Generating survival times to simulate Cox proportional hazards models

Simulation studies present an important statistical tool to investigate the performance, properties and adequacy of statistical models in pre-specified situations. One of the most important statistical models in medical research is the proportional hazards model of Cox. In this paper, techniques to generate survival times for simulation studies regarding Cox proportional hazards models are presented. A general formula describing the relation between the hazard and the corresponding survival time of the Cox model is derived, which is useful in simulation studies. It is shown how the exponential, the Weibull and the Gompertz distribution can be applied to generate appropriate survival times for simulation studies. Additionally, the general relation between hazard and survival time can be used to develop own distributions for special situations and to handle flexibly parameterized proportional hazards models. The use of distributions other than the exponential distribution is indispensable to investigate the characteristics of the Cox proportional hazards model, especially in non-standard situations, where the partial likelihood depends on the baseline hazard. A simulation study investigating the effect of measurement errors in the German Uranium Miners Cohort Study is considered to illustrate the proposed simulation techniques and to emphasize the importance of a careful modelling of the baseline hazard in Cox models.     

Semiparametric linear transformation models: Effect measures,estimators, and applications

Semiparametric linear transformation models form a versatile class of regression models with the Cox proportional hazards model being the most well-known member. These models are well studied for right censored outcomes and are typically used in survival analysis. We consider transformation models as a tool for situations with uncensored continuous outcomes where linear regression is not appropriate. We introduce the probabilistic index as a uniform effect measure for the class of transformation models. We discuss and compare three estimators using a working Cox regression model: the partial likelihood estimator, an estimator based on binary generalized linear models and one based on probabilistic index model estimating equations. The latter has a superior performance in terms of bias and variance when the working model is misspecified. For the purpose of illustration, we analyze data that were collected at an urban alcohol and drug detoxification unit.     

Parametric models for accelerated and long-term survival: a comment on proportional hazards

The Cox proportional hazards model (CPH) is routinely used in clinical trials, but it may encounter serious difficulties with departures from the proportional hazards assumption, even when the departures are not readily detected by commonly used diagnostics. We consider the Gamel-Boag (GB) model, a log-normal model for accelerated failure in which a proportion of subjects are long-term survivors. When the CPH model is fit to simulated data generated from this model, the results can range from gross overstatement of the effect size, to a situation where increasing follow-up may cause a decline in power. We implement a fitting algorithm for the GB model that permits separate covariate effects on the rapidity of early failure and the fraction of long-term survivors. When effects are detected by both the CPH and GB methods, the attribution of the effect to long-term or short-term survival may change the interpretation of the data. We believe these examples motivate more frequent use of parametric survival models in conjunction with the semi-parametric Cox proportional hazards model.     

Improving testing and description of treatment effect in clinical trials with survival outcomes

Cox model inference and the log-rank test have been the cornerstones for design and analysis of clinical trials with survival outcomes. In this article, we summarize some recently developed methods for analyzing survival data when the hazards may possibly be nonproportional and also propose some new estimators for summary measures of the treatment effect. These methods utilize the short-term and long-term hazard ratio model proposed in Yang and Prentice (2005), which contains the Cox model and also accommodates various nonproportional hazards scenarios. Without the proportional hazards assumption, these methods often improve the log-rank test and inference procedures based on the Cox model, as well as nonparametric procedures currently available in the literature. The proposed methods have sound theoretical justifications and can be computed quickly. R codes for implementing them are available. Detailed illustrations with 3 clinical trials are provided.     

Likelihood approaches for proportional likelihood ratio model with right‐censored data

Regression methods for survival data with right censoring have been extensively studied under semiparametric transformation models such as the Cox regression model and the proportional odds model. However, their practical application could be limited because of possible violation of model assumption or lack of ready interpretation for the regression coefficients in some cases. As an alternative, in this paper, the proportional likelihood ratio model introduced by Luo and Tsai is extended to flexibly model the relationship between survival outcome and covariates. This model has a natural connection with many important semiparametric models such as generalized linear model and density ratio model and is closely related to biased sampling problems. Compared with the semiparametric transformation model, the proportional likelihood ratio model is appealing and practical in many ways because of its model flexibility and quite direct clinical interpretation. We present two likelihood approaches for the estimation and inference on the target regression parameters under independent and dependent censoring assumptions. Based on a conditional likelihood approach using uncensored failure times, a numerically simple estimation procedure is developed by maximizing a pairwise pseudo‐likelihood. We also develop a full likelihood approach, and the most efficient maximum likelihood estimator is obtained by a profile likelihood. Simulation studies are conducted to assess the finite‐sample properties of the proposed estimators and compare the efficiency of the two likelihood approaches. An application to survival data for bone marrow transplantation patients of acute leukemia is provided to illustrate the proposed method and other approaches for handling non‐proportionality. The relative merits of these methods are discussed in concluding remarks. Copyright © 2014 John Wiley & Sons, Ltd.     

