N. Balakrishnan, E. Castilla, N. Martin, L. Pardo
In this paper, we develop robust estimators and tests for one-shot device testing under proportional hazards assumption based on divergence measures. Through a detailed Monte Carlo simulation study and a numerical example, the developed inferential procedures are shown to be more robust than the classical procedures, based on maximum likelihood estimators.
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Efficient and Debiased Learning of Average Hazard Under Non-Proportional Hazards
本文提出了一种半参数双重鲁棒框架，用于高效且去偏地估计非比例危害下的平均危害，解决了传统方法在高维干扰估计和sqrt{n}推断中的不足。
The semi-parametric Cox proportional hazards regression model has been widely used for many years in several applied sciences. However, a fully parametric proportional hazards model, if appropriately assumed, can often lead to more efficient inference. To tackle the extreme non-robustness of the traditional maximum likelihood estimator in the presence of outliers in the data under such fully parametric proportional hazard models, a robust estimation procedure has recently been proposed extending the concept of the minimum density power divergence estimator (MDPDE) under this set-up. In this pa…
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…
A New Class of Mark-Specific Proportional Hazards Models for Recurrent Events: Application to Opioid Refills Among Post-Surgical Patients
针对术后阿片类药物续配问题，提出一类将剂量作为连续变化标记的复发事件特定标记比例风险模型，采用双重加权边际模型并建立渐近性质，应用于真实数据揭示了影响不同剂量续配风险的因素，为个性化疼痛管理提供参考。
This paper proposes a decorrelation-based approach to test hypotheses and construct confidence intervals for the low dimensional component of high dimensional proportional hazards models. Motivated by the geometric projection principle, we propose new decorrelated score, Wald and partial likelihood ratio statistics. Without assuming model selection consistency, we prove the asymptotic normality of these test statistics, establish their semiparametric optimality. We also develop new procedures for constructing pointwise confidence intervals for the baseline hazard function and baseline survival…
Smooth backfitting has proven to have a number of theoretical and practical advantages in structured regression. Smooth backfitting projects the data down onto the structured space of interest providing a direct link between data and estimator. This paper introduces the ideas of smooth backfitting to survival analysis in a proportional hazard model, where we assume an underlying conditional hazard with multiplicative components. We develop asymptotic theory for the estimator and we use the smooth backfitter in a practical application, where we extend recent advances of in-sample forecasting me…
Cox proportional hazards model with measurement error is investigated. In Kukush et al. (2011) [Journal of Statistical Research 45, 77-94] and Chimisov and Kukush (2014) [Modern Stochastics: Theory and Applications 1, 13-32] asymptotic properties of simultaneous estimator $λ_n(\cdot)$, $β_n$ were studied for baseline hazard rate $λ(\cdot)$ and regression parameter $β$, at that the parameter set $Θ=Θ_λ\times Θ_β$ was assumed bounded. In the present paper, the set $Θ_λ$ is unbounded from above and not separated away from $0$. We construct the estimator in two steps: first we derive a strongly co…
Many insurance premium principles are defined and various estimation procedures introduced in the literature. In this paper, we focus on the estimation of the excess-of-loss reinsurance premium when the risks are randomly right-censored. The asymptotic normality of the proposed estimator is established under suitable conditions and its performance evaluated through sets of simulated data.
The use of massive survival data has become common in survival analysis. In this study, a subsampling algorithm is proposed for the Cox proportional hazards model with time-dependent covariates when the sample is extraordinarily large but computing resources are relatively limited. A subsample estimator is developed by maximizing the weighted partial likelihood; it is shown to have consistency and asymptotic normality. By minimizing the asymptotic mean squared error of the subsample estimator, the optimal subsampling probabilities are formulated with explicit expressions. Simulation studies sh…
We investigate nonparametric estimation of a monotone baseline hazard and a decreasing baseline density within the Cox model. Two estimators of a nondecreasing baseline hazard function are proposed. We derive the nonparametric maximum likelihood estimator and consider a Grenander type estimator, defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. We demonstrate that the two estimators are strong consistent and asymptotically equivalent and derive their common limit distribution at a fixed point. Both estimators of a nonincreasing baseline hazard and their a…
In this paper we propose a Multiple kernel testing procedure to infer survival data when several factors (e.g. different treatment groups, gender, medical history) and their interaction are of interest simultaneously. Our method is able to deal with complex data and can be seen as an alternative to the omnipresent Cox model when assumptions such as proportionality cannot be justified. Our methodology combines well-known concepts from Survival Analysis, Machine Learning and Multiple Testing: differently weighted log-rank tests, kernel methods and multiple contrast tests. By that, complex hazard…
