulation, under a broad range of measurement error structures.     

Tests of trait-haplotype association when linkage phase is ambiguous, appropriate for matched case-control and cohort studies with competing risks

The impact of competing risks on tests of association between disease and haplotypes has been largely ignored. We consider situations in which linkage phase is ambiguous and show that tests for disease-haplotype association can lead to rejection of the null hypothesis, even when true, with more than the nominal 5 per cent frequency. This problem tends to occur if a haplotype is associated with overall mortality, even if the haplotype is not associated with disease risk. A small simulation study illustrates the magnitude of bias (high type I error rate) in the context of a cohort study in which a modest number of disease cases (about 350) occur over time. The bias remains even if the score test is based on a logistic model that includes age as a covariate. For cohort studies, we propose a new test based on a modification of the proportional hazards model and for case-control studies, a test based on a conditional likelihood that have the correct size under the null even in the presence of competing risks, and that can be used when haplotype is ambiguous.     

Double-robust estimation of an exposure-outcome odds ratio adjusting for confounding in cohort and case-control studies

Modern epidemiologic studies often aim to evaluate the causal effect of a point exposure on the risk of a disease from cohort or case-control observational data. Because confounding bias is of serious concern in such non-experimental studies, investigators routinely adjust for a large number of potential confounders in a logistic regression analysis of the effect of exposure on disease outcome. Unfortunately, when confounders are not correctly modeled, standard logistic regression is likely biased in its estimate of the effect of exposure, potentially leading to erroneous conclusions. We partially resolve this serious limitation of standard logistic regression analysis with a new iterative approach that we call ProRetroSpective estimation, which carefully combines standard logistic regression with a logistic regression analysis in which exposure is the dependent variable and the outcome and confounders are the independent variables. As a result, we obtain a correct estimate of the exposure-outcome odds ratio, if either thestandard logistic regression of the outcome given exposure and confounding factors is correct, or the regression model of exposure given the outcome and confounding factors is correct but not necessarily both, that is, it is double-robust. In fact, it also has certain advantadgeous efficiency properties. The approach is general in that it applies to both cohort and case-control studies whether the design of the study is matched or unmatched on a subset of covariates. Finally, an application illustrates the methods using data from the National Cancer Institute's Black/White Cancer Survival Study.     

Distribution‐free models for longitudinal count responses with overdispersion and structural zeros

Overdispersion and structural zeros are two major manifestations of departure from the Poisson assumption when modeling count responses using Poisson log‐linear regression. As noted in a large body of literature, ignoring such departures could yield bias and lead to wrong conclusions. Different approaches have been developed to tackle these two major problems. In this paper, we review available methods for dealing with overdispersion and structural zeros within a longitudinal data setting and propose a distribution‐free modeling approach to address the limitations of these methods by utilizing a new class of functional response models. We illustrate our approach with both simulated and real study data. Copyright © 2012 John Wiley & Sons, Ltd.     

Augmented generalized estimating equations for improving efficiency and validity of estimation in cluster randomized trials by leveraging cluster-level and individual-level covariates

Recent methodological advances in covariate adjustment in randomized clinical trials have used semiparametric theory to improve efficiency of inferences by incorporating baseline covariates; these methods have focused on independent outcomes. We modify one of these approaches, augmentation of standard estimators, for use within cluster randomized trials in which treatments are assigned to groups of individuals, thereby inducing correlation. We demonstrate the potential for imbalance correction and efficiency improvement through consideration of both cluster-level covariates and individual-level covariates. To improve small-sample estimation, we consider several variance adjustments. We evaluate this approach for continuous and binary outcomes through simulation and apply it to data from a cluster randomized trial of a community behavioral intervention related to HIV prevention in Tanzania.