Competing Risk Analysis of Censored Survival Data

A competing risk model is developed to accommodate both planned Type I censoring and random withdrawals. MLE's, their properties, confidence regions for parameters and mean lifetimes are obtained for a model regarding random censoring as a competing risk and compared to those obtained for the model in which withdrawals are regarded as random censoring. Estimated net and crude probabilities are calculated and compared for the two models. The model is developed for two competing risks, one following a Weibull distribution and the other a Rayleigh distribution, and random withdrawals following a Weibull distribution.     

Inference on Risk Rates Based on Mortality Data Under Censoring and Competing Risks Using Parametric Models

In this paper we consider the competing risks model where the risks may not be independent. We assume both fixed and random censoring. The random censoring mechanism could have either a parametric or a non-parametric form. The life distributions and the parametric censoring distribution considered are exponential or Weibull. The expressions for the asymptotic confidence intervals for various parameters of interest under different models, using the estimated Fisher information matrix and parametric bootstrap techniques have been derived. Monte Carlo simulation studies for some of these cases have been carried out.     

Regression modeling of semicompeting risks data

Semicompeting risks data are often encountered in clinical trials with intermediate endpoints subject to dependent censoring from informative dropout. Unlike with competing risks data, dropout may not be dependently censored by the intermediate event. There has recently been increased attention to these data, in particular inferences about the marginal distribution of the intermediate event without covariates. In this article, we incorporate covariates and formulate their effects on the survival function of the intermediate event via a functional regression model. To accommodate informative censoring, a time-dependent copula model is proposed in the observable region of the data which is more flexible than standard parametric copula models for the dependence between the events. The model permits estimation of the marginal distribution under weaker assumptions than in previous work on competing risks data. New nonparametric estimators for the marginal and dependence models are derived from nonlinear estimating equations and are shown to be uniformly consistent and to converge weakly to Gaussian processes. Graphical model checking techniques are presented for the assumed models. Nonparametric tests are developed accordingly, as are inferences for parametric submodels for the time-varying covariate effects and copula parameters. A novel time-varying sensitivity analysis is developed using the estimation procedures. Simulations and an AIDS data analysis demonstrate the practical utility of the methodology.     

A joint model for longitudinal measurements and survival data in the presence of multiple failure types

Summary .   In this article we study a joint model for longitudinal measurements and competing risks survival data. Our joint model provides a flexible approach to handle possible nonignorable missing data in the longitudinal measurements due to dropout. It is also an extension of previous joint models with a single failure type, offering a possible way to model informatively censored events as a competing risk. Our model consists of a linear mixed effects submodel for the longitudinal outcome and a proportional cause-specific hazards frailty submodel ( Prentice et al., 1978 , Biometrics 34, 541–554) for the competing risks survival data, linked together by some latent random effects. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their standard errors using a profile likelihood method. The developed method works well in our simulation studies and is applied to a clinical trial for the scleroderma lung disease.     

Human disease mortality kinetics are explored through a chain model embodying principles of extreme value theory and competing risks.

The distributions for human disease-specific mortality exhibit two striking characteristics: survivorship curves that intersect near the longevity limit; and, the clustering of best-fitting Weibull shape parameter values into groups centered on integers. Correspondingly, we have hypothesized that the distribution intersections result from either competitive processes or population partitioning and the integral clustering in the shape parameter results from the occurrence of a small number of rare, rate-limiting events in disease progression. In this report we initiate a theoretical examination of these questions by exploring serial chain model dynamics and parameteric competing risks theory. The links in our chain models are composed of more than one bond, where the number of bonds in a link are denoted the link size and are the number of events necessary to break the link and, hence, the chain. We explored chains with all links of the same size or with segments of the chain composed of different size links (competition). Simulations showed that chain breakage dynamics depended on the weakest-link principle and followed kinetics of extreme-values which were very similar to human mortality kinetics. In particular, failure distributions for simple chains were Weibull-type extreme-value distributions with shape parameter values that were identifiable with the integral link size in the limit of infinite chain length. Furthermore, for chains composed of several segments of differing link size, the survival distributions for the various segments converged at a point in the S(t) tails indistinguishable from human data. This was also predicted by parameteric competing risks theory using Weibull underlying distributions. In both the competitive chain simulations and the parametric competing risks theory, however, the shape values for the intersecting distributions deviated from the integer values typical of human data. We conclude that rare events can be the source of integral shapes in human mortality, that convergence is a salient feature of multiple endpoints, but that pure competition may not be the best explanation for the exact type of convergence observable in human mortality. Finally, while the chain models were not motivated by any specific biological structures, interesting biological correlates to them may be useful in gerontological research.     