The proportional hazards assumption in the commonly used Cox model for censored failure time data is often violated in scientific studies. Yang and Prentice (2005) proposed a novel semiparametric two-sample model that includes the proportional hazards model and the proportional odds model as sub-models, and accommodates crossing survival curves. The model leaves the baseline hazard unspecified and the two model parameters can be interpreted as the short-term and long-term hazard ratios. Inference procedures were developed based on a pseudo score approach. Although extension to accommodate cova…
Robust inference for an interval-monitored step-stress experiment under proportional hazards
本文针对比例风险下的区间监测步进应力加速寿命试验，提出稳健的最小密度幂散度估计器族，用于估计可靠性和寿命特征，并通过模拟验证其稳健性与效率。
For statistical inference on regression models with a diverging number of covariates, the existing literature typically makes sparsity assumptions on the inverse of the Fisher information matrix. Such assumptions, however, are often violated under Cox proportion hazards models, leading to biased estimates with under-coverage confidence intervals. We propose a modified debiased lasso approach, which solves a series of quadratic programming problems to approximate the inverse information matrix without posing sparse matrix assumptions. We establish asymptotic results for the estimated regression…
Let $X_1, \ldots , X_n$ be mutually independent exponential random variables with distinct hazard rates $λ_1, \ldots , λ_n > 0$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lmd = \sum_{i=1}^n \lmd_i/n$. Also let $X_{1:n} < \cdots < X_{n:n}$ and $Y_{1:n} < \cdots < Y_{n:n}$ be their associated order statistics. It is shown that for $1\le i <j \le n$, the generalized spacing $X_{j:\, n} - X_{i:\, n}$ is more dispersed than $Y_{j:\,n} - Y_{i:\, n}$ according to dispersive ordering. This result is used to solve a long standing open problem…
Thomas' partial likelihood estimator of regression parameters is widely used in the analysis of nested case-control data with Cox's model. This paper proposes a new estimator of the regression parameters, which is consistent and asymptotically normal. Its asymptotic variance is smaller than that of Thomas' estimator away from the null. Unlike some other existing estimators, the proposed estimator does not rely on any more data than strictly necessary for Thomas' estimator and is easily computable from a closed form estimating equation with a unique solution. The variance estimation is obtained…
Measuring Inaccuracies in the Proportional Hazard Rate Model based on Extropy using a Length-Biased Weighted Residual approach
本文提出基于Extropy的加权残差不精确性度量，研究其在比例风险率模型中的性质、判别准则与不等式，并给出非参数估计及模拟和实例验证。
Severe Ionosphere magnetic storms are feared events for integrity and continuity of navigation systems such as EGNOS, the European SBAS (Satellite-Based Augmentation System) complementing GPS and an accurate modelling of this event probability is necessary. Our aim for the work presented in this paper is to give an estimation of the frequency of such extreme magnetic storms per time unit (year) throughout a solar cycle. Thus, we develop an innovative approach based on a proportional hazard model, inspired by the Cox model, with time dependent covariates. The number of storms during a cycle is …
Estimating customer lifetime value (CLV or LTV) is extremely important for making better business decisions. The proposed flexible proportional hazards model allows an estimation of lifetime value in contractual settings. This approach takes advantage of a churn model, which is assumed to be available.
Stationary Proportional Hazard Processes via Complementary Power Function Distribution Processes
该研究通过对互补幂函数分布过程做边际变换提出新的平稳比例风险过程，还推导了两种新Pareto过程，分析其分布特性与推断方法，并给出实际数据应用示例。
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 …
Proportional hazards are a common assumption when designing confirmatory clinical trials in oncology. With the emergence of immunotherapy and novel targeted therapies, departure from the proportional hazard assumption is not rare in nowadays clinical research. Under non-proportional hazards, the hazard ratio does not have a straightforward clinical interpretation, and the log-rank test is no longer the most powerful statistical test even though it is still valid. Nevertheless, the log-rank test and the hazard ratio are still the primary analysis tools, and traditional approaches such as sample…
Cox's proportional hazards model is one of the most popular statistical models to evaluate associations of exposure with a censored failure time outcome. When confounding factors are not fully observed, the exposure hazard ratio estimated using a Cox model is subject to unmeasured confounding bias. To address this, we propose a novel approach for the identification and estimation of the causal hazard ratio in the presence of unmeasured confounding factors. Our approach is based on a binary instrumental variable, and an additional no-interaction assumption in a first stage regression of the tre…
