For time-to-event data with finitely many competing risks, the proportional hazards model has been a popular tool for relating the cause-specific outcomes to covariates [Prentice et al. Biometrics 34 (1978) 541--554]. This article studies an extension of this approach to allow a continuum of competing risks, in which the cause of failure is replaced by a continuous mark only observed at the failure time. We develop inference for the proportional hazards model in which the regression parameters depend nonparametrically on the mark and the baseline hazard depends nonparametrically on both time and mark. This work is motivated by the need to assess HIV vaccine efficacy, while taking into account the genetic divergence of infecting HIV viruses in trial participants from the HIV strain that is contained in the vaccine, and adjusting for covariate effects. Mark-specific vaccine efficacy is expressed in terms of one of the regression functions in the mark-specific proportional hazards model. The new approach is evaluated in simulations and applied to the first HIV vaccine efficacy trial.
Three statistical studies, all published between 2004 and 2008 but without referring to one another, assert a useful equivalence involving the hazard ratio, a parameter estimated for time to event data by the frequently used proportional hazards model. Stated generally, when the assumption of proportional hazards is justified, the hazard ratio is equivalent to the odds that a randomly chosen person from the group whose hazard is represented in the ratio's numerator will experience the event of interest before a randomly chosen person from the group represented by the ratio's denominator.
Maximum approximate Bernstein likelihood estimates of the baseline density function and the regression coefficients in the proportional hazard regression models based on interval-censored event time data are proposed. This results in not only a smooth estimate of the survival function which enjoys faster convergence rate but also improved estimates of the regression coefficients. Simulation shows that the finite sample performance of the proposed method is better than the existing ones. The proposed method is illustrated by real data applications.
This work presents novel and powerful tests for comparing non-proportional hazard functions, based on sample-space partitions. Right censoring introduces two major difficulties which make the existing sample-space partition tests for uncensored data non-applicable: (i) the actual event times of censored observations are unknown; and (ii) the standard permutation procedure is invalid in case the censoring distributions of the groups are unequal. We overcome these two obstacles, introduce invariant tests, and prove their consistency. Extensive simulations reveal that under non-proportional alter…
The Cox regression, a semi-parametric method of survival analysis, is extremely popular in biomedical applications. The proportional hazards assumption is a key requirement in the Cox model. To accommodate non-proportional hazards, we propose to parameterise the shape parameter of the baseline hazard function using the additional, separate Cox-regression term which depends on the vector of the covariates. We call this model the double-Cox model. The R programs for fitting the double-Cox model are available on Github. We formally introduce the double-Cox model with shared frailty and investigat…
The classical approach to analyze time-to-event data, e.g. in clinical trials, is to fit Kaplan-Meier curves yielding the treatment effect as the hazard ratio between treatment groups. Afterwards commonly a log-rank test is performed in order to investigate whether there is a difference in survival, or, depending on additional covariates, a Cox proportional hazard model is used. However, in numerous trials these approaches fail due to the presence of non-proportional hazards, resulting in difficulties of interpreting the hazard ratio and a loss of power. When considering equivalence or non-inf…
Loss of power and clear description of treatment differences are key issues in designing and analyzing a clinical trial where non-proportional hazard is a possibility. A log-rank test may be very inefficient and interpretation of the hazard ratio estimated using Cox regression is potentially problematic. In this case, the current ICH E9 (R1) addendum would suggest designing a trial with a clinically relevant estimand, e.g., expected life gain. This approach considers appropriate analysis methods for supporting the chosen estimand. However, such an approach is case specific and may suffer lack …
Latent class analysis (LCA) is a useful tool to investigate the heterogeneity of a disease population with time-to-event data. We propose a new method based on non-parametric maximum likelihood estimator (NPMLE), which facilitates theoretically validated inference procedure for covariate effects and cumulative hazard functions. We assess the proposed method via extensive simulation studies and demonstrate improved predictive performance over standard Cox regression model. We further illustrate the practical utility of the proposed method through an application to a mild cognitive impairment (M…
We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the …