Robust Joint Modeling of Longitudinal Measurements and Competing Risks Failure Time Data

Existing methods for joint modeling of longitudinal measurements and survival data can be highly influenced by outliers in the longitudinal outcome. We propose a joint model for analysis of longitudinal measurements and competing risks failure time data which is robust in the presence of outlying longitudinal observations during follow‐up. Our model consists of a linear mixed effects sub‐model for the longitudinal outcome and a proportional cause‐specific hazards frailty sub‐model for the competing risks data, linked together by latent random effects. Instead of the usual normality assumption for measurement errors in the linear mixed effects sub‐model, we adopt a t ‐distribution which has a longer tail and thus is more robust to outliers. We derive an EM algorithm for the maximum likelihood estimates of the parameters and estimate their standard errors using a profile likelihood method. The proposed method is evaluated by simulation studies and is applied to a scleroderma lung study (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Bayesian multi-risks survival (MRS) model in the presence of double censorings

Semi-competing risks data include the time to a nonterminating event and the time to a terminating event, while competing risks data include the time to more than one terminating event. Our work is motivated by a prostate cancer study, which has one nonterminating event and two terminating events with both semi-competing risks and competing risks present as well as two censoring times. In this paper, we propose a new multi-risks survival (MRS) model for this type of data. In addition, the proposed MRS model can accommodate noninformative right-censoring times for nonterminating and terminating events. Properties of the proposed MRS model are examined in detail. Theoretical and empirical results show that the estimates of the cumulative incidence function for a nonterminating event may be biased if the information on a terminating event is ignored. A Markov chain Monte Carlo sampling algorithm is also developed. Our methodology is further assessed using simulations and also an analysis of the real data from a prostate cancer study. As a result, a prostate-specific antigen velocity greater than 2.0 ng/mL per year and higher biopsy Gleason scores are positively associated with a shorter time to death due to prostate cancer.     

On a probability Model for Classifying the Unknown Causes of Death Due to Two Competing Risks into Assignable Causes

A probability model to classify the otherwise unclassified causes of death due to two competing risks R₁ and R₂ has been evolved in this paper. A simple model based on Poisson inputs has been illustrated by numerical illustrations. Further generalization of the model with more than two competing risks is straightforward for the given model.     

High-dimensional feature selection in competing risks modeling: A stable approach using a split-and-merge ensemble algorithm

Variable selection is critical in competing risks regression with high-dimensional data. Although penalized variable selection methods and other machine learning-based approaches have been developed, many of these methods often suffer from instability in practice. This paper proposes a novel method named Random Approximate Elastic Net (RAEN). Under the proportional subdistribution hazards model, RAEN provides a stable and generalizable solution to the large-p-small-n variable selection problem for competing risks data. Our general framework allows the proposed algorithm to be applicable to other time-to-event regression models, including competing risks quantile regression and accelerated failure time models. We show that variable selection and parameter estimation improved markedly using the new computationally intensive algorithm through extensive simulations. A user-friendly R package RAEN is developed for public use. We also apply our method to a cancer study to identify influential genes associated with the death or progression from bladder cancer.     

Competing risk analysis for life table data with known observation times

"Several models have been proposed for the analysis of cohort mortality in the presence of competing risks.... This paper describes a maximum likelihood approach to the analysis of follow up data in life table format for the case of two competing risks--a specific cause and its competing complement. The model developed uses a robust survivorship assumption--the piecewise exponential--and takes into account information on time to death and time to withdrawal." (summary in GER)     

Kinetic modeling of native Cassava starch thermo‐oxidative degradation using Weibull and Weibull‐derived models

A new approach in kinetic modeling of thermo‐oxidative degradation process of starch granules extracted from the Cassava roots was developed. Based on the thermoanalytical measurements, three reaction stages were detected. Using Weibull and Weibull‐derived (inverse) models, it was found that the first two reaction stages could be described with the change of apparent activation energy (E_a) on conversion fraction (α(T)) (using “Model‐free” analysis). It was found that first reaction stage, which involves dehydration and evaporation of lower molecular mass fractions, can be described with an inverse Weibull model. This model with its distribution of E_a values and derived distribution parameters includes the occurrence of three‐dimensional diffusion mechanism. The second reaction stage is very complex, and it was found to contain the system of simultaneous reactions (where depolymerization occurs), and can be described with standard Weibull model. Identified statistical model with its distribution of E_a values and derived distribution parameters includes the kinetic model that gives the variable reaction order values. Based on the established models, shelf‐life studies for first two stages were carried out. Shelf‐life testing has shown that optimal dehydration time is achieved by a programmed heating at medium heating rate, whereas optimal time of degradation is achieved at highest heating rate. © 2013 Wiley Periodicals, Inc. Biopolymers 101: 41–57, 2014.     

