Comparing first hitting time and proportional hazards regression models

Cox's widely used semi-parametric proportional hazards (PH) regression model places restrictions on the possible shapes of the hazard function. Models based on the first hitting time (FHT) of a stochastic process are among the alternatives and have the attractive feature of being based on a model of the underlying process. We review and compare the PH model and an FHT model based on a Wiener process which leads to an inverse Gaussian (IG) regression model. This particular model can also represent a “cured fraction” or long-term survivors. A case study of survival after coronary artery bypass grafting is used to examine the interpretation of the IG model, especially in relation to covariates that affect both of its parameters.     

Risk patterns in drug safety study using relative times by accelerated failure time models when proportional hazards assumption is questionable: an illustrative case study of cancer risk of patients on glucose‐lowering therapies

Observational drug safety studies may be susceptible to confounding or protopathic bias. This bias may cause a spurious relationship between drug exposure and adverse side effect when none exists and may lead to unwarranted safety alerts. The spurious relationship may manifest itself through substantially different risk levels between exposure groups at the start of follow‐up when exposure is deemed too short to have any plausible biological effect of the drug. The restrictive proportional hazards assumption with its arbitrary choice of baseline hazard function renders the commonly used Cox proportional hazards model of limited use for revealing such potential bias. We demonstrate a fully parametric approach using accelerated failure time models with an illustrative safety study of glucose‐lowering therapies and show that its results are comparable against other methods that allow time‐varying exposure effects. Our approach includes a wide variety of models that are based on the flexible generalized gamma distribution and allows direct comparisons of estimated hazard functions following different exposure‐specific distributions of survival times. This approach lends itself to two alternative metrics, namely relative times and difference in times to event, allowing physicians more ways to communicate patient's prognosis without invoking the concept of risks, which some may find hard to grasp. In our illustrative case study, substantial differences in cancer risks at drug initiation followed by a gradual reduction towards null were found. This evidence is compatible with the presence of protopathic bias, in which undiagnosed symptoms of cancer lead to switches in diabetes medication. Copyright © 2015 John Wiley & Sons, Ltd.     

The analysis of transition times in multistate stochastic processes using proportional hazard regression models

Extensions to Cox's proportional hazards regression model (Cox, 1972) for the analysis of survival data are considered for a more general multistate framework. This framework allows several transient disease states between initial entry state and death as well as incorporating possible competing causes of death. Methods for parameter and function estimation within this extension are presented and applied to the analysis of data from the Stanford Heart Transplantation Program (Crowley and Hu,1977).     

Asymptotic expansions and the reliability of tests in accelerated failure time models

This paper gives matrix formilae for the O(n^-1 ) cerrecti0n applicable to asymptotically efficient conditional moment tests. These formulae only require expectations of functions involving, at most, second order derivatives of the log-likelihood; unlike those previously providcd by Ferrari and Corddro(1994). The correction is used to assess the reliability of first order asymptotic theory for arbitrary residual-based diagnostics in a class of accelerated failure time models: this correction is always parameter free, depending only on the number of included covariates in the regression design. For all but one of the tests considered, first order theory is found to be extremely unreliable, even in quite large samples, although this may not be widely appreciated by applied workers.     

Maximum likelihood methods for complex sample data: logistic regression and discrete proportional hazards models

To estimate model parameters from complex sample data. we apply maximum likelihood techniques to the complex sample data from the finite population, which is treated as a sample from an i nfinite superpopulation. General asymptotic distribution theory is developed and then applied to both logistic regression and discrete proportional hazards models. Data from the Lipid Research Clinics Program areused to illustrate each model, demonstrating the effects on inference of neglecting the sampling design during parameter estimation. These empirical results also shed light on the issue of model-based vs. design-based inferences.     

On gehan's wilcoxon and the accelerated failure time model

A proof is provided to show that Gehan's 1965 generalization of the two sample Wilcoxon test lies outside the class of efficient score procedures for right censored data (Prentice 1978).     

Bayesian parametric accelerated failure time spatial model and its application to prostate cancer

Prostate cancer (PrCA) is the most common cancer diagnosed in American men and the second leading cause of death from malignancies. There are large geographical variation and racial disparities existing in the survival rate of PrCA. Much work on the spatial survival model is based on the proportional hazards (PH) model, but few focused on the accelerated failure time (AFT) model. In this paper, we investigate the PrCA data of Louisiana from the Surveillance, Epidemiology, and End Results program and the violation of the PH assumption suggests that the spatial survival model based on the AFT model is more appropriate for this data set. To account for the possible extra-variation, we consider spatially referenced independent or dependent spatial structures. The deviance information criterion is used to select a best-fitting model within the Bayesian frame work. The results from our study indicate that age, race, stage, and geographical distribution are significant in evaluating PrCA survival.     

