A dependent Dirichlet process model for survival data with competing risks

In this paper, we first propose a dependent Dirichlet process (DDP) model using a mixture of Weibull models with each mixture component resembling a Cox model for survival data. We then build a Dirichlet process mixture model for competing risks data without regression covariates. Next we extend this model to a DDP model for competing risks regression data by using a multiplicative covariate effect on subdistribution hazards in the mixture components. Though built on proportional hazards (or subdistribution hazards) models, the proposed nonparametric Bayesian regression models do not require the assumption of constant hazard (or subdistribution hazard) ratio. An external time-dependent covariate is also considered in the survival model. After describing the model, we discuss how both cause-specific and subdistribution hazard ratios can be estimated from the same nonparametric Bayesian model for competing risks regression. For use with the regression models proposed, we introduce an omnibus prior that is suitable when little external information is available about covariate effects. Finally we compare the models’ performance with existing methods through simulations. We also illustrate the proposed competing risks regression model with data from a breast cancer study. An R package “DPWeibull” implementing all of the proposed methods is available at CRAN.

     

Smooth Semi‐nonparametric Analysis for Mixture Cure Models and Its Application to Breast Cancer

Mixture cure models are widely used when a proportion of patients are cured. The proportional hazards mixture cure model and the accelerated failure time mixture cure model are the most popular models in practice. Usually the expectation–maximisation (EM) algorithm is applied to both models for parameter estimation. Bootstrap methods are used for variance estimation. In this paper we propose a smooth semi‐nonparametric (SNP) approach in which maximum likelihood is applied directly to mixture cure models for parameter estimation. The variance can be estimated by the inverse of the second derivative of the SNP likelihood. A comprehensive simulation study indicates good performance of the proposed method. We investigate stage effects in breast cancer by applying the proposed method to breast cancer data from the South Carolina Cancer Registry.     

A note on the asymptotic relative efficiency of estimators from the additive hazards model

In this note, the asymptotic variance formulas are explicitly derived and compared between the parametric and semiparametric estimators of a regression parameter and survival probability under the additive hazards model. To obtain explicit formulas, it is assumed that the covariate term including a regression coefficient follows a gamma distribution and the baseline hazard function is constant. The results show that the semiparametric estimator of the regression coefficient parameter is fully efficient relative to the parametric counterpart when the survival time and a covariate are independent, as in the proportional hazards model. Relative to a more realistic case of the parametric additive hazards model with a Weibull baseline, the loss of efficiency of the semiparametric estimator of survival probability is moderate.     

Semiparametric Efficient Estimation for a Class of Generalized Proportional Odds Cure Models

We present a mixture cure model with the survival time of the "uncured" group coming from a class of linear transformation models, which is an extension of the proportional odds model. This class of model, first proposed by Dabrowska and Doksum (1988), which we term "generalized proportional odds model," is well suited for the mixture cure model setting due to a clear separation between long-term and short-term effects. A standard expectation-maximization algorithm can be employed to locate the nonparametric maximum likelihood estimators, which are shown to be consistent and semiparametric efficient. However, there are difficulties in the M-step due to the nonparametric component. We overcome these difficulties by proposing two different algorithms. The first is to employ an majorize-minimize (MM) algorithm in the M-step instead of the usual Newton-Raphson method, and the other is based on an alternative form to express the model as a proportional hazards frailty model. The two new algorithms are compared in a simulation study with an existing estimating equation approach by Lu and Ying (2004). The MM algorithm provides both computational stability and efficiency. A case study of leukemia data is conducted to illustrate the proposed procedures.     

Frailty Models and Rank Tests

Connections are established between the theories of weighted logrank tests and of frailty models. These connections arise because omission of a balanced covariate from a proportional hazards model generally leads to a model with non-proportional hazards, for which the simple logrank test is no longer optimal. The optimal weighting function and the asymptotic relative efficiencies of the simple logrank test and of the optimally weighted logrank test relative to the adjusted test that would be used if the covariate values were known, are expressible in terms of the Laplace transform of the hazard ratio for the distribution of the omitted covariate. For example if this hazard ratio has a gamma distribution, the optimal test is a member of the G class introduced by Harrington and Fleming (1982). We also consider positive stable, inverse Gaussian, displaced Poisson and two-point frailty distribution. Results are obtained for parametric and nonparametric tests and are extended to include random censoring. We show that the loss of efficiency from omitting a covariate is generally more important than the additional loss due to misspecification of the resulting non-proportional hazards model as a proportional hazards model. However two-point frailty distributions can provide exceptions to this rule. Censoring generally increases the efficiency of the simple logrank test to the adjusted logrank test.     

