Discrimination measures for survival outcomes: connection between the AUC and the predictiveness curve

Finding out biomarkers and building risk scores to predict the occurrence of survival outcomes is a major concern of clinical epidemiology, and so is the evaluation of prognostic models. In this paper, we are concerned with the estimation of the time-dependent AUC--area under the receiver-operating curve--which naturally extends standard AUC to the setting of survival outcomes and enables to evaluate the discriminative power of prognostic models. We establish a simple and useful relation between the predictiveness curve and the time-dependent AUC--AUC(t). This relation confirms that the predictiveness curve is the key concept for evaluating calibration and discrimination of prognostic models. It also highlights that accurate estimates of the conditional absolute risk function should yield accurate estimates for AUC(t). From this observation, we derive several estimators for AUC(t) relying on distinct estimators of the conditional absolute risk function. An empirical study was conducted to compare our estimators with the existing ones and assess the effect of model misspecification--when estimating the conditional absolute risk function--on the AUC(t) estimation. We further illustrate the methodology on the Mayo PBC and the VA lung cancer data sets.     

A latent variable approach to study gene-environment interactions in the presence of multiple correlated exposures

Many existing cohort studies initially designed to investigate disease risk as a function of environmental exposures have collected genomic data in recent years with the objective of testing for gene-environment interaction (G × E) effects. In environmental epidemiology, interest in G × E arises primarily after a significant effect of the environmental exposure has been documented. Cohort studies often collect rich exposure data; as a result, assessing G × E effects in the presence of multiple exposure markers further increases the burden of multiple testing, an issue already present in both genetic and environment health studies. Latent variable (LV) models have been used in environmental epidemiology to reduce dimensionality of the exposure data, gain power by reducing multiplicity issues via condensing exposure data, and avoid collinearity problems due to presence of multiple correlated exposures. We extend the LV framework to characterize gene-environment interaction in presence of multiple correlated exposures and genotype categories. Further, similar to what has been done in case-control G × E studies, we use the assumption of gene-environment (G-E) independence to boost the power of tests for interaction. The consequences of making this assumption, or the issue of how to explicitly model G-E association has not been previously investigated in LV models. We postulate a hierarchy of assumptions about the LV model regarding the different forms of G-E dependence and show that making such assumptions may influence inferential results on the G, E, and G × E parameters. We implement a class of shrinkage estimators to data adaptively trade-off between the most restrictive to most flexible form of G-E dependence assumption and note that such class of compromise estimators can serve as a benchmark of model adequacy in LV models. We demonstrate the methods with an example from the Early Life Exposures in Mexico City to Neuro-Toxicants Study of lead exposure, iron metabolism genes, and birth weight.     

Fast approximations of pseudo-observations in the context of right censoring and interval censoring

In the context of right-censored and interval-censored data, we develop asymptotic formulas to compute pseudo-observations for the survival function and the restricted mean survival time (RMST). These formulas are based on the original estimators and do not involve computation of the jackknife estimators. For right-censored data, Von Mises expansions of the Kaplan–Meier estimator are used to derive the pseudo-observations. For interval-censored data, a general class of parametric models for the survival function is studied. An asymptotic representation of the pseudo-observations is derived involving the Hessian matrix and the score vector. Theoretical results that justify the use of pseudo-observations in regression are also derived. The formula is illustrated on the piecewise-constant-hazard model for the RMST. The proposed approximations are extremely accurate, even for small sample sizes, as illustrated by Monte Carlo simulations and real data. We also study the gain in terms of computation time, as compared to the original jackknife method, which can be substantial for a large dataset.     