Ridge estimation in generalized linear models and proportional hazards regressions

Conventionally, a ridge parameter is estimated as a function of regression parameters based on ordinary least squares. In this article, we proposed an iterative procedure instead of the one-step or conventional ridge method. Additionally, we construct an indicator that measures the potential degree of improvement in mean squared error when ridge estimates are employed. Simulations show that our methods are appropriate for a wide class of non linear models including generalized linear models and proportional hazards (PHs) regressions. The method is applied to a PH regression with highly collinear covariates in a cancer recurrence study.     

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.     

Empirical likelihood method for the multivariate accelerated failure time models

In applications, multivariate failure time data appears when each study subject may potentially experience several types of failures or recurrences of a certain phenomenon, or failure times may be clustered. Three types of marginal accelerated failure time models dealing with multiple events data, recurrent events data and clustered events data are considered. We propose a unified empirical likelihood inferential procedure for the three types of models based on rank estimation method. The resulting log-empirical likelihood ratios are shown to possess chi-squared limiting distributions. The properties can be applied to do tests and construct confidence regions without the need to solve the rank estimating equations nor to estimate the limiting variance-covariance matrices. The related computation is easy to implement. The proposed method is illustrated by extensive simulation studies and a real example.     

Some estimators and tests for accelerated hazards model using weighted cumulative hazard difference

For a censored two-sample problem, Chen and Wang [Y.Q. Chen and M.-C. Wang, Analysis of accelerated hazards models, J. Am. Statist. Assoc. 95 (2000), pp. 608–618] introduced the accelerated hazards model. The scale-change parameter in this model characterizes the association of two groups. However, its estimator involves the unknown density in the asymptotic variance. Thus, to make an inference on the parameter, numerically intensive methods are needed. The goal of this article is to propose a simple estimation method in which estimators are asymptotically normal with a density-free asymptotic variance. Some lack-of-fit tests are also obtained from this. These tests are related to Gill–Schumacher type tests [R.D. Gill and M. Schumacher, A simple test of the proportional hazards assumption, Biometrika 74 (1987), pp. 289–300] in which the estimating functions are evaluated at two different weight functions yielding two estimators that are close to each other. Numerical studies show that for some weight functions, the estimators and tests perform well. The proposed procedures are illustrated in two applications.     

Assessing gamma frailty models for clustered failure time data

Proportional hazards frailty models use a random effect, so called frailty, to construct association for clustered failure time data. It is customary to assume that the random frailty follows a gamma distribution. In this paper, we propose a graphical method for assessing adequacy of the proportional hazards frailty models. In particular, we focus on the assessment of the gamma distribution assumption for the frailties. We calculate the average of the posterior expected frailties at several followup time points and compare it at these time points to 1, the known mean frailty. Large discrepancies indicate lack of fit. To aid in assessing the goodness of fit, we derive and estimate the standard error of the mean of the posterior expected frailties at each time point examined. We give an example to illustrate the proposed methodology and perform sensitivity analysis by simulations.     

Comparison of goodness of fit tests for the cox proportional hazards model

The proportional hazards regression model of Cox(1972) is widely used in analyzing survival data. We examine several goodness of fit tests for checking the proportionality of hazards in the Cox model with two-sample censored data, and compare the performance of these tests by a simulation study. The strengths and weaknesses of the tests are pointed out. The effects of the extent of random censoring on the size and power are also examined. Results of a simulation study demonstrate that Gill and Schumacher's test is most powerful against a broad range of monotone departures from the proportional hazards assumption, but it may not perform as well fail for alternatives of nonmonotone hazard ratio. For the latter kind of alternatives, Andersen's test may detect patterns of irregular changes in hazards.     

Semiparametric Bayesian accelerated failure time model with interval-censored data

Many existing approaches to analysing interval-censored data lack flexibility or efficiency. In this paper, we propose an efficient, easy to implement approach on accelerated failure time model with a logarithm transformation of the failure time and flexible specifications on the error distribution. We use exact inference for the Dirichlet process without approximation in imputation. Our algorithm can be implemented with simple Gibbs sampling which produces exact posterior distributions on the features of interest. Simulation and real data analysis demonstrate the advantage of our method compared to some other methods.     

Generalized M-estimation for the accelerated failure time model

The accelerated failure time (AFT) model is an important regression tool to study the association between failure time and covariates. In this paper, we propose a robust weighted generalized M (GM) estimation for the AFT model with right-censored data by appropriately using the Kaplan–Meier weights in the GM–type objective function to estimate the regression coefficients and scale parameter simultaneously. This estimation method is computationally simple and can be implemented with existing software. Asymptotic properties including the root-n consistency and asymptotic normality are established for the resulting estimator under suitable conditions. We further show that the method can be readily extended to handle a class of nonlinear AFT models. Simulation results demonstrate satisfactory finite sample performance of the proposed estimator. The practical utility of the method is illustrated by a real data example.     