A Semi-parametric Bayesian Analysis of Survival Data Based on Lévy-driven Processes

In the presence of covariate information, the proportional hazards model is one of the most popular models. In this paper, in a Bayesian nonparametric framework, we use a Markov (Lévy-driven) process to model the baseline hazard rate. Previous Bayesian nonparametric models have been based on neutral to the right processes, which have a number of drawbacks, such as discreteness of the cumulative hazard function. We allow the covariates to be time dependent functions and develop a full posterior analysis via substitution sampling. A detailed illustration is presented.     

Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches

The joint modeling of longitudinal and survival data has received extraordinary attention in the statistics literature recently, with models and methods becoming increasingly more complex. Most of these approaches pair a proportional hazards survival with longitudinal trajectory modeling through parametric or nonparametric specifications. In this paper we closely examine one data set previously analyzed using a two parameter parametric model for Mediterranean fruit fly (medfly) egg-laying trajectories paired with accelerated failure time and proportional hazards survival models. We consider parametric and nonparametric versions of these two models, as well as a proportional odds rate model paired with a wide variety of longitudinal trajectory assumptions reflecting the types of analyses seen in the literature. In addition to developing novel nonparametric Bayesian methods for joint models, we emphasize the importance of model selection from among joint and non joint models. The default in the literature is to omit at the outset non joint models from consideration. For the medfly data, a predictive diagnostic criterion suggests that both the choice of survival model and longitudinal assumptions can grossly affect model adequacy and prediction. Specifically for these data, the simple joint model used in by Tseng et al. (Biometrika 92:587–603, 2005) and models with much more flexibility in their longitudinal components are predictively outperformed by simpler analyses. This case study underscores the need for data analysts to compare on the basis of predictive performance different joint models and to include non joint models in the pool of candidates under consideration.     

Testing for Covariate Effect in the Cox Proportional Hazards Regression Model

This article presents methods for testing covariate effect in the Cox proportional hazards model based on Kullback–Leibler divergence and Renyi's information measure. Renyi's measure is referred to as the information divergence of order γ (γ ≠ 1) between two distributions. In the limiting case γ → 1, Renyi's measure becomes Kullback–Leibler divergence. In our case, the distributions correspond to the baseline and one possibly due to a covariate effect. Our proposed statistics are simple transformations of the parameter vector in the Cox proportional hazards model, and are compared with the Wald, likelihood ratio and score tests that are widely used in practice. Finally, the methods are illustrated using two real-life data sets.     

Data-driven smooth tests of the proportional hazards assumption

A new test of the proportional hazards assumption in the Cox model is proposed. The idea is based on Neyman’s smooth tests. The Cox model with proportional hazards (i.e. time-constant covariate effects) is embedded in a model with a smoothly time-varying covariate effect that is expressed as a combination of some basis functions (e.g., Legendre polynomials, cosines). Then the smooth test is the score test for significance of these artificial covariates. Furthermore, we apply a modification of Schwarz’s selection rule to choosing the dimension of the smooth model (the number of the basis functions). The score test is then used in the selected model. In a simulation study, we compare the proposed tests with standard tests based on the score process.     

On proportional hazards assumption under the random effects models

The proportional hazards mixed-effects model (PHMM) was proposed to handle dependent survival data. Motivated by its application in genetic epidemiology, we study the interpretation of its parameter estimates under violations of the proportional hazards assumption. The estimated fixed effect turns out to be an averaged regression effect over time, while the estimated variance component could be unaffected, inflated or attenuated depending on whether the random effect is on the baseline hazard, and whether the non-proportional regression effect decreases or increases over time. Using the conditional distribution of the covariates we define the standardized covariate residuals, which can be used to check the proportional hazards assumption. The model checking technique is illustrated on a multi-center lung cancer trial.     

Global Partial Likelihood for Nonparametric Proportional Hazards Models

As an alternative to the local partial likelihood method of Tibshirani and Hastie and Fan, Gijbels, and King, a global partial likelihood method is proposed to estimate the covariate effect in a nonparametric proportional hazards model, λ(t|x) = exp{ψ(x)}λ(0)(t). The estimator, ψ?(x), reduces to the Cox partial likelihood estimator if the covariate is discrete. The estimator is shown to be consistent and semiparametrically efficient for linear functionals of ψ(x). Moreover, Breslow-type estimation of the cumulative baseline hazard function, using the proposed estimator ψ?(x), is proved to be efficient. The asymptotic bias and variance are derived under regularity conditions. Computation of the estimator involves an iterative but simple algorithm. Extensive simulation studies provide evidence supporting the theory. The method is illustrated with the Stanford heart transplant data set. The proposed global approach is also extended to a partially linear proportional hazards model and found to provide efficient estimation of the slope parameter. This article has the supplementary materials online.     