Modeling clustered long‐term survivors using marginal mixture cure model

There is a great deal of recent interests in modeling right‐censored clustered survival time data with a possible fraction of cured subjects who are nonsusceptible to the event of interest using marginal mixture cure models. In this paper, we consider a semiparametric marginal mixture cure model for such data and propose to extend an existing generalized estimating equation approach by a new unbiased estimating equation for the regression parameters in the latency part of the model. The large sample properties of the regression effect estimators in both incidence and the latency parts are established. The finite sample properties of the estimators are studied in simulation studies. The proposed method is illustrated with a bone marrow transplantation data and a tonsil cancer data.     

Estimating the proportion cured of cancer: Some practical advice for users

BackgroundCure models can provide improved possibilities for inference if used appropriately, but there is potential for misleading results if care is not taken. In this study, we compared five commonly used approaches for modelling cure in a relative survival framework and provide some practical advice on the use of these approaches.Patients and methodsData for colon, female breast, and ovarian cancers were used to illustrate these approaches. The proportion cured was estimated for each of these three cancers within each of three age groups. We then graphically assessed the assumption of cure and the model fit, by comparing the predicted relative survival from the cure models to empirical life table estimates.ResultsWhere both cure and distributional assumptions are appropriate (e.g., for colon or ovarian cancer patients aged <75 years), all five approaches led to similar estimates of the proportion cured. The estimates varied slightly when cure was a reasonable assumption but the distributional assumption was not (e.g., for colon cancer patients ≥75 years). Greater variability in the estimates was observed when the cure assumption was not supported by the data (breast cancer).ConclusionsIf the data suggest cure is not a reasonable assumption then we advise against fitting cure models. In the scenarios where cure was reasonable, we found that flexible parametric cure models performed at least as well, or better, than the other modelling approaches. We recommend that, regardless of the model used, the underlying assumptions for cure and model fit should always be graphically assessed.     

Bias analysis and the simulation‐extrapolation method for survival data with covariate measurement error under parametric proportional odds models

It has been well known that ignoring measurement error may result in substantially biased estimates in many contexts including linear and nonlinear regressions. For survival data with measurement error in covariates, there has been extensive discussion in the literature with the focus on proportional hazards (PH) models. Recently, research interest has extended to accelerated failure time (AFT) and additive hazards (AH) models. However, the impact of measurement error on other models, such as the proportional odds model, has received relatively little attention, although these models are important alternatives when PH, AFT, or AH models are not appropriate to fit data. In this paper, we investigate this important problem and study the bias induced by the naive approach of ignoring covariate measurement error. To adjust for the induced bias, we describe the simulation‐extrapolation method. The proposed method enjoys a number of appealing features. Its implementation is straightforward and can be accomplished with minor modifications of existing software. More importantly, the proposed method does not require modeling the covariate process, which is quite attractive in practice. As the precise values of error‐prone covariates are often not observable, any modeling assumption on such covariates has the risk of model misspecification, hence yielding invalid inferences if this happens. The proposed method is carefully assessed both theoretically and empirically. Theoretically, we establish the asymptotic normality for resulting estimators. Numerically, simulation studies are carried out to evaluate the performance of the estimators as well as the impact of ignoring measurement error, along with an application to a data set arising from the Busselton Health Study. Sensitivity of the proposed method to misspecification of the error model is studied as well.     

Bayesian cure rate frailty models with application to a root canal therapy study

Due to natural or artificial clustering, multivariate survival data often arise in biomedical studies, for example, a dental study involving multiple teeth from each subject. A certain proportion of subjects in the population who are not expected to experience the event of interest are considered to be "cured" or insusceptible. To model correlated or clustered failure time data incorporating a surviving fraction, we propose two forms of cure rate frailty models. One model naturally introduces frailty based on biological considerations while the other is motivated from the Cox proportional hazards frailty model. We formulate the likelihood functions based on piecewise constant hazards and derive the full conditional distributions for Gibbs sampling in the Bayesian paradigm. As opposed to the Cox frailty model, the proposed methods demonstrate great potential in modeling multivariate survival data with a cure fraction. We illustrate the cure rate frailty models with a root canal therapy data set.     