As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables $T_{1},...,T_{r}$. In any case, we assume that $T_{1},...,T_{r}$ are identically distributed, with a common survival function $\overline{G}$ and their survival copula is denoted by $K$. The diagonal's and subdiagonals' sections of $K$, along with $\overline{G}$, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the a…
In engineering systems, it is usually assumed that lifetimes of components are independent and identically distributed (iid). But, the failure of a component results in a higher load on the remaining components and hence causes the distribution of the surviving components change. For modeling this kind of systems, the theory of sequential order statistics (SOS) can be used. Assuming Weibull distribution for lifetimes of components and conditionally proportional hazard rates model as a special case of the SOS theory, the maximum likelihood estimates of the unknown parameters are obtained in dif…
In this paper, we introduce a new extension of the generalized linear failure rate distributions. It includes some well-known lifetime distributions such as extension of generalized exponential and generalized linear failure rate distributions as special sub-models. In addition, it can have a constant, decreasing, increasing, upside-down bathtub (unimodal), and bathtub-shaped hazard rate function depending on its parameters. We provide some of its statistical properties such as moments, quantiles, skewness, kurtosis, hazard rate function, and reversible hazard rate function. The maximum likeli…
A Review of EMA Public Assessment Reports where Non-Proportional Hazards were Identified
本文综述 16 项 EMA 上市许可程序及 18 项试验，分析非比例风险的来源、处理与分析方法，并提取和数字化生存数据以支持后续模拟研究。
A fundamental concept in two-arm non-parametric survival analysis is the comparison of observed versus expected numbers of events on one of the treatment arms (the choice of which arm is arbitrary), where the expectation is taken assuming that the true survival curves in the two arms are identical. This concept is at the heart of the counting-process theory that provides a rigorous basis for methods such as the log-rank test. It is natural, therefore, to maintain this perspective when extending the log-rank test to deal with non-proportional hazards, for example by considering a weighted sum o…
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…
One of the most common ways researchers compare survival outcomes across treatments when confounding is present is using Cox regression. This model is limited by its underlying assumption of proportional hazards; in some cases, substantial violations may occur. Here we present and compare approaches which attempt to address this issue, including Cox models with time-varying hazard ratios; parametric accelerated failure time models; Kaplan-Meier curves; and pseudo-observations. To adjust for differences between treatment groups, we use Inverse Probability of Treatment Weighting based on the pro…
To better understand the interplay of censoring and sparsity we develop finite sample properties of nonparametric Cox proportional hazard's model. Due to high impact of sequencing data, carrying genetic information of each individual, we work with over-parametrized problem and propose general class of group penalties suitable for sparse structured variable selection and estimation. Novel non-asymptotic sandwich bounds for the partial likelihood are developed. We establish how they extend notion of local asymptotic normality (LAN) of Le Cam's. Such non-asymptotic LAN principles are further exte…
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…
We use a discrete-time proportional hazards model of time to involuntary employment termination. This model enables us to examine both the continuous effect of the age of an employee and whether that effect has varied over time, generalizing earlier work [Kadane and Woodworth J. Bus. Econom. Statist. 22 (2004) 182--193]. We model the log hazard surface (over age and time) as a thin-plate spline, a Bayesian smoothness-prior implementation of penalized likelihood methods of surface-fitting [Wahba (1990) Spline Models for Observational Data. SIAM]. The nonlinear component of the surface has only …
CauchyCP: a powerful test under non-proportional hazards using Cauchy combination of change-point Cox regressions
针对非比例风险数据，提出CauchyCP检验，结合变点Cox回归的Cauchy组合，在控制I类错误、提高功效和计算效率上优于现有方法，并已在CRAN发布R包。
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…
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…
Assuming some regression model, it is common to study the conditional distribution of survival given covariates. Here, we consider the impact of further conditioning, specifically conditioning on a marginal survival function, known or estimated. We investigate to what purposes any such information can be used in a proportional or non-proportional hazards regression analysis of time on the covariates. It does not lead to any improvement in efficiency when the form of the assumed proportional hazards model is valid. However, when the proportional hazards model is not valid, the usual partial lik…
