A new proposal for multivariable modelling of time-varying effects in survival data based on fractional polynomial time-transformation

The Cox proportional hazards model has become the standard for the analysis of survival time data in cancer and other chronic diseases. In most studies, proportional hazards (PH) are assumed for covariate effects. With long-term follow-up, the PH assumption may be violated, leading to poor model fit. To accommodate non-PH effects, we introduce a new procedure, MFPT, an extension of the multivariable fractional polynomial (MFP) approach, to do the following: (1) select influential variables; (2) determine a sensible dose-response function for continuous variables; (3) investigate time-varying effects; (4) model such time-varying effects on a continuous scale. Assuming PH initially, we start with a detailed model-building step, including a search for possible non-linear functions for continuous covariates. Sometimes a variable with a strong short-term effect may appear weak or non-influential if 'averaged' over time under the PH assumption. To protect against omitting such variables, we repeat the analysis over a restricted time-interval. Any additional prognostic variables identified by this second analysis are added to create our final time-fixed multivariable model. Using a forward-selection algorithm we search for possible improvements in fit by adding time-varying covariates. The first part to create a final time-fixed model does not require the use of MFP. A model may be given from 'outside' or a different strategy may be preferred for this part. This broadens the scope of the time-varying part. To motivate and illustrate the methodology, we create prognostic models from a large database of patients with primary breast cancer. Non-linear time-fixed effects are found for progesterone receptor status and number of positive lymph nodes. Highly statistically significant time-varying effects are present for progesterone receptor status and tumour size.     

A partial likelihood approach to smooth estimation of dynamic covariate effects using penalised splines

Survival data are often modelled by the Cox proportional hazards model, which assumes that covariate effects are constant over time. In recent years however, several new approaches have been suggested which allow covariate effects to vary with time. Non-proportional hazard functions, with covariate effects changing dynamically, can be fitted using penalised spline (P-spline) smoothing. By utilising the link between P-spline smoothing and generalised linear mixed models, the smoothing parameters steering the amount of smoothing can be selected. A hybrid routine, combining the mixed model approach with a classical Akaike criterion, is suggested. This approach is evaluated with simulations and applied to data from the West of Scotland Coronary Prevention Study.     

An estimator for the proportional hazards model with multiple longitudinal covariates measured with error

In many longitudinal studies, it is of interest to characterize the relationship between a time-to-event (e.g. survival) and several time-dependent and time-independent covariates. Time-dependent covariates are generally observed intermittently and with error. For a single time-dependent covariate, a popular approach is to assume a joint longitudinal data-survival model, where the time-dependent covariate follows a linear mixed effects model and the hazard of failure depends on random effects and time-independent covariates via a proportional hazards relationship. Regression calibration and likelihood or Bayesian methods have been advocated for implementation; however, generalization to more than one time-dependent covariate may become prohibitive. For a single time-dependent covariate, Tsiatis and Davidian (2001) have proposed an approach that is easily implemented and does not require an assumption on the distribution of the random effects. This technique may be generalized to multiple, possibly correlated, time-dependent covariates, as we demonstrate. We illustrate the approach via simulation and by application to data from an HIV clinical trial.     

Approximate nonparametric corrected-score method for joint modeling of survival and longitudinal data measured with error

We consider the problem of jointly modeling survival time and longitudinal data subject to measurement error. The survival times are modeled through the proportional hazards model and a random effects model is assumed for the longitudinal covariate process. Under this framework, we propose an approximate nonparametric corrected-score estimator for the parameter, which describes the association between the time-to-event and the longitudinal covariate. The term nonparametric refers to the fact that assumptions regarding the distribution of the random effects and that of the measurement error are unnecessary. The finite sample size performance of the approximate nonparametric corrected-score estimator is examined through simulation studies and its asymptotic properties are also developed. Furthermore, the proposed estimator and some existing estimators are applied to real data from an AIDS clinical trial.     

Estimation and interpretation of incidence rate difference for recurrent events when the estimation model is misspecified

Recurrent events data are common in experimental and observational studies. It is often of interest to estimate the effect of an intervention on the incidence rate of the recurrent events. The incidence rate difference is a useful measure of intervention effect. A weighted least squares estimator of the incidence rate difference for recurrent events was recently proposed for an additive rate model in which both the baseline incidence rate and the covariate effects were constant over time. In this article, we relax this model assumption and examine the properties of the estimator under the additive and multiplicative rate models assumption in which the baseline incidence rate and covariate effects may vary over time. We show analytically and numerically that the estimator gives an appropriate summary measure of the time‐varying covariate effects. In particular, when the underlying covariate effects are additive and time‐varying, the estimator consistently estimates the weighted average of the covariate effects over time. When the underlying covariate effects are multiplicative and time‐varying, and if there is only one binary covariate indicating the intervention status, the estimator consistently estimates the weighted average of the underlying incidence rate difference between the intervention and control groups over time. We illustrate the method with data from a randomized vaccine trial.     

