Applications of a parametric model for informative censoring

In this article, we explore the use of a parametric model (for analyzing survival data) which is defined to allow sensitivity analysis for the presence of informative censoring. The dependence between the failure and the censoring processes is expressed through a parameter delta and a general bias function B(t, theta). We calculate the expectation of the potential bias due to informative censoring, which is an overall measure of how misleading our results might be if censoring is actually nonignorable. Bounds are also calculated for quantities of interest, e.g., parameter of the distribution of the failure process, which do not depend on the choice of the bias function for fixed delta. An application that relates to systematic lupus erythematosus data illustrates how additional information can result in reducing the uncertainty on estimates of the location parameter. Sensitivity analysis on a relative risk parameter is also explored.     

Regression survival analysis with an assumed copula for dependent censoring: a sensitivity analysis approach

SUMMARY: In clinical studies, when censoring is caused by competing risks or patient withdrawal, there is always a concern about the validity of treatment effect estimates that are obtained under the assumption of independent censoring. Because dependent censoring is nonidentifiable without additional information, the best we can do is a sensitivity analysis to assess the changes of parameter estimates under different assumptions about the association between failure and censoring. This analysis is especially useful when knowledge about such association is available through literature review or expert opinions. In a regression analysis setting, the consequences of falsely assuming independent censoring on parameter estimates are not clear. Neither the direction nor the magnitude of the potential bias can be easily predicted. We provide an approach to do sensitivity analysis for the widely used Cox proportional hazards models. The joint distribution of the failure and censoring times is assumed to be a function of their marginal distributions. This function is called a copula. Under this assumption, we propose an iteration algorithm to estimate the regression parameters and marginal survival functions. Simulation studies show that this algorithm works well. We apply the proposed sensitivity analysis approach to the data from an AIDS clinical trial in which 27% of the patients withdrew due to toxicity or at the request of the patient or investigator.     

An approach to model clustered survival data with dependent censoring

In this study we introduce a likelihood-based method, via the Weibull and piecewise exponential distributions, capable of accommodating the dependence between failure and censoring times. The methodology is developed for the analysis of clustered survival data and it assumes that failure and censoring times are mutually independent conditional on a latent frailty. The dependent censoring mechanism is accounted through the frailty effect and this is accomplished by means of a key parameter accommodating the correlation between failure and censored observations. The full specification of the likelihood in our work simplifies the inference procedures with respect to Huang and Wolfe since it reduces the computation burden of working with the profile likelihood. In addition, the assumptions made for the baseline distributions lead to models with continuous survival functions. In order to carry out inferences, we devise a Monte Carlo EM algorithm. The performance of the proposed models is investigated through a simulation study. Finally, we explore a real application involving patients from the Dialysis Outcomes and Practice Patterns Study observed between 1996 and 2015.     

A frailty model for informative censoring

To account for the correlation between failure and censoring, we propose a new frailty model for clustered data. In this model, the risk to be censored is affected by the risk of failure. This model allows flexibility in the direction and degree of dependence between failure and censoring. It includes the traditional frailty model as a special case. It allows censoring by some causes to be analyzed as informative while treating censoring by other causes as noninformative. It can also analyze data for competing risks. To fit the model, the EM algorithm is used with Markov chain Monte Carlo simulations in the E-steps. Simulation studies and analysis of data for kidney disease patients are provided. Consequences of incorrectly assuming noninformative censoring are investigated.     

Nonparametric inference on median residual life function

Summary .  A simple approach to the estimation of the median residual lifetime is proposed for a single group by inverting a function of the Kaplan–Meier estimators. A test statistic is proposed to compare two median residual lifetimes at any fixed time point. The test statistic does not involve estimation of the underlying probability density function of failure times under censoring. Extensive simulation studies are performed to validate the proposed test statistic in terms of type I error probabilities and powers at various time points. One of the oldest data sets from the National Surgical Adjuvant Breast and Bowel Project (NSABP), which has more than a quarter century of follow-up, is used to illustrate the method. The analysis results indicate that, without systematic post-operative therapy, a significant difference in median residual lifetimes between node-negative and node-positive breast cancer patients persists for about 10 years after surgery. The new estimates of the median residual lifetime could serve as a baseline for physicians to explain any incremental effects of post-operative treatments in terms of delaying breast cancer recurrence or prolonging remaining lifetimes of breast cancer patients.     