Methods for non-proportional hazards in clinical trials: A systematic review
本研究针对临床试验非比例风险分析方法缺乏最优共识的问题，通过系统检索纳入211篇文献，梳理了相关统计方法与软件，提炼了应用指导，为该领域方法选择提供参考。
On the Proportional Principal Stratum Hazards Model
针对临床试验中死亡竞争风险下无法准确估计治疗对非致死事件直接效应的问题，提出比例主层风险模型，通过共享脆弱模型识别主层归属，经模拟验证可靠性并在心力衰竭临床试验中完成演示。
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…
\noindent Randomized nomination sampling (RNS) is a rank-based sampling technique which has been shown to be effective in several nonparametric studies involving environmental and ecological applications. In this paper, we investigate parametric inference using RNS design for estimating the unknown vector of parameters $\boldsymbolθ$ in the proportional hazard rate and proportional reverse hazard rate models. We examine both maximum likelihood (ML) and method of moments (MM) methods and investigate the relative precision of our proposed RNS-based estimators compared with those based on simple …
Analyzing outcomes in long-term cancer survivor studies can be complex. The effects of predictors on the failure process may be difficult to assess over longer periods of time, as the commonly used assumption of proportionality of hazards holding over an extended period is often questionable. In this manuscript, we compare seven different survival models that estimate the hazard rate and the effects of proportional and non-proportional covariates. In particular, we focus on an extension of the the multi-resolution hazard (MRH) estimator, combining a non-proportional hierarchical MRH approach w…
One-sample survival tests in the presence of non-proportional hazards in oncology clinical trials
本文提出了一种结合分段指数模型和加速危害模型的生存检验方法，用于评估单臂试验中治疗效果，通过模拟研究验证了其在不同场景下的优越性。
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, …
Statistical Considerations for Evaluating Treatment Effect under Various Non-proportional Hazard Scenarios
本研究系统性比较了Log-rank检验与MaxCombo、dRMST等方法在比例与非比例风险场景下分析生存结局的性能，通过真实肿瘤试验数据和模拟交叉风险场景评估其效能、I类错误率等，提出各场景下的推荐方案。
Statistical tests for comparing the associations of multiple exposures with a common outcome in Cox proportional hazard models
提出一种基于复制方法的 Cox 模型多变量 Wald 检验，用于正式比较多种暴露与共同结局的关联差异，并通过膳食模式与慢性病实例及 R 代码验证其灵活性。
Bayesian non-parametric specification of bathtub shaped hazard rate functions
本文扩展Lo和Weng方法，提出四种基于Gamma过程先验的非参数浴缸形风险率函数，并实现模拟验证。
In cancer epidemiology using population-based data, regression models for the excess mortality hazard is a useful method to estimate cancer survival and to describe the association between prognosis factors and excess mortality. This method requires expected mortality rates from general population life tables: each cancer patient is assigned an expected (background) mortality rate obtained from the life tables, typically at least according to their age and sex, from the population they belong to. However, those life tables may be insufficiently stratified, as some characteristics such as depri…
Markov Renewal Proportional Hazards is All You Need
针对多状态建模中的转移概率估计问题，将半马尔可夫与马尔可夫更新框架应用于EBMT数据集，对比AJ和DSH估计量，证明半马尔可夫模型更优且DSH估计量更平滑，还证明了其弱收敛性。
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…
Addressing Data Heterogeneity in Federated Learning of Cox Proportional Hazards Models
针对联邦生存分析中Cox比例风险模型的数据异质性问题，提出基于特征聚类和事件报告策略的联邦学习方法，提升模型性能，并在合成数据及SEER数据库上验证有效性。
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 …
We consider the conditional treatment effect for competing risks data in observational studies. While it is described as a constant difference between the hazard functions given the covariates, we do not assume specific functional forms for the covariates. We derive the efficient score for the treatment effect using modern semiparametric theory, as well as two doubly robust scores with respect to 1) the assumed propensity score for treatment and the censoring model, and 2) the outcome models for the competing risks. An important asymptotic result regarding the estimators is rate double robustn…
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…
Privacy enhanced collaborative inference in the Cox proportional hazards model for distributed data
针对多中心分布式生存分析中的数据共享与隐私泄露问题，提出仅交换低维汇总统计量的协作式 Cox 比例风险模型，并用理论、模拟和 OPTN 肾移植数据验证其近集中式性能。
Bayesian variable selection in a Cox proportional hazards model with the "Sum of Single Effects" prior
本文提出CoxPH-SuSiE方法，将SuSiE扩展到Cox比例风险模型，用于解决遗传精细定位中高相关性和大规模数据下的时间-事件结局贝叶斯变量选择问题，并在模拟数据和UK Biobank哮喘数据中验证了其有效性。
Time-to-Event Modeling with Pseudo-Observations in Federated Settings
本文提出了一种基于伪观测的联邦生存分析框架，能够在保护隐私的前提下处理非比例风险假设下的时间到事件数据，通过灵活的链接函数和去偏制程实现稳健的推断和适应性偏差-方差权衡。
A Cosine-Projection Framework for Choosing Among Cox HR, D-RMST, and Average-Hazard Tests in Survival Trials
本文提出余弦投影框架，推导Cox HR、D-RMST、RAH检验的渐近相对效率，揭示删失和入组对效率排序的影响，经CheckMate 057试验和模拟验证，为生存试验检验选择提供统一方法。
lifelines-hc: Higher Criticism testing for sparse non-proportional hazard departures in Python
本文提出lifelines-hc包，实现HCHG检验，用于检测生存分析中稀疏非比例风险偏离，在四个临床数据集上显著优于对数秩检验，且不依赖预设时间模式。
Efficiency of nonparametric superiority tests based on restricted mean survival time versus the log-rank test under proportional hazards
针对比例风险下时间-事件终点RCT的分析效率争议，通过渐近分析与模拟探究事件率和删失分布的联合效应，厘清了RMST非参数检验与log-rank检验的相对效率表现。
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