Applying the Cox proportional hazards model when the change time of a binary time-varying covariate is interval censored

This paper develops methodology for estimation of the effect of a binary time-varying covariate on failure times when the change time of the covariate is interval censored. The motivating example is a study of cytomegalovirus (CMV) disease in patients with human immunodeficiency virus (HIV) disease. We are interested in determining whether CMV shedding predicts an increased hazard for developing active CMV disease. Since a clinical screening test is needed to detect CMV shedding, the time that shedding begins is only known to lie in an interval bounded by the patient's last negative and first positive tests. In a Cox proportional hazards model with a time-varying covariate for CMV shedding, the partial likelihood depends on the covariate status of every individual in the risk set at each failure time. Due to interval censoring, this is not always known. To solve this problem, we use a Monte Carlo EM algorithm with a Gibbs sampler embedded in the E-step. We generate multiple completed data sets by drawing imputed exact shedding times based on the joint likelihood of the shedding times and event times under the Cox model. The method is evaluated using a simulation study and is applied to the data set described above.     

Semiparametric approaches for joint modeling of longitudinal and survival data with time-varying coefficients

Summary .   We study joint modeling of survival and longitudinal data. There are two regression models of interest. The primary model is for survival outcomes, which are assumed to follow a time-varying coefficient proportional hazards model. The second model is for longitudinal data, which are assumed to follow a random effects model. Based on the trajectory of a subject's longitudinal data, some covariates in the survival model are functions of the unobserved random effects. Estimated random effects are generally different from the unobserved random effects and hence this leads to covariate measurement error. To deal with covariate measurement error, we propose a local corrected score estimator and a local conditional score estimator. Both approaches are semiparametric methods in the sense that there is no distributional assumption needed for the underlying true covariates. The estimators are shown to be consistent and asymptotically normal. However, simulation studies indicate that the conditional score estimator outperforms the corrected score estimator for finite samples, especially in the case of relatively large measurement error. The approaches are demonstrated by an application to data from an HIV clinical trial.     

Variable selection for nonparametric additive Cox model with interval-censored data

The standard Cox model is perhaps the most commonly used model for regression analysis of failure time data but it has some limitations such as the assumption on linear covariate effects. To relax this, the nonparametric additive Cox model, which allows for nonlinear covariate effects, is often employed, and this paper will discuss variable selection and structure estimation for this general model. For the problem, we propose a penalized sieve maximum likelihood approach with the use of Bernstein polynomials approximation and group penalization. To implement the proposed method, an efficient group coordinate descent algorithm is developed and can be easily carried out for both low- and high-dimensional scenarios. Furthermore, a simulation study is performed to assess the performance of the presented approach and suggests that it works well in practice. The proposed method is applied to an Alzheimer's disease study for identifying important and relevant genetic factors.     

Validation of the alternating conditional estimation algorithm for estimation of flexible extensions of Cox's proportional hazards model with nonlinear constraints on the parameters

Standard optimization algorithms for maximizing likelihood may not be applicable to the estimation of those flexible multivariable models that are nonlinear in their parameters. For applications where the model's structure permits separating estimation of mutually exclusive subsets of parameters into distinct steps, we propose the alternating conditional estimation (ACE) algorithm. We validate the algorithm, in simulations, for estimation of two flexible extensions of Cox's proportional hazards model where the standard maximum partial likelihood estimation does not apply, with simultaneous modeling of (1) nonlinear and time‐dependent effects of continuous covariates on the hazard, and (2) nonlinear interaction and main effects of the same variable. We also apply the algorithm in real‐life analyses to estimate nonlinear and time‐dependent effects of prognostic factors for mortality in colon cancer. Analyses of both simulated and real‐life data illustrate good statistical properties of the ACE algorithm and its ability to yield new potentially useful insights about the data structure.     