A regularized variable selection procedure in additive hazards model with stratified case-cohort design

Case-cohort designs are commonly used in large epidemiological studies to reduce the cost associated with covariate measurement. In many such studies the number of covariates is very large. An efficient variable selection method is needed for case-cohort studies where the covariates are only observed in a subset of the sample. Current literature on this topic has been focused on the proportional hazards model. However, in many studies the additive hazards model is preferred over the proportional hazards model either because the proportional hazards assumption is violated or the additive hazards model provides more relevent information to the research question. Motivated by one such study, the Atherosclerosis Risk in Communities (ARIC) study, we investigate the properties of a regularized variable selection procedure in stratified case-cohort design under an additive hazards model with a diverging number of parameters. We establish the consistency and asymptotic normality of the penalized estimator and prove its oracle property. Simulation studies are conducted to assess the finite sample performance of the proposed method with a modified cross-validation tuning parameter selection methods. We apply the variable selection procedure to the ARIC study to demonstrate its practical use.     

Maximum Likelihood Estimations and EM Algorithms with Length-biased Data

Length-biased sampling has been well recognized in economics, industrial reliability, etiology applications, epidemiological, genetic and cancer screening studies. Length-biased right-censored data have a unique data structure different from traditional survival data. The nonparametric and semiparametric estimations and inference methods for traditional survival data are not directly applicable for length-biased right-censored data. We propose new expectation-maximization algorithms for estimations based on full likelihoods involving infinite dimensional parameters under three settings for length-biased data: estimating nonparametric distribution function, estimating nonparametric hazard function under an increasing failure rate constraint, and jointly estimating baseline hazards function and the covariate coefficients under the Cox proportional hazards model. Extensive empirical simulation studies show that the maximum likelihood estimators perform well with moderate sample sizes and lead to more efficient estimators compared to the estimating equation approaches. The proposed estimates are also more robust to various right-censoring mechanisms. We prove the strong consistency properties of the estimators, and establish the asymptotic normality of the semi-parametric maximum likelihood estimators under the Cox model using modern empirical processes theory. We apply the proposed methods to a prevalent cohort medical study. Supplemental materials are available online.     

Case-cohort analysis with semiparametric transformation models

Semiparametric transformation models provide flexible regression models for survival analysis, including the Cox proportional hazards and the proportional odds models as special cases. We consider the application of semiparametric transformation models in case-cohort studies, where the covariate data are observed only on cases and on a subcohort randomly sampled from the full cohort. We first propose an approximate profile likelihood approach with full-cohort data, which amounts to the pseudo-partial likelihood approach of Zucker [2005. A pseudo-partial likelihood method for semiparametric survival regression with covariate errors. J. Amer. Statist. Assoc. 100, 1264–1277]. Simulation results show that our proposal is almost as efficient as the nonparametric maximum likelihood estimator. We then extend this approach to the case-cohort design, applying the Horvitz–Thompson weighting method to the estimating equations from the approximated profile likelihood. Two levels of weights can be utilized to achieve unbiasedness and to gain efficiency. The resulting estimator has a closed-form asymptotic covariance matrix, and is found in simulations to be substantially more efficient than the estimator based on martingale estimating equations. The extension to left-truncated data will be discussed. We illustrate the proposed method on data from a cardiovascular risk factor study conducted in Taiwan.     

Marginal Likelihood Estimation for Proportional Odds Models with Right Censored Data

One majoraspect in medical research is to relate the survival times ofpatients with the relevant covariates or explanatory variables.The proportional hazards model has been used extensively in thepast decades with the assumption that the covariate effects actmultiplicatively on the hazard function, independent of time.If the patients become more homogeneous over time, say the treatmenteffects decrease with time or fade out eventually, then a proportionalodds model may be more appropriate. In the proportional oddsmodel, the odds ratio between patients can be expressed as afunction of their corresponding covariate vectors, in which,the hazard ratio between individuals converges to unity in thelong run. In this paper, we consider the estimation of the regressionparameter for a semiparametric proportional odds model at whichthe baseline odds function is an arbitrary, non-decreasing functionbut is left unspecified. Instead of using the exact survivaltimes, only the rank order information among patients is used.A Monte Carlo method is used to approximate the marginal likelihoodfunction of the rank invariant transformation of the survivaltimes which preserves the information about the regression parameter.The method can be applied to other transformation models withcensored data such as the proportional hazards model, the generalizedprobit model or others. The proposed method is applied to theVeteran's Administration lung cancer trial data.     