Estimating and modeling the cure fraction in population-based cancer survival analysis

In population-based cancer studies, cure is said to occur when the mortality (hazard) rate in the diseased group of individuals returns to the same level as that expected in the general population. The cure fraction (the proportion of patients cured of disease) is of interest to patients and is a useful measure to monitor trends in survival of curable disease. There are 2 main types of cure fraction model, the mixture cure fraction model and the non-mixture cure fraction model, with most previous work concentrating on the mixture cure fraction model. In this paper, we extend the parametric non-mixture cure fraction model to incorporate background mortality, thus providing estimates of the cure fraction in population-based cancer studies. We compare the estimates of relative survival and the cure fraction between the 2 types of model and also investigate the importance of modeling the ancillary parameters in the selected parametric distribution for both types of model.     

Generalized parametric cure models for relative survival

Cure models are used in time-to-event analysis when not all individuals are expected to experience the event of interest, or when the survival of the considered individuals reaches the same level as the general population. These scenarios correspond to a plateau in the survival and relative survival function, respectively. The main parameters of interest in cure models are the proportion of individuals who are cured, termed the cure proportion, and the survival function of the uncured individuals. Although numerous cure models have been proposed in the statistical literature, there is no consensus on how to formulate these. We introduce a general parametric formulation of mixture cure models and a new class of cure models, termed latent cure models, together with a general estimation framework and software, which enable fitting of a wide range of different models. Through simulations, we assess the statistical properties of the models with respect to the cure proportion and the survival of the uncured individuals. Finally, we illustrate the models using survival data on colon cancer, which typically display a plateau in the relative survival. As demonstrated in the simulations, mixture cure models which are not guaranteed to be constant after a finite time point, tend to produce accurate estimates of the cure proportion and the survival of the uncured. However, these models are very unstable in certain cases due to identifiability issues, whereas LC models generally provide stable results at the price of more biased estimates.     

A frailty mixture cure model with application to hospital admission data [corrected

Mixture cure models have been utilized to analyze survival data with possible cure. This paper considers the inclusion of frailty into the mixture cure model to model recurrent event data with a cure fraction. An attractive feature of the proposed model is the allowance for heterogeneity in risk among those individuals experiencing the event of interest in addition to the incorporation of a cured component. Maximum likelihood estimates can be obtained using the Expectation Maximization algorithm and standard errors are calculated from the Bootstrap method. The model is applied to hospital readmission data among colorectal cancer patients.     

Robust alternatives to the F-Test in mixed linear models based on MM-estimates

Mixed linear models are commonly used to analyze data in many settings. These models are generally fitted by means of (restricted) maximum likelihood techniques relying heavily on normality. The sensitivity of the resulting estimators and related tests to this underlying assumption has been identified as a weakness that can even lead to wrong interpretations. Very recently a highly robust estimator based on a scale estimate, that is, an S-estimator, has been proposed for general mixed linear models. It has the advantage of being easy to compute and allows the computation of a robust score test. However, this proposal cannot be used to define a likelihood ratio type test that is certainly the most direct route to robustify an F-test. As the latter is usually a key tool of hypothesis testing in mixed linear models, we propose two new robust estimators that allow the desired extension. They also lead to resistant Wald-type tests useful for testing contrasts and covariate effects. We study their properties theoretically and by means of simulations. The analysis of a real data set illustrates the advantage of the new approach in the presence of outlying observations.     

A bivariate cure-mixture approach for modeling familial association in diseases

For modeling correlation in familial diseases with variable ages at onset, we propose a bivariate model that incorporates two types of pairwise association, one between the lifetime risk or the overall susceptibility of two individuals and one between the ages at onset between two susceptible individuals. For estimation, we consider a two-stage estimation procedure similar to that of Shih (1998, Biometrics 54, 1115-1128). We evaluate the properties of the estimators through simulations and compare the performance with that from a bivariate survival model that allows correlation between ages at onset only. We apply the methodology to breast cancer using the kinship data from the Washington Ashkenazi Study. We also discuss potential applications of the proposed method in the area of cure modeling.     