Smoothing spline-based score tests for proportional hazards models

We propose "score-type" tests for the proportional hazards assumption and for covariate effects in the Cox model using the natural smoothing spline representation of the corresponding nonparametric functions of time or covariate. The tests are based on the penalized partial likelihood and are derived by viewing the inverse of the smoothing parameter as a variance component and testing an equivalent null hypothesis that the variance component is zero. We show that the tests have a size close to the nominal level and good power against general alternatives, and we apply them to data from a cancer clinical trial.     

Cox regression with survival-time-dependent missing covariate values

Analysis with time-to-event data in clinical and epidemiological studies often encounters missing covariate values, and the missing at random assumption is commonly adopted, which assumes that missingness depends on the observed data, including the observed outcome which is the minimum of survival and censoring time. However, it is conceivable that in certain settings, missingness of covariate values is related to the survival time but not to the censoring time. This is especially so when covariate missingness is related to an unmeasured variable affected by the patient's illness and prognosis factors at baseline. If this is the case, then the covariate missingness is not at random as the survival time is censored, and it creates a challenge in data analysis. In this article, we propose an approach to deal with such survival-time-dependent covariate missingness based on the well known Cox proportional hazard model. Our method is based on inverse propensity weighting with the propensity estimated by nonparametric kernel regression. Our estimators are consistent and asymptotically normal, and their finite-sample performance is examined through simulation. An application to a real-data example is included for illustration.     

A nonparametric mixture model for cure rate estimation

Nonparametric methods have attracted less attention than their parametric counterparts for cure rate analysis. In this paper, we study a general nonparametric mixture model. The proportional hazards assumption is employed in modeling the effect of covariates on the failure time of patients who are not cured. The EM algorithm, the marginal likelihood approach, and multiple imputations are employed to estimate parameters of interest in the model. This model extends models and improves estimation methods proposed by other researchers. It also extends Cox's proportional hazards regression model by allowing a proportion of event-free patients and investigating covariate effects on that proportion. The model and its estimation method are investigated by simulations. An application to breast cancer data, including comparisons with previous analyses using a parametric model and an existing nonparametric model by other researchers, confirms the conclusions from the parametric model but not those from the existing nonparametric model.     

Influence of trial duration on the bias of the estimated treatment effect in clinical trials when individual heterogeneity is ignored

In randomized clinical trials where the times to event of two treatment groups are compared under a proportional hazards assumption, it has been established that omitting prognostic factors from the model entails an underestimation of the hazards ratio. Heterogeneity due to unobserved covariates in cancer patient populations is a concern since genomic investigations have revealed molecular and clinical heterogeneity in these populations. In HIV prevention trials, heterogeneity is unavoidable and has been shown to decrease the treatment effect over time. This article assesses the influence of trial duration on the bias of the estimated hazards ratio resulting from omitting covariates from the Cox analysis. The true model is defined by including an unobserved random frailty term in the individual hazard that reflects the omitted covariate. Three frailty distributions are investigated: gamma, log‐normal, and binary, and the asymptotic bias of the hazards ratio estimator is calculated. We show that the attenuation of the treatment effect resulting from unobserved heterogeneity strongly increases with trial duration, especially for continuous frailties that are likely to reflect omitted covariates, as they are often encountered in practice. The possibility of interpreting the long‐term decrease in treatment effects as a bias induced by heterogeneity and trial duration is illustrated by a trial in oncology where adjuvant chemotherapy in stage 1B NSCLC was investigated.     

Model misspecification in proportional hazards regression

The proportional hazards model is frequently used to evaluatethe effect of treatment on failure time events in randomisedclinical trials. Concomitant variables are usually availableand may be considered for use in the primary analyses underthe assumption that incorporating them may reduce bias or improveefficiency. In this paper we consider two approaches to includingcovariate information: regression modelling and stratification.We focus on the setting where covariate effects are nonproportionaland we compare the bias, efficiency and coverage propertiesof these approaches. These results indicate that our intuitionbased on linear model analysis of covariance is misleading.Covariate adjustment in proportional hazards models has littleeffect on the variance but may significantly improve the accuracyof the treatment effect estimator.     