Accelerated hazards mixture cure model

We propose a new cure model for survival data with a surviving or cure fraction. The new model is a mixture cure model where the covariate effects on the proportion of cure and the distribution of the failure time of uncured patients are separately modeled. Unlike the existing mixture cure models, the new model allows covariate effects on the failure time distribution of uncured patients to be negligible at time zero and to increase as time goes by. Such a model is particularly useful in some cancer treatments when the treat effect increases gradually from zero, and the existing models usually cannot handle this situation properly. We develop a rank based semiparametric estimation method to obtain the maximum likelihood estimates of the parameters in the model. We compare it with existing models and methods via a simulation study, and apply the model to a breast cancer data set. The numerical studies show that the new model provides a useful addition to the cure model literature.     

Parameter Orthogonality in Mixed Regression Models for Survival Data

The implications of parameter orthogonality for the robustness of survival regression models are considered. The question of which of the proportional hazards or the accelerated life families of models would be more appropriate for analysis is usually ignored, and the proportional hazards family is applied, particularly in medicine, for convenience. Accelerated life models have conventionally been used in reliability applications. We propose a one-parameter family mixture survival model which includes both the accelerated life and the proportional hazards models. By orthogonalizing relative to the mixture parameter, we can show that, for small effects of the covariates, the regression parameters under the alternative families agree to within a constant. This recovers a known misspecification result. We use notions of parameter orthogonality to explore robustness to other types of misspecification including misspecified base-line hazards. The results hold in the presence of censoring. We also study the important question of when proportionality matters.     

A gradient-based algorithm for semiparametric models with missing covariates

In the parametric regression model, the covariate missing problem under missing at random is considered. It is often desirable to use flexible parametric or semiparametric models for the covariate distribution, which can reduce a potential misspecification problem. Recently, a completely nonparametric approach was developed by [H.Y. Chen, Nonparametric and semiparametric models for missing covariates in parameter regression, J. Amer. Statist. Assoc. 99 (2004), pp. 1176–1189; Z. Zhang and H.E. Rockette, On maximum likelihood estimation in parametric regression with missing covariates, J. Statist. Plann. Inference 47 (2005), pp. 206–223]. Although it does not require a model for the covariate distribution or the missing data mechanism, the proposed method assumes that the covariate distribution is supported only by observed values. Consequently, their estimator is a restricted maximum likelihood estimator (MLE) rather than the global MLE. In this article, we show the restricted semiparametric MLE could be very misleading in some cases. We discuss why this problem occurs and suggest an algorithm to obtain the global MLE. Then, we assess the performance of the proposed method via some simulation experiments.     

A Proportional Hazards Regression Model for the Subdistribution with Covariates‐adjusted Censoring Weight for Competing Risks Data

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate‐dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate‐dependent censoring. We consider a covariate‐adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate‐adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate‐adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research. Here, cancer relapse and death in complete remission are two competing risks.     

Flexible competing risks regression modeling and goodness-of-fit

In this paper we consider different approaches for estimation and assessment of covariate effects for the cumulative incidence curve in the competing risks model. The classic approach is to model all cause-specific hazards and then estimate the cumulative incidence curve based on these cause-specific hazards. Another recent approach is to directly model the cumulative incidence by a proportional model (Fine and Gray, J Am Stat Assoc 94:496–509, 1999), and then obtain direct estimates of how covariates influences the cumulative incidence curve. We consider a simple and flexible class of regression models that is easy to fit and contains the Fine–Gray model as a special case. One advantage of this approach is that our regression modeling allows for non-proportional hazards. This leads to a new simple goodness-of-fit procedure for the proportional subdistribution hazards assumption that is very easy to use. The test is constructive in the sense that it shows exactly where non-proportionality is present. We illustrate our methods to a bone marrow transplant data from the Center for International Blood and Marrow Transplant Research (CIBMTR). Through this data example we demonstrate the use of the flexible regression models to analyze competing risks data when non-proportionality is present in the data.