Testing goodness of fit of a uniform truncation model

Several goodness-of-fit tests of a lifetime distribution have been suggested in the literature; many take into account censoring and/or truncation of event times. In some contexts, a goodness-of-fit test for the truncation distribution is of interest. In particular, better estimates of the lifetime distribution can be obtained when knowledge of the truncation law is exploited. In cross-sectional sampling, for example, there are theoretical justifications for the assumption of a uniform truncation distribution, and several studies have used it to improve the efficiency of their survival estimates. The duality of lifetime and truncation in the absence of censoring enables methods for testing goodness of fit of the lifetime distribution to be used for testing goodness of fit of the truncation distribution. However, under random censoring, this duality does not hold and different tests are required. In this article, we introduce several goodness-of-fit tests for the truncation distribution and investigate their performance in the presence of censored event times using simulation. We demonstrate the use of our tests on two data sets.     

One- and two-sample nonparametric inference procedures in the presence of a mixture of independent and dependent censoring

In survival analysis, the event time T is often subject to dependent censorship. Without assuming a parametric model between the failure and censoring times, the parameter Theta of interest, for example, the survival function of T, is generally not identifiable. On the other hand, the collection Omega of all attainable values for Theta may be well defined. In this article, we present nonparametric inference procedures for Omega in the presence of a mixture of dependent and independent censoring variables. By varying the criteria of classifying censoring to the dependent or independent category, our proposals can be quite useful for the so-called sensitivity analysis of censored failure times. The case that the failure time is subject to possibly dependent interval censorship is also discussed in this article. The new proposals are illustrated with data from two clinical studies on HIV-related diseases.     

Semiparametric Analysis for Recurrent Event Data with Time-Dependent Covariates and Informative Censoring

Summary .  Recurrent event data analyses are usually conducted under the assumption that the censoring time is independent of the recurrent event process. In many applications the censoring time can be informative about the underlying recurrent event process, especially in situations where a correlated failure event could potentially terminate the observation of recurrent events. In this article, we consider a semiparametric model of recurrent event data that allows correlations between censoring times and recurrent event process via frailty. This flexible framework incorporates both time-dependent and time-independent covariates in the formulation, while leaving the distributions of frailty and censoring times unspecified. We propose a novel semiparametric inference procedure that depends on neither the frailty nor the censoring time distribution. Large sample properties of the regression parameter estimates and the estimated baseline cumulative intensity functions are studied. Numerical studies demonstrate that the proposed methodology performs well for realistic sample sizes. An analysis of hospitalization data for patients in an AIDS cohort study is presented to illustrate the proposed method.     

Locally efficient estimation of the quality-adjusted lifetime distribution with right-censored data and covariates

Zhao and Tsiatis (1997) consider the problem of estimation of the distribution of the quality-adjusted lifetime when the chronological survival time is subject to right censoring. The quality-adjusted lifetime is typically defined as a weighted sum of the times spent in certain states up until death or some other failure time. They propose an estimator and establish the relevant asymptotics under the assumption of independent censoring. In this paper we extend the data structure with a covariate process observed until the end of follow-up and identify the optimal estimation problem. Because of the curse of dimensionality, no globally efficient nonparametric estimators, which have a good practical performance at moderate sample sizes, exist. Given a correctly specified model for the hazard of censoring conditional on the observed quality-of-life and covariate processes, we propose a closed-form one-step estimator of the distribution of the quality-adjusted lifetime whose asymptotic variance attains the efficiency bound if we can correctly specify a lower-dimensional working model for the conditional distribution of quality-adjusted lifetime given the observed quality-of-life and covariate processes. The estimator remains consistent and asymptotically normal even if this latter submodel is misspecified. The practical performance of the estimators is illustrated with a simulation study. We also extend our proposed one-step estimator to the case where treatment assignment is confounded by observed risk factors so that this estimator can be used to test a treatment effect in an observational study.     