Combining patient-level data from clinical trials can connect rare phenomena with clinical endpoints, but statistical techniques applied to a single trial may become problematical when trials are pooled. Estimating the hazard of a binary variable unevenly distributed across trials showcases a common pooled database issue. We studied how an unevenly distributed binary variable can compromise the integrity of fixed and random effects Cox proportional hazards models. We compared fixed effect and random effects Cox proportional hazards models on a set of simulated datasets inspired by a 17-trial p…
Recent research has shown the potential for neural networks to improve upon classical survival models such as the Cox model, which is widely used in clinical practice. Neural networks, however, typically rely on data that are centrally available, whereas healthcare data are frequently held in secure silos. We present a federated Cox model that accommodates this data setting and also relaxes the proportional hazards assumption, allowing time-varying covariate effects. In this latter respect, our model does not require explicit specification of the time-varying effects, reducing upfront organisa…
In two harmonized efficacy studies to prevent HIV infection through multiple infusions of the monoclonal antibody VRC01, a key objective is to evaluate whether the serum concentration of VRC01, which changes cyclically over time along with the infusion schedule, is associated with the rate of HIV infection. Simulation studies are needed in the development of such survival models. In this paper, we consider simulating event time data with a continuous time-varying covariate whose values vary with time through multiple drug administration cycles, and whose effect on survival changes differently …
Antibodies, an essential part of our immune system, develop through an intricate process to bind a wide array of pathogens. This process involves randomly mutating DNA sequences encoding these antibodies to find variants with improved binding, though mutations are not distributed uniformly across sequence sites. Immunologists observe this nonuniformity to be consistent with "mutation motifs", which are short DNA subsequences that affect how likely a given site is to experience a mutation. Quantifying the effect of motifs on mutation rates is challenging: a large number of possible motifs makes…
In this paper, our proposal consists of incorporating frailty into a statistical methodology for modeling time-to-event data, based on non-proportional hazards regression model. Specifically, we use the generalized time-dependent logistic (GTDL) model with a frailty term introduced in the hazard function to control for unobservable heterogeneity among the sampling units. We also add a regression in the parameter that measures the effect of time, since it can directly reflect the influence of covariates on the effect of time-to-failure. The practical relevance of the proposed model is illustrat…
Adaptive weight selection for time-to-event data under non-proportional hazards
Institute of Biostatistics and Clinical Research, University of Münster（明斯特大学生物统计与临床研究研究所）；Department of Statistics, TU Dortmund University（多特蒙德工业大学统计系）
CoxSE: Exploring the Potential of Self-Explaining Neural Networks with Cox Proportional Hazards Model for Survival Analysis
Center for Applied Intelligent Systems Research (CAISR)（应用智能系统研究中心）；Halmstad University（哈马斯特德大学）；Centre for Environmental and Climate Science（环境与气候科学中心）；Lund University（吕勒奥大学）；Research Institutes of Sweden (RISE)（瑞典研究机构）
School of Electrical and Computer Engineering, Purdue University（普渡大学电气与计算机工程学院）；Department of Medicine, Indiana University School of Medicine（印第安纳大学医学院医学系）；School of Electrical and Computer Engineering, Indiana University - Purdue University Indianapolis（印第安纳大学-普渡大学印第安纳波利斯分校电气与计算机工程学院）
One of the central goals in precision health is the understanding and interpretation of high-dimensional biological data to identify genes and markers associated with disease initiation, development, and outcomes. Though significant effort has been committed to harness gene expression data for multiple analyses while accounting for time-to-event modeling by including survival times, many traditional analyses have focused separately on non-negative matrix factorization (NMF) of the gene expression data matrix and survival regression with Cox proportional hazards model. In this work, Cox proport…
One-sample survival tests in the presence of non-proportional hazards in oncology clinical trials
Oncostat, CESP, INSERM U1018（INSERM U1018）；University Paris-Saclay（巴黎萨克雷大学）；Department of Statistics and Acturial Science（统计与精算学系）；University of Hong Kong（香港大学）
The proportional hazards (PH) model is arguably one of the most popular models used to analyze time to event data arising from clinical trials and longitudinal studies, among many others. In many such studies, the event time of interest is not directly observed but is known relative to periodic examination times; i.e., practitioners observe either current status or interval-censored data. The analysis of data of this structure is often fraught with many difficulties. Further exacerbating this issue, in some such studies the observed data also consists of instantaneous failures; i.e., the event…
The lifetimes of subjects which are left-censored lie below a threshold value or a limit of detection. A popular tool used to handle left-censored data is the reversed hazard rate. In this work, we study the properties and develop characterizations of a class of distributions based on proportional reversed hazard rates used for analyzing left censored data. These characterizations are applied to simulate samples as well as analyze real data using distributions belonging to this class.