Comparing proportional hazards and accelerated failure time models for survival analysis

This paper describes a method proposed for a censored linear regression model that can be used in the context of survival analysis. The method has the important characteristic of allowing estimation and inference without knowing the distribution of the duration variable. Moreover, it does not need the assumption of proportional hazards. Therefore, it can be an interesting alternative to the Cox proportional hazards models when this assumption does not hold. In addition, implementation and interpretation of the results is simple. In order to analyse the performance of this methodology, we apply it to two real examples and we carry out a simulation study. We present its results together with those obtained with the traditional Cox model and AFT parametric models. The new proposal seems to lead to more precise results.     

Mixtures of proportional hazards regression models.

This paper presents a mixture model which combines features of the usual Cox proportional hazards model with those of a class of models, known as mixtures-of-experts. The resulting model is more flexible than the usual Cox model in the sense that the log hazard ratio is allowed to vary non-linearly as a function of the covariates. Thus it provides a flexible approach to both modelling survival data and model checking. The method is illustrated with simulated data, as well as with multiple myeloma data.     

Bayesian transformation cure frailty models with multivariate failure time data

We propose a class of transformation cure frailty models to accommodate a survival fraction in multivariate failure time data. Established through a general power transformation, this family of cure frailty models includes the proportional hazards and the proportional odds modeling structures as two special cases. Within the Bayesian paradigm, we obtain the joint posterior distribution and the corresponding full conditional distributions of the model parameters for the implementation of Gibbs sampling. Model selection is based on the conditional predictive ordinate statistic and deviance information criterion. As an illustration, we apply the proposed method to a real data set from dentistry.     

Joint models for efficient estimation in proportional hazards regression models

In survival studies, information lost through censoring can be partially recaptured through repeated measures data which are predictive of survival. In addition, such data may be useful in removing bias in survival estimates, due to censoring which depends upon the repeated measures. Here we investigate joint models for survival T and repeated measurements Y, given a vector of covariates Z. Mixture models indexed as f (T/Z) f (Y/T,Z) are well suited for assessing covariate effects on survival time. Our objective is efficiency gains, using non-parametric models for Y in order to avoid introducing bias by misspecification of the distribution for Y. We model (T/Z) as a piecewise exponential distribution with proportional hazards covariate effect. The component (Y/T,Z) has a multinomial model. The joint likelihood for survival and longitudinal data is maximized, using the EM algorithm. The estimate of covariate effect is compared to the estimate based on the standard proportional hazards model and an alternative joint model based estimate. We demonstrate modest gains in efficiency when using the joint piecewise exponential joint model. In a simulation, the estimated efficiency gain over the standard proportional hazards model is 6.4 per cent. In clinical trial data, the estimated efficiency gain over the standard proportional hazards model is 10.2 per cent.     

A comparison of estimated proportional hazards models and regression trees

We present examples of the usage of regression trees for censored response via two real world datasets, one a rheumatoid arthritis survival study and the other a hip replacement study, and draw comparisons with the results of Cox proportional hazards modelling. The two methods pursue different goals. Motivation of the tree techniques is the desire to extract meaningful prognostic groups while the proportional hazards model enables assessment of the impact of risk factors. The methods are thus complementary. For the arthritis study the two techniques corroborate one another, although the flavour of the conclusions derived differ. For the hip replacement study, however, the regression tree approach reveals structure that would not emerge from a routine proportional hazards analysis. We also discuss the treatment of data analytic issues such as the handling of missing values and influence in the presence of non-uniform censoring.     

The restricted cubic spline as baseline hazard in the proportional hazards model with step function time-dependent covariables

We incorporate a cubic spline function where the tails are linearly constrained, as the baseline hazard, into the proportional hazards model. We show estimation of covariable coefficients and survival probabilities with this model to be as efficient statistically as with the Cox proportional hazards model when covariables are fixed. Examples show that the inclusion of time-dependent covariables defined as step functions into the restricted cubic spline proportional hazards model reduces computation time by a factor of 213 over the Cox model. Advantages of the spline model also include flexibility of the hazard, smooth survival curves, and confidence limits for the survival and hazard estimates when there are time-dependent covariables present.     