A two-sample comparison for multiple ordered event data

A longitudinal study is conducted to compare the process of particular disease between two groups. The process of the disease is monitored according to which of several ordered events occur. In the paper, the sojourn time between two successive events is considered as the outcome of interest. The group effects on the sojourn times of the multiple events are parameterized by scale changes in a semiparametric accelerated failure time model where the dependence structure among the multivariate sojourn times is unspecified. Suppose that the sojourn times are subject to dependent censoring and the censoring times are observed for all subjects. A log-rank-type estimating approach by rescaling the sojourn times and the dependent censoring times into the same distribution is constructed to estimate the group effects and the corresponding estimators are consistent and asymptotically normal. Without the dependent censoring, the independent censoring times in general are not available for the uncensored data. In order to complete the censoring information, pseudo-censoring times are generated from the corresponding nonparametrically estimated survival function in each group, and we can still obtained unbiased estimating functions for the group effects. A real application and a simulation study are conducted to illustrate the proposed methods.     

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.     

Analysis of the anisotropy decay of trans-parinaric acid in lipid bilayers.

An analysis is presented of the complex anisotropy behavior of trans-parinaric acid in single component DEPC lipid bilayers. It is shown that a model involving two species with distinct lifetime and motional behavior is required, and is adequate, to explain the observed data. In particular, the observed increase in the anisotropy at long times demonstrates the presence of a species with a long fluorescence lifetime that has a high anisotropy. The time dependence of the anisotropy for these two environments is treated using both a purely mathematical sum of exponentials and a constrained fit based on an approximate solution of the anisotropic diffusion problem. In this latter model the anisotropy is described in terms of the second and fourth rank order parameters, (P2) and (P4), and a single dynamical parameter, D1, the perpendicular diffusion coefficient for this uniaxial probe. The parameters of both models are accurately determined from the fits to the data when two environments coexist and an association is made between lifetime components and distinct rotational sites. The values of the parameters obtained demonstrate the "solid-like" and "fluidlike" nature of these two coexisting environments.     

One-step targeted maximum likelihood estimation for time-to-event outcomes

Researchers in observational survival analysis are interested in not only estimating survival curve nonparametrically but also having statistical inference for the parameter. We consider right-censored failure time data where we observe n independent and identically distributed observations of a vector random variable consisting of baseline covariates, a binary treatment at baseline, a survival time subject to right censoring, and the censoring indicator. We assume the baseline covariates are allowed to affect the treatment and censoring so that an estimator that ignores covariate information would be inconsistent. The goal is to use these data to estimate the counterfactual average survival curve of the population if all subjects are assigned the same treatment at baseline. Existing observational survival analysis methods do not result in monotone survival curve estimators, which is undesirable and may lose efficiency by not constraining the shape of the estimator using the prior knowledge of the estimand. In this paper, we present a one-step Targeted Maximum Likelihood Estimator (TMLE) for estimating the counterfactual average survival curve. We show that this new TMLE can be executed via recursion in small local updates. We demonstrate the finite sample performance of this one-step TMLE in simulations and an application to a monoclonal gammopathy data.     

General right censoring and its impact on the analysis of survival data.

This paper concerns general right censoring and some of the difficulties it creates in the analysis of survival data. A general formulation of censored-survival processes leads to the partition of all models into those based on noninformative and informative censoring. Nearly all statistical methods for censored data assume that censoring is noninformative. Topics considered within this class include: the relationships between three models for noninformative censoring, the use of likelihood methods for inferences about the distribution of survival time, the effects of censoring on the K-sample problem, and the effects of censoring on model testing. Also considered are several topics which relate to informative censoring models. These include: problems of nonidentifiability that can be encountered when attempting to assess a set of data for the type of censoring in effect, the consequences of falsely assuming that censoring is noninformative, and classes of informative censoring models.     

Lifetime of Organic Salt Photovoltaics

Solar energy deployment can be augmented with the use of wavelength‐selective transparent photovoltaics (PVs). Moving forward, operating lifetime is arguably among the most important challenge that must be addressed to increase commercial viability of these emerging technologies. In this work, the lifetimes of PVs with organic near‐infrared selective small molecules and molecular salts are investigated. This is the first comprehensive lifetime study on devices featuring organic salts with varied counterions. Based on the tunability afforded by anion exchange, an extrapolated lifetime of 7 ± 2 years from continuous illumination measurements on organic salt devices held at the maximum power point is demonstrated. These lifetimes are compared with changes in external quantum efficiency, hydrophobicity, molecular orbital levels, and optical absorption to determine the limiting characteristics and failure mechanisms of PV devices utilizing each donor. A key correlation between the lifetime and the hydrophobicity of the donor layer is uncovered. This could provide a targeted parameter for designing organic molecules and salts with exceptional lifetime and enhanced commercial viability.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

pH dependence of the urea and guanidine hydrochloride denaturation of ribonuclease A and ribonuclease T1