Parameter Estimation for Partially Complete Time and Type of Failure Data

The theory of competing risks has been developed to asses a specific risk in presence of other risk factors. In this paper we consider the parametric estimation of different failure modes under partially complete time and type of failure data using latent failure times and cause specific hazard functions models. Uniformly minimum variance unbiased estimators and maximum likelihood estimators are obtained when latent failure times and cause specific hazard functions are exponentially distributed. We also consider the case when they follow Weibull distributions. One data set is used to illustrate the proposed techniques. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Testing homogeneity in Weibull-regression models

In survival studies with families or geographical units it may be of interest testing whether such groups are homogeneous for given explanatory variables. In this paper we consider score type tests for group homogeneity based on a mixing model in which the group effect is modelled as a random variable. As opposed to hazard-based frailty models, this model presents survival times that conditioned on the random effect, has an accelerated failure time representation. The test statistics requires only estimation of the conventional regression model without the random effect and does not require specifying the distribution of the random effect. The tests are derived for a Weibull regression model and in the uncensored situation, a closed form is obtained for the test statistic. A simulation study is used for comparing the power of the tests. The proposed tests are applied to real data sets with censored data.     

A model for early death caused by radiation pneumonitis and pulmonary fibrosis after inhaling insoluble radioactive particles

Large radiation doses to the lung can cause early death from cardiopulmonary insufficiency resulting from radiation pneumonitis and pulmonary fibrosis. A model for early death following inhalation of insoluble radioactive particles is propose. The model is based on three assumptions: (1) early death results from damage to a cluster of cells from a large number of cell clusters at risk, (2) the dose that causes early death depends on how the radiation is delivered in time and (3) the cell clusters at risk to damage are equally sensitive ro radiation. Results from asymptotic theory of extreme values, along with biophysical considerations, suggest that the cumultive distribution function for the absorbed radiation dose to the production of pulmonary injury sufficient to cause early death is best estimated by the third asymptotic distribution without a threshold. This distribution function is identical to the Weibull cumulative distribution function. Data for Beagle dogs after inhaling relatively insoluble forms of alpha- or beta-gamma-emitting particles are shown to support the Weibull model.     

Multiple Decrement Analysis with Dependent Risks

In a competing risks problem where a well-defined population is exposed simultaneously to several causes of death, interest has centered on the estimation of the probability of death from a given cause when one or more other causes have been eliminated. A basic component of all available procedures for estimating these probabilities is the assumption that the several causes of death act independently—an unrealistic assumption in the context of human and animal populations. This article considers the estimation of these probabilities assuming the existence ofinterdependencies among the various causes of death. A general formula is derived based on a given set of crude probabilities of death as well as the characteristics of the joint distribution of random variables indicating death from the various causes. This formula identifies alternative assumptions, less restrictive than that of independent risks, which may he used for estimation purposes.     

Analysis of cell surface light chain immunoglobulin expression by flow cytometry in normal controls: a new mathematical approach

It has been observed in flow cytometric studies that in normal individuals there are proportionally more kappa+ than lambda+ bearing light chain B-cells. Overt predominance of one type of light chain bearing cell over the other is a characteristic of B-cell neoplasias, a phenomenon called clonal excess (CE). A mathematical model using the Weibull distribution is proposed for studying such an excess. The new approach is desirable for two reasons: First, it is parametric and hence offers a more sensitive and versatile analysis than its nonparametric counterparts. Second, it utilizes only the relevant information from the upper tails of the distributions of the fluorescence intensity of the kappa+ and lambda+ cells. Two measures of CE based on the Weibull model are proposed, and a normal range of variability was determined for each measure using a random sample of 48 normal controls. Such normal ranges are particularly useful in detecting cancer patients with minimal B-cell neoplasias. A comparative study of the new measures, Ault's maximum difference measure, and a measure based on Ligler's method showed that the parametric approach provides much more sensitivity than both the nonparametric ones.     