Non-proportional hazards (NPH) have been observed recently in many immuno-oncology clinical trials. Weighted log-rank tests (WLRT) with suitably chosen weights can be used to improve the power of detecting the difference of the two survival curves in the presence of NPH. However, it is not easy to choose a proper WLRT in practice when both robustness and efficiency are considered. A versatile maxcombo test was proposed to achieve the balance of robustness and efficiency and has received increasing attentions in both methodology development and application. However, survival trials often warran…
The proportional hazards model represents the most commonly assumed hazard structure when analysing time to event data using regression models. We study a general hazard structure which contains, as particular cases, proportional hazards, accelerated hazards, and accelerated failure time structures, as well as combinations of these. We propose an approach to apply these different hazard structures, based on a flexible parametric distribution (Exponentiated Weibull) for the baseline hazard. This distribution allows us to cover the basic hazard shapes of interest in practice: constant, bathtub, …
Comparison of the Cox proportional hazards model and Random Survival Forest algorithm for predicting patient-specific survival probabilities in clinical trial data
University of Augsburg（奥格斯堡大学）；University of Reading（雷丁大学）
The Cox model, which remains as the first choice in analyzing time-to-event data even for large datasets, relies on the proportional hazards (PH) assumption. When survival data arrive sequentially in chunks, a fast and minimally storage intensive approach to test the PH assumption is desirable. We propose an online updating approach that updates the standard test statistic as each new block of data becomes available, and greatly lightens the computational burden. Under the null hypothesis of PH, the proposed statistic is shown to have the same asymptotic distribution as the standard version co…
Understanding the factors that explain differences in survival times is an important issue for establishing policies to improve national health systems. Motivated by breast cancer data arising from the Surveillance Epidemiology and End Results program, we propose a covariate-adjusted proportional hazards frailty model for the analysis of clustered right-censored data. Rather than incorporating exchangeable frailties in the linear predictor of commonly-used survival models, we allow the frailty distribution to flexibly change with both continuous and categorical cluster-level covariates and mod…
An individualized risk prediction model that dynamically updates the probability of a clinical event from a specific cause is valuable for physicians to be able to optimize personalized treatment strategies in real-time by incorporating all available information collected over the follow-up. However, this is more complex and challenging when competing risks are present, because it requires simultaneously updating the overall survival and the cumulative incidence functions (CIFs) while adjusting for the time-dependent covariates and time-varying covariate effects. In this study, we developed a …
Medical practitioners use survival models to explore and understand the relationships between patients' covariates (e.g. clinical and genetic features) and the effectiveness of various treatment options. Standard survival models like the linear Cox proportional hazards model require extensive feature engineering or prior medical knowledge to model treatment interaction at an individual level. While nonlinear survival methods, such as neural networks and survival forests, can inherently model these high-level interaction terms, they have yet to be shown as effective treatment recommender system…
The log-rank test is most powerful under proportional hazards (PH). In practice, non-PH patterns are often observed in clinical trials, such as in immuno-oncology; therefore, alternative methods are needed to restore the efficiency of statistical testing. Three categories of testing methods were evaluated, including weighted log-rank tests, Kaplan-Meier curve-based tests (including weighted Kaplan-Meier and Restricted Mean Survival Time, RMST), and combination tests (including Breslow test, Lee's combo test, and MaxCombo test). Nine scenarios representing the PH and various non-PH patterns wer…
IMPORTANCE: Time-to-event outcomes are commonly used in clinical trials and biomarker discovery studies and have been primarily analyzed using Cox proportional hazards models. But it's unclear which statistical models should be recommended for feature selection tasks when time-to-event outcomes are of the primary interest. OBJECTIVE: To explore if Gaussian regression of log-transformed survival time could outperform Cox proportional hazards models in feature selection. DESIGN: In this simulation study, the true models are multivariate Cox proportional hazards models with 10 covariates. For all…