Estimating the area under the ROC curve when transporting a prediction model to a target population

We propose methods for estimating the area under the receiver operating characteristic (ROC) curve (AUC) of a prediction model in a target population that differs from the source population that provided the data used for original model development. If covariates that are associated with model performance, as measured by the AUC, have a different distribution in the source and target populations, then AUC estimators that only use data from the source population will not reflect model performance in the target population. Here, we provide identification results for the AUC in the target population when outcome and covariate data are available from the sample of the source population, but only covariate data are available from the sample of the target population. In this setting, we propose three estimators for the AUC in the target population and show that they are consistent and asymptotically normal. We evaluate the finite-sample performance of the estimators using simulations and use them to estimate the AUC in a nationally representative target population from the National Health and Nutrition Examination Survey for a lung cancer risk prediction model developed using source population data from the National Lung Screening Trial.     

An Estimate of the Variance of Estimators for Lead Time and Screening Benefit Time in Randomized Cancer Screening Trials

Variance estimators are derived for estimators of the average lead time and average benefit time due to screening in a randomized screening trial via influence functions. The influence functions demonstrate that these estimators are asymptotically equivalent to the mean difference, between the study and control case groups, in the appropriate survival times. For estimating benefit time, the survival time is measured since start of study; for estimating lead time, the survival time is measured since time of diagnosis. Asymptotic variances of these estimators can be calculated in a straightforward manner from the influence functions, and these variances can be estimated from actual trial data. The performance of the variance estimators is assessed via a simulated screening trial. The situation involving censored data is also discussed.     

A General Class of Semiparametric Transformation Frailty Models for Nonproportional Hazards Survival Data

Summary We propose a semiparametrically efficient estimation of a broad class of transformation regression models for nonproportional hazards data. Classical transformation models are to be viewed from a frailty model paradigm, and the proposed method provides a unified approach that is valid for both continuous and discrete frailty models. The proposed models are shown to be flexible enough to model long‐term follow‐up survival data when the treatment effect diminishes over time, a case for which the PH or proportional odds assumption is violated, or a situation in which a substantial proportion of patients remains cured after treatment. Estimation of the link parameter in frailty distribution, considered to be unknown and possibly dependent on a time‐independent covariates, is automatically included in the proposed methods. The observed information matrix is computed to evaluate the variances of all the parameter estimates. Our likelihood‐based approach provides a natural way to construct simple statistics for testing the PH and proportional odds assumptions for usual survival data or testing the short‐ and long‐term effects for survival data with a cure fraction. Simulation studies demonstrate that the proposed inference procedures perform well in realistic settings. Applications to two medical studies are provided.     

Promotion time models with time-changing exposure and heterogeneity: application to infectious diseases

Promotion time models have been recently adapted to the context of infectious diseases to take into account discrete and multiple exposures. However, Poisson distribution of the number of pathogens transmitted at each exposure was a very strong assumption and did not allow for inter-individual heterogeneity. Bernoulli, the negative binomial, and the compound Poisson distributions were proposed as alternatives to Poisson distribution for the promotion time model with time-changing exposure. All were derived within the frailty model framework. All these distributions have a point mass at zero to take into account non-infected people. Bernoulli distribution, the two-component cure rate model, was extended to multiple exposures. Contrary to the negative binomial and the compound Poisson distributions, Bernoulli distribution did not enable to connect the number of pathogens transmitted to the delay between transmission and infection detection. Moreover, the two former distributions enable to account for inter-individual heterogeneity. The delay to surgical site infection was an example of single exposure. The probability of infection was very low; thus, estimation of the effect of selected risk factors on that probability obtained with Bernoulli and Poisson distributions were very close. The delay to nosocomial urinary tract infection was a multiple exposure example. The probabilities of pathogen transmission during catheter placement and catheter presence were estimated. Inter-individual heterogeneity was very high, and the fit was better with the compound Poisson and the negative binomial distributions. The proposed models proved to be also mechanistic. The negative binomial and the compound Poisson distributions were useful alternatives to account for inter-individual heterogeneity.     