A test for proportional hazards assumption within the class of exponential conditional mean models

Two classes of econometric estimators are popular for modeling outcomes with idiosyncratic characteristics such as those present in medical costs data: (1) estimators based on the exponential conditional mean models where the mean function of the outcome is equal to exponential of the linear predictor and (2) estimators based on the proportional hazard assumption where hazard function of the outcome is equal to exponential of the linear predictor. Recent work has provided guidance both on choosing between the two classes of estimators and also on choosing among alternative estimators within the exponential conditional mean framework. The present work extends this literature by proposing a test for identifying the proportional hazards assumption within the class of exponential conditional mean models, thereby eliminating the need to run both classes of models in order to make informative choices. We implement this test using the generalized gamma regression model, thereby allowing the analyst to select between both parametric alternatives and also the semi-parametric Cox model from one cohesive framework. Our simulation results indicate that the proposed test perform as well as the traditional test of proportional hazards assumption following a Cox regression based on power and Type I error under a variety of data generating mechanisms. We illustrate its use in an analysis of physician visits.     

Efficiency of the logistic regression and Cox proportional hazards models in longitudinal studies

Both logistic regression and Cox proportional hazards models are used widely in longitudinal epidemiologic studies for analysing the relationship between several risk factors and a time-related dichotomous event. The two models yield similar estimates of regression coefficients in studies with short follow-up and low incidence of event occurrence. Further, with just one dichotomous covariate and identical censoring times for all subjects, the asymptotic relative efficiency of the two models is very close to 1 unless the duration of follow-up is extended. We generalize this result to several qualitative or quantitative covariates. This was motivated by the analysis of mortality data from a study where all subjects are followed up during the same fixed period without loss except by death. Logistic and Cox models were applied to these data. Similar results were obtained for the two models in shorter periods of follow-up of five years or less, but not in longer periods of ten years or more, where the survival rate was lower.     

Using instrumental variables to estimate a Cox’s proportional hazards regression subject to additive confounding

The estimation of treatment effects is one of the primary goals of statistics in medicine. Estimation based on observational studies is subject to confounding. Statistical methods for controlling bias due to confounding include regression adjustment, propensity scores and inverse probability weighted estimators. These methods require that all confounders are recorded in the data. The method of instrumental variables (IVs) can eliminate bias in observational studies even in the absence of information on confounders. We propose a method for integrating IVs within the framework of Cox’s proportional hazards model and demonstrate the conditions under which it recovers the causal effect of treatment. The methodology is based on the approximate orthogonality of an instrument with unobserved confounders among those at risk. We derive an estimator as the solution to an estimating equation that resembles the score equation of the partial likelihood in much the same way as the traditional IV estimator resembles the normal equations. To justify this IV estimator for a Cox model we perform simulations to evaluate its operating characteristics. Finally, we apply the estimator to an observational study of the effect of coronary catheterization on survival.     

Assessing time-by-covariate interactions in proportional hazards regression models using cubic spline functions

Proportional hazards (or Cox) regression is a popular method for modelling the effects of prognostic factors on survival. Use of cubic spline functions to model time-by-covariate interactions in Cox regression allows investigation of the shape of a possible covariate-time dependence without having to specify a specific functional form. Cubic spline functions allow one to graph such time-by-covariate interactions, to test formally for the proportional hazards assumption, and also to test for non-linearity of the time-by-covariate interaction. The functions can be fitted with existing software using relatively few parameters; the regression coefficients are estimated using standard maximum likelihood methods.     

多结局生存分析模型与Cox模型的随机模拟比较

目的通过随机模拟评价不同多结局生存分析模型的特点。方法利用随机模拟数据,比较多结局生存分析模型、将多个结局视为一种结局的单结局Cox模型及将各个结局分开拟合多个单结局Cox模型的回归系数的估计精度。结果多结局生存分析模型回归系数的估计最准确,95%可信区间包含回归系数的百分比最高,95%可信区间不包含0的百分比也最高。结论采用将各个结局分开单独拟合Cox模型和将多种结局视为单一结局拟合Cox模型分析多结局生存数据会导致回归系数估计不准及检验效能的降低。     

Variable selection for proportional odds model

In this paper we study the problem of variable selection for the proportional odds model, which is a useful alternative to the proportional hazards model and might be appropriate when the proportional hazards assumption is not satisfied. We propose to fit the proportional odds model by maximizing the marginal likelihood subject to a shrinkage-type penalty, which encourages sparse solutions and hence facilitates the process of variable selection. Two types of shrinkage penalties are considered: the LASSO and the adaptive-LASSO (ALASSO) penalty. In the ALASSO penalty, different weights are imposed on different coefficients such that important variables are more protectively retained in the final model while unimportant ones are more likely to be shrunk to zeros. We further provide an efficient computation algorithm to implement the proposed methods, and demonstrate their performance through simulation studies and an application to real data. Numerical results indicate that both methods can produce accurate and interpretable models, and the ALASSO tends to work better than the usual LASSO.     