While analysing time-to-event data, it is possible that a certain fraction of subjects will never experience the event of interest and they are said to be cured. When this feature of survival models is taken into account, the models are commonly referred to as cure models. In the presence of covariates, the conditional survival function of the population can be modelled by using cure model which depends on the probability of being uncured (incidence) and the conditional survival function of the uncured subjects (latency), and a combination of logistic regression and Cox proportional hazards (P…
Studies to compare the survival of two or more groups using time-to-event data are of high importance in medical research. The gold standard is the log-rank test, which is optimal under proportional hazards. As the latter is no simple regularity assumption, we are interested in evaluating the power of various statistical tests under different settings including proportional and non-proportional hazards with a special emphasize on crossing hazards. This challenge has been going on for many years now and multiple methods have already been investigated in extensive simulation studies. However, in…
We develop a functional proportional hazards mixture cure (FPHMC) model with scalar and functional covariates measured at the baseline. The mixture cure model, useful in studying populations with a cure fraction of a particular event of interest is extended to functional data. We employ the EM algorithm and develop a semiparametric penalized spline-based approach to estimate the dynamic functional coefficients of the incidence and the latency part. The proposed method is computationally efficient and simultaneously incorporates smoothness in the estimated functional coefficients via roughness …
In this manuscript, we study stochastic comparisons of the second-order statistics from dependent or independent observations with modified proportional hazard rates models. First, we establish the usual stochastic order of the second-order statistics from dependent and heterogeneous observations. Second, sufficient conditions are provided in the hazard rate order of the second-order statistics from independent observations. Then, we investigate the hazard rate order of the second-order statistics arising from two sets of independent multiple-outlier modified proportional hazard rates observat…
We introduce new methods of analysing time to event data via extended versions of the proportional hazards and accelerated failure time (AFT) models. In many time to event studies, the time of first observation is arbitrary, in the sense that no risk modifying event occurs. This is particularly common in epidemiological studies. We show formally that, in these situations, it is not sensible to take the first observation as the time origin, either in AFT or proportional hazards type models. Instead, we advocate using age of the subject as the time scale. We account for the fact that baseline ob…
Time-to-event analysis often relies on prior parametric assumptions, or, if a non-parametric approach is chosen, Cox's model. This is inherently tied to the assumption of proportional hazards, with the analysis potentially invalidated if this assumption is not fulfilled. In addition, most interpretations focus on the hazard ratio, that is often misinterpreted as the relative risk. In this paper, we introduce an alternative to current methodology for assessing a treatment effect in a two-group situation, not relying on the proportional hazards assumption but assuming proportional risks. Precise…
This paper deals with the proportional hazards model proposed by D. R. Cox in a high-dimensional and sparse setting for a regression parameter. To estimate the regression parameter, the Dantzig selector is applied. The variable selection consistency of the Dantzig selector for the model will be proved. This property enables us to reduce the dimension of the parameter and to construct asymptotically normal estimators for the regression parameter and the cumulative baseline hazard function.
We study large sample properties of Bayesian analysis of the proportional hazard model with neutral to the right process priors on the baseline hazard function. We show that the posterior distribution of the baseline cumulative hazard function and regression coefficients centered at the maximum likelihood estimator is jointly asymptotically equivalent to the sampling distribution of the maximum likelihood estimator.
Semi-parametric survival analysis methods like the Cox Proportional Hazards (CPH) regression (Cox, 1972) are a popular approach for survival analysis. These methods involve fitting of the log-proportional hazard as a function of the covariates and are convenient as they do not require estimation of the baseline hazard rate. Recent approaches have involved learning non-linear representations of the input covariates and demonstrate improved performance. In this paper we argue against such deep parameterizations for survival analysis and experimentally demonstrate that more interpretable semi-par…
Single-index Semiparametric Transformation Cure Models with Interval-censored Data
School of Economics, Jinan University（暨南大学经济学院）；International Center for Interdisciplinary Statistics, School of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院交叉统计国际中心）