Estimation of a common effect parameter from sparse follow-up data

Breslow (1981, Biometrika 68, 73-84) has shown that the Mantel-Haenszel odds ratio is a consistent estimator of a common odds ratio in sparse stratifications. For cohort studies, however, estimation of a common risk ratio or risk difference can be of greater interest. Under a binomial sparse-data model, the Mantel-Haenszel risk ratio and risk difference estimators are consistent in sparse stratifications, while the maximum likelihood and weighted least squares estimators are biased. Under Poisson sparse-data models, the Mantel-Haenszel and maximum likelihood rate ratio estimators have equal asymptotic variances under the null hypothesis and are consistent, while the weighted least squares estimators are again biased; similarly, of the common rate difference estimators the weighted least squares estimators are biased, while the estimator employing "Mantel-Haenszel" weights is consistent in sparse data. Variance estimators that are consistent in both sparse data and large strata can be derived for all the Mantel-Haenszel estimators.     

Semiparametric transformation models for interval‐censored data in the presence of a cure fraction

Mixed case interval‐censored data arise when the event of interest is known only to occur within an interval induced by a sequence of random examination times. Such data are commonly encountered in disease research with longitudinal follow‐up. Furthermore, the medical treatment has progressed over the last decade with an increasing proportion of patients being cured for many types of diseases. Thus, interest has grown in cure models for survival data which hypothesize a certain proportion of subjects in the population are not expected to experience the events of interest. In this article, we consider a two‐component mixture cure model for regression analysis of mixed case interval‐censored data. The first component is a logistic regression model that describes the cure rate, and the second component is a semiparametric transformation model that describes the distribution of event time for the uncured subjects. We propose semiparametric maximum likelihood estimation for the considered model. We develop an EM type algorithm for obtaining the semiparametric maximum likelihood estimators (SPMLE) of regression parameters and establish their consistency, efficiency, and asymptotic normality. Extensive simulation studies indicate that the SPMLE performs satisfactorily in a wide variety of settings. The proposed method is illustrated by the analysis of the hypobaric decompression sickness data from National Aeronautics and Space Administration.     

Semiparametric Bayesian estimation of quantile function for breast cancer survival data with cured fraction

Existing cure‐rate survival models are generally not convenient for modeling and estimating the survival quantiles of a patient with specified covariate values. This paper proposes a novel class of cure‐rate model, the transform‐both‐sides cure‐rate model (TBSCRM), that can be used to make inferences about both the cure‐rate and the survival quantiles. We develop the Bayesian inference about the covariate effects on the cure‐rate as well as on the survival quantiles via Markov Chain Monte Carlo (MCMC) tools. We also show that the TBSCRM‐based Bayesian method outperforms existing cure‐rate models based methods in our simulation studies and in application to the breast cancer survival data from the National Cancer Institute's Surveillance, Epidemiology, and End Results (SEER) database.     

Generalized Linear Mixed-Effects Models with a Finite-Support Random-Effects Distribution: A Maximum-Penalized-Likelihood Approach

We discuss a method for simultaneously estimating the fixed parameters of a generalized linear mixed-effects model and the random-effects distribution of which no parametric assumption is made. In addition, classifying subjects into clusters according to the random regression coefficients is a natural by-product of the proposed method. An alternative approach to maximum-likelihood method, maximum-penalized-likelihood method, is used to avoid estimating “too many” clusters. Consistency and asymptotic normality properties of the estimators are presented. We also provide robust variance estimators of the fixed parameters estimators which remain consistent even in presence of misspecification. The methodology is illustrated by an application to a weight loss study.