Generalized additive models with interval-censored data and time-varying covariates: application to human immunodeficiency virus infection in hemophiliacs

We describe a method for extending smooth nonparametric modeling methods to time-to-event data where the event may be known only to lie within a window of time. Maximum penalized likelihood is used to fit a discrete proportional hazards model that also models the baseline hazard, and left-truncation and time-varying covariates are accommodated. The implementation follows generalized additive modeling conventions, allowing both parametric and smooth terms and specifying the amount of smoothness in terms of the effective degrees of freedom. We illustrate the method on a well-known interval-censored data set on time of human immunodeficiency virus infection in a multicenter study of hemophiliacs. The ability to examine time-varying covariates, not available with previous methods, allows detection and modeling of nonproportional hazards and use of a time-varying covariate that fits the data better and is more plausible than a fixed alternative.     

Confidence bands for cumulative incidence curves under the additive risk model

In the context of competing risks, the cumulative incidence function is often used to summarize the cause-specific failure-time data. As an alternative to the proportional hazards model, the additive risk model is used to investigate covariate effects by specifying that the subject-specific hazard function is the sum of a baseline hazard function and a regression function of covariates. Based on such a formulation, we present an approach to constructing simultaneous confidence intervals for the cause-specific cumulative incidence function of patients with given risk factors. A melanoma data set is used for the purpose of illustration.     

Estimating a Treatment Effect in Residual Time Quantiles Under the Additive Hazards Model

For randomized clinical trials where the endpoint of interest is a time-to-event subject to censoring, estimating the treatment effect has mostly focused on the hazard ratio from the Cox proportional hazards model. Since the model’s proportional hazards assumption is not always satisfied, a useful alternative, the so-called additive hazards model, may instead be used to estimate a treatment effect on the difference of hazard functions. Still, the hazards difference may be difficult to grasp intuitively, particularly in a clinical setting of, e.g., patient counseling, or resource planning. In this paper, we study the quantiles of a covariate’s conditional survival function in the additive hazards model. Specifically, we estimate the residual time quantiles, i.e., the quantiles of survival times remaining at a given time t, conditional on the survival times greater than t, for a specific covariate in the additive hazards model. We use the estimates to translate the hazards difference into the difference in residual time quantiles, which allows a more direct clinical interpretation. We determine the asymptotic properties, assess the performance via Monte-Carlo simulations, and demonstrate the use of residual time quantiles in two real randomized clinical trials.     

Cox regression model with randomly censored covariates

This paper deals with a Cox proportional hazards regression model, where some covariates of interest are randomly right‐censored. While methods for censored outcomes have become ubiquitous in the literature, methods for censored covariates have thus far received little attention and, for the most part, dealt with the issue of limit‐of‐detection. For randomly censored covariates, an often‐used method is the inefficient complete‐case analysis (CCA) which consists in deleting censored observations in the data analysis. When censoring is not completely independent, the CCA leads to biased and spurious results. Methods for missing covariate data, including type I and type II covariate censoring as well as limit‐of‐detection do not readily apply due to the fundamentally different nature of randomly censored covariates. We develop a novel method for censored covariates using a conditional mean imputation based on either Kaplan–Meier estimates or a Cox proportional hazards model to estimate the effects of these covariates on a time‐to‐event outcome. We evaluate the performance of the proposed method through simulation studies and show that it provides good bias reduction and statistical efficiency. Finally, we illustrate the method using data from the Framingham Heart Study to assess the relationship between offspring and parental age of onset of cardiovascular events.     

Reduced rank hazard regression with fixed and time‐varying effects of the covariates

Modelling survival data from long‐term follow‐up studies presents challenges. The commonly used proportional hazards model should be extended to account for dynamic behaviour of the effects of fixed covariates. This work illustrates the use of reduced rank models in survival data, where some of the covariate effects are allowed to behave dynamically in time and some as fixed. Time‐varying effects of the covariates can be fitted by using interactions of the fixed covariates with flexible transformations of time based on b‐splines. To avoid overfitting, a reduced rank model will restrict the number of parameters, resulting in a more sensible fit to the data. This work presents the basic theory and the algorithm to fit such models. An application to breast cancer data is used for illustration of the suggested methods.     

Goodness-of-fit tests in proportional hazards models with random effects

This paper deals with testing the functional form of the covariate effects in a Cox proportional hazards model with random effects. We assume that the responses are clustered and incomplete due to right censoring. The estimation of the model under the null (parametric covariate effect) and the alternative (nonparametric effect) is performed using the full marginal likelihood. Under the alternative, the nonparametric covariate effects are estimated using orthogonal expansions. The test statistic is the likelihood ratio statistic, and its distribution is approximated using a bootstrap method. The performance of the proposed testing procedure is studied through simulations. The method is also applied on two real data sets one from biomedical research and one from veterinary medicine.