Restricted mean survival time as a summary measure of time‐to‐event outcome

Many clinical research studies evaluate a time‐to‐event outcome, illustrate survival functions, and conventionally report estimated hazard ratios to express the magnitude of the treatment effect when comparing between groups. However, it may not be straightforward to interpret the hazard ratio clinically and statistically when the proportional hazards assumption is invalid. In some recent papers published in clinical journals, the use of restricted mean survival time (RMST) or τ ‐year mean survival time is discussed as one of the alternative summary measures for the time‐to‐event outcome. The RMST is defined as the expected value of time to event limited to a specific time point corresponding to the area under the survival curve up to the specific time point. This article summarizes the necessary information to conduct statistical analysis using the RMST, including the definition and statistical properties of the RMST, adjusted analysis methods, sample size calculation, information fraction for the RMST difference, and clinical and statistical meaning and interpretation. Additionally, we discuss how to set the specific time point to define the RMST from two main points of view. We also provide developed SAS codes to determine the sample size required to detect an expected RMST difference with appropriate power and reconstruct individual survival data to estimate an RMST reference value from a reported survival curve.     

Estimation in the Cox proportional hazards model with doubly censored and truncated data

In longitudinal studies, the proportional hazard model is often used to analyse covariate effects on the duration time, defined as the elapsed time between the first and second event. In this article, we consider the situation when the first event suffers partly interval-censoring and the second event suffers left-truncation and right-censoring. We proposed a two-step estimation procedure for estimating the regression coefficients of the proportional model. A simulation study is conducted to investigate the performance of the proposed estimator.     

Assessing the prediction accuracy of cure in the Cox proportional hazards cure model: an application to breast cancer data

A cure rate model is a survival model incorporating the cure rate with the assumption that the population contains both uncured and cured individuals. It is a powerful statistical tool for prognostic studies, especially in cancer. The cure rate is important for making treatment decisions in clinical practice. The proportional hazards (PH) cure model can predict the cure rate for each patient. This contains a logistic regression component for the cure rate and a Cox regression component to estimate the hazard for uncured patients. A measure for quantifying the predictive accuracy of the cure rate estimated by the Cox PH cure model is required, as there has been a lack of previous research in this area. We used the Cox PH cure model for the breast cancer data; however, the area under the receiver operating characteristic curve (AUC) could not be estimated because many patients were censored. In this study, we used imputation‐based AUCs to assess the predictive accuracy of the cure rate from the PH cure model. We examined the precision of these AUCs using simulation studies. The results demonstrated that the imputation‐based AUCs were estimable and their biases were negligibly small in many cases, although ordinary AUC could not be estimated. Additionally, we introduced the bias‐correction method of imputation‐based AUCs and found that the bias‐corrected estimate successfully compensated the overestimation in the simulation studies. We also illustrated the estimation of the imputation‐based AUCs using breast cancer data. Copyright © 2014 John Wiley & Sons, Ltd.     

Choice of Parametric Accelerated Life and Proportional Hazards Models for Survival Data: Asymptotic Results

We discuss the impact of misspecifying fully parametric proportional hazards and accelerated life models. For the uncensored case, misspecified accelerated life models give asymptotically unbiased estimates of covariate effect, but the shape and scale parameters depend on the misspecification. The covariate, shape and scale parameters differ in the censored case. Parametric proportional hazards models do not have a sound justification for general use: estimates from misspecified models can be very biased, and misleading results for the shape of the hazard function can arise. Misspecified survival functions are more biased at the extremes than the centre. Asymptotic and first order results are compared. If a model is misspecified, the size of Wald tests will be underestimated. Use of the sandwich estimator of standard error gives tests of the correct size, but misspecification leads to a loss of power. Accelerated life models are more robust to misspecification because of their log-linear form. In preliminary data analysis, practitioners should investigate proportional hazards and accelerated life models; software is readily available for several such models.     

Generalizing the standardized hazard ratio to multivariate proportional hazards regression,with an application to clinical~genomic studies

The standardized hazard ratio for univariate proportional hazards regression is generalized as a scalar to multivariate proportional hazards regression. Estimators of the standardized log hazard ratio are developed, with corrections for bias and for regression to the mean in high-dimensional analyses. Tests of point and interval null hypotheses and confidence intervals are constructed. Cohort sampling study designs, commonly used in prospective–retrospective clinical genomic studies, are accommodated.