Robust and efficient estimation in the parametric proportional hazards model under random censoring

Cox proportional hazard regression model is a popular tool to analyze the relationship between a censored lifetime variable with other relevant factors. The semiparametric Cox model is widely used to study different types of data arising from applied disciplines such as medical science, biology, and reliability studies. A fully parametric version of the Cox regression model, if properly specified, can yield more efficient parameter estimates, leading to better insight generation. However, the existing maximum likelihood approach of generating inference under the fully parametric proportional hazards model is highly nonrobust against data contamination (often manifested through outliers), which restricts its practical usage. In this paper, we develop a robust estimation procedure for the parametric proportional hazards model based on the minimum density power divergence approach. The proposed minimum density power divergence estimator is seen to produce highly robust estimates under data contamination with only a slight loss in efficiency under pure data. Further, it is always seen to generate more precise inference than the likelihood based estimates under the semiparametric Cox models or their existing robust versions. We also justify their robustness theoretically through the influence function analysis. The practical applicability and usefulness of the proposal are illustrated through simulations and real data examples.     

On hazard ratio estimators by proportional hazards models in matched-pair cohort studies

Background

In matched-pair cohort studies with censored events, the hazard ratio (HR) may be of main interest. However, it is lesser known in epidemiologic literature that the partial maximum likelihood estimator of a common HR conditional on matched pairs is written in a simple form, namely, the ratio of the numbers of two pair-types. Moreover, because HR is a noncollapsible measure and its constancy across matched pairs is a restrictive assumption, marginal HR as “average” HR may be targeted more than conditional HR in analysis.

Methods

Based on its simple expression, we provided an alternative interpretation of the common HR estimator as the odds of the matched-pair analog of C-statistic for censored time-to-event data. Through simulations assuming proportional hazards within matched pairs, the influence of various censoring patterns on the marginal and common HR estimators of unstratified and stratified proportional hazards models, respectively, was evaluated. The methods were applied to a real propensity-score matched dataset from the Rotterdam tumor bank of primary breast cancer.

Results

We showed that stratified models unbiasedly estimated a common HR under the proportional hazards within matched pairs. However, the marginal HR estimator with robust variance estimator lacks interpretation as an “average” marginal HR even if censoring is unconditionally independent to event, unless no censoring occurs or no exposure effect is present. Furthermore, the exposure-dependent censoring biased the marginal HR estimator away from both conditional HR and an “average” marginal HR irrespective of whether exposure effect is present. From the matched Rotterdam dataset, we estimated HR for relapse-free survival of absence versus presence of chemotherapy; estimates (95% confidence interval) were 1.47 (1.18–1.83) for common HR and 1.33 (1.13–1.57) for marginal HR.

Conclusion

The simple expression of the common HR estimator would be a useful summary of exposure effect, which is less sensitive to censoring patterns than the marginal HR estimator. The common and the marginal HR estimators, both relying on distinct assumptions and interpretations, are complementary alternatives for each other.

     

Corrected score estimation in the proportional hazards model with misclassified discrete covariates

We consider Cox proportional hazards regression when the covariate vector includes error-prone discrete covariates along with error-free covariates, which may be discrete or continuous. The misclassification in the discrete error-prone covariates is allowed to be of any specified form. Building on the work of Nakamura and his colleagues, we present a corrected score method for this setting. The method can handle all three major study designs (internal validation design, external validation design, and replicate measures design), both functional and structural error models, and time-dependent covariates satisfying a certain 'localized error' condition. We derive the asymptotic properties of the method and indicate how to adjust the covariance matrix of the regression coefficient estimates to account for estimation of the misclassification matrix. We present the results of a finite-sample simulation study under Weibull survival with a single binary covariate having known misclassification rates. The performance of the method described here was similar to that of related methods we have examined in previous works. Specifically, our new estimator performed as well as or, in a few cases, better than the full Weibull maximum likelihood estimator. We also present simulation results for our method for the case where the misclassification probabilities are estimated from an external replicate measures study. Our method generally performed well in these simulations. The new estimator has a broader range of applicability than many other estimators proposed in the literature, including those described in our own earlier work, in that it can handle time-dependent covariates with an arbitrary misclassification structure. We illustrate the method on data from a study of the relationship between dietary calcium intake and distal colon cancer.