Competing Risks and Time‐Dependent Covariates

Time‐dependent covariates are frequently encountered in regression analysis for event history data and competing risks. They are often essential predictors, which cannot be substituted by time‐fixed covariates. This study briefly recalls the different types of time‐dependent covariates, as classified by Kalbfleisch and Prentice [The Statistical Analysis of Failure Time Data, Wiley, New York, 2002] with the intent of clarifying their role and emphasizing the limitations in standard survival models and in the competing risks setting. If random (internal) time‐dependent covariates are to be included in the modeling process, then it is still possible to estimate cause‐specific hazards but prediction of the cumulative incidences and survival probabilities based on these is no longer feasible. This article aims at providing some possible strategies for dealing with these prediction problems. In a multi‐state framework, a first approach uses internal covariates to define additional (intermediate) transient states in the competing risks model. Another approach is to apply the landmark analysis as described by van Houwelingen [Scandinavian Journal of Statistics 2007, 34 , 70–85] in order to study cumulative incidences at different subintervals of the entire study period. The final strategy is to extend the competing risks model by considering all the possible combinations between internal covariate levels and cause‐specific events as final states. In all of those proposals, it is possible to estimate the changes/differences of the cumulative risks associated with simple internal covariates. An illustrative example based on bone marrow transplant data is presented in order to compare the different methods.     

Statistical Analysis of Illness–Death Processes and Semicompeting Risks Data

Summary In many instances, a subject can experience both a nonterminal and terminal event where the terminal event (e.g., death) censors the nonterminal event (e.g., relapse) but not vice versa. Typically, the two events are correlated. This situation has been termed semicompeting risks (e.g., Fine, Jiang, and Chappell, 2001 , Biometrika 88, 907–939; Wang, 2003 , Journal of the Royal Statistical Society, Series B 65, 257–273), and analysis has been based on a joint survival function of two event times over the positive quadrant but with observation restricted to the upper wedge. Implicitly, this approach entertains the idea of latent failure times and leads to discussion of a marginal distribution of the nonterminal event that is not grounded in reality. We argue that, similar to models for competing risks, latent failure times should generally be avoided in modeling such data. We note that semicompeting risks have more classically been described as an illness–death model and this formulation avoids any reference to latent times. We consider an illness–death model with shared frailty, which in its most restrictive form is identical to the semicompeting risks model that has been proposed and analyzed, but that allows for many generalizations and the simple incorporation of covariates. Nonparametric maximum likelihood estimation is used for inference and resulting estimates for the correlation parameter are compared with other proposed approaches. Asymptotic properties, simulations studies, and application to a randomized clinical trial in nasopharyngeal cancer evaluate and illustrate the methods. A simple and fast algorithm is developed for its numerical implementation.     

Time-dependent covariates in the proportional subdistribution hazards model for competing risks

Separate Cox analyses of all cause-specific hazards are the standard technique of choice to study the effect of a covariate in competing risks, but a synopsis of these results in terms of cumulative event probabilities is challenging. This difficulty has led to the development of the proportional subdistribution hazards model. If the covariate is known at baseline, the model allows for a summarizing assessment in terms of the cumulative incidence function. black Mathematically, the model also allows for including random time-dependent covariates, but practical implementation has remained unclear due to a certain risk set peculiarity. We use the intimate relationship of discrete covariates and multistate models to naturally treat time-dependent covariates within the subdistribution hazards framework. The methodology then straightforwardly translates to real-valued time-dependent covariates. As with classical survival analysis, including time-dependent covariates does not result in a model for probability functions anymore. Nevertheless, the proposed methodology provides a useful synthesis of separate cause-specific hazards analyses. We illustrate this with hospital infection data, where time-dependent covariates and competing risks are essential to the subject research question.     

Parametric regression on cumulative incidence function

We propose parametric regression analysis of cumulative incidence function with competing risks data. A simple form of Gompertz distribution is used for the improper baseline subdistribution of the event of interest. Maximum likelihood inferences on regression parameters and associated cumulative incidence function are developed for parametric models, including a flexible generalized odds rate model. Estimation of the long-term proportion of patients with cause-specific events is straightforward in the parametric setting. Simple goodness-of-fit tests are discussed for evaluating a fixed odds rate assumption. The parametric regression methods are compared with an existing semiparametric regression analysis on a breast cancer data set where the cumulative incidence of recurrence is of interest. The results demonstrate that the likelihood-based parametric analyses for the cumulative incidence function are a practically useful alternative to the semiparametric analyses.