To investigate the pH dependence of the conformational stability of ribonucleases A and T1, urea and guanidine hydrochloride denaturation curves have been determined over the pH range 2-10. The maximum conformational stability of both proteins is about 9 kcal/mol and occurs near pH 4.5 for ribonuclease T1 and between pH 7 and 9 for ribonuclease A. The pH dependence suggests that electrostatic interactions among the charged groups make a relatively small contribution to the conformational stability of these proteins. The dependence of delta G on urea concentration increases from about 1200 cal mol-1 M-1 at high pH to about 2400 cal mol-1 M-1 at low pH for ribonuclease A. This suggests that the unfolded conformations of RNase A become more accessible to urea as the net charge on the molecule increases. For RNase T1, the dependence of delta G on urea concentration is minimal near pH 6 and increases at both higher and lower pH. An analysis of information of this type for several proteins in terms of a model developed by Tanford [Tanford, C. (1964) J. Am. Chem. Soc. 86, 2050-2059] suggests that the unfolded states of proteins in urea and GdnHCl solutions may differ significantly in the extent of their interaction with denaturants. Thus, the conformations assumed by unfolded proteins may depend to at least some extent on the amino acid sequence of the protein.     

Estimation of morbid risk and age at onset with missing information.

Investigators of genetic illnesses are currently employing life-table techniques to estimate the lifetime risk of disease and the age-at-onset distribution. This methodology assumes that onset ages are known for affected individuals and that censoring ages are known for unaffected individuals. We extend these methods to incorporate affected individuals with unknown onset ages and unaffected persons with unknown censoring ages and illustrate how conventional life-table methods can produce seriously biased estimates, particularly of lifetime risk. The methodology is not restricted to genetic illnesses and can be applied to more complex illnesses with unknown etiology. We present an example for Huntington disease, which is generally assumed to be a Mendelian autosomal dominant disease, yielding estimates of lifetime risk of .503 +/- .70 and mean onset age of 47.7 +/- 3.1 years for offspring with a single affected parent. When conventional life-table techniques are employed, these estimates are .238 +/- .032 and 43.2 +/- 2.2.     

Double Inverse‐Weighted Estimation of Cumulative Treatment Effects Under Nonproportional Hazards and Dependent Censoring

Summary In medical studies of time‐to‐event data, nonproportional hazards and dependent censoring are very common issues when estimating the treatment effect. A traditional method for dealing with time‐dependent treatment effects is to model the time‐dependence parametrically. Limitations of this approach include the difficulty to verify the correctness of the specified functional form and the fact that, in the presence of a treatment effect that varies over time, investigators are usually interested in the cumulative as opposed to instantaneous treatment effect. In many applications, censoring time is not independent of event time. Therefore, we propose methods for estimating the cumulative treatment effect in the presence of nonproportional hazards and dependent censoring. Three measures are proposed, including the ratio of cumulative hazards, relative risk, and difference in restricted mean lifetime. For each measure, we propose a double inverse‐weighted estimator, constructed by first using inverse probability of treatment weighting (IPTW) to balance the treatment‐specific covariate distributions, then using inverse probability of censoring weighting (IPCW) to overcome the dependent censoring. The proposed estimators are shown to be consistent and asymptotically normal. We study their finite‐sample properties through simulation. The proposed methods are used to compare kidney wait‐list mortality by race.     

A copula approach for detecting prognostic genes associated with survival outcome in microarray studies

A challenging and crucial issue in clinical studies in cancer involving gene microarray experiments is the discovery, among a large number of genes, of a relatively small panel of genes whose elements are associated with a relevant clinical outcome variable such as time-to-death or time-to-recurrence of disease. A semiparametric approach, using dependence functions known as copulas, is considered to quantify and estimate the pairwise association between the outcome and each gene expression. These time-to-event type endpoints are typically subject to censoring as not all events are realized at the time of the analysis. Furthermore, given that the total number of genes is typically large, it is imperative to control a relevant error rate in any gene discovery procedure. The proposed method addresses the two aforementioned issues by direct incorporation of the censoring mechanism and by appropriate statistical adjustment for multiplicity. The performance of the proposed method is studied through simulation and illustrated with an application using a case study in lung cancer.