In survival data analysis, the stratified Cox model becomes a popular option when the proportional hazards assumption of the conventional Cox model does not hold for certain covariates. For a stratified Cox model, when the observed survival times contain only right censoring, the method of maximum partial likelihood can still be implemented. However, if survival times include interval-censored observations, the method of maximum partial likelihood is not viable, and the partial likelihood approach cannot be applied. Furthermore, partial likelihood analysis does not supply a smooth estimate of the baseline hazard. In this paper, we consider the stratified Cox model under partly interval-censored survival times. We present a penalized likelihood method for estimating the model parameters, including the baseline hazards. Penalty functions are used to produce smoothed baseline hazards estimates, and also to relax the requirement on optimal number and location of the knots used in the baseline hazards estimates. We also derive a large sample normality result for the estimates, which can be used to make inferences on quantities of interest, such as survival probabilities, without relying on computing-intensive resampling methods.

In survival analysis, accelerated failure time (AFT) models are often used as an alternative to the Cox model. A benefit of the AFT model is that it provides a direct link between covariates and event times, and therefore offers meaningful interpretations of the regression coefficients directly on the event times. This paper is motivated by a randomised clinical trial dataset on advanced melanoma patients, in which the event times are interval-censored and one of the covariates in the model is time-varying. To allow direct assessment of the effects of covariates on the event time, we aim to fit an AFT model. However, there are no readily available computational packages for AFT models with time-varying covariates and interval censoring. In this paper, we propose a maximum penalised likelihood approach for fitting such models. The effectiveness of our method is demonstrated through extensive simulations. Also, an application to the above mentioned melanoma dataset is used to demonstrate practical utility of our method.

Falls among the elderly represent a significant public health issue, posing substantial risks to their independence and quality of life. Given the aging global population, developing effective fall risk assessment tools is a critical research priority. In this context, plantar pressure sensors have emerged as crucial devices. They provide valuable insights into gait and balance, key indicators of fall risk. This study aims to evaluate the role of these sensors within a hybrid neural network model for fall risk assessment. Utilizing data from 48 older adults characterized by their balance capabilities, we developed a model integrating feature extraction from sensor data and a novel data fusion approach combining a Walking Neuron Model (WNM) and a Multi-Layer Perceptron (MLP). Our results demonstrate that the hybrid model, which classifies individuals into high-risk and low-risk fall categories, significantly outperforms traditional assessment methods. The introduction of novel gait-biometric parameters and optimized feature selection contributed to high accuracy and precision. These findings underscore the utility of plantar pressure sensors in enhancing predictive models, offering substantial theoretical and practical value in preventive healthcare.

Clinical trials often show treatment curves that diverge early and converge later, or vice versa patterns that are poorly captured by the proportional-hazards assumption. We develop a joint inferential framework for two nonparametric functionals of censored survival data: the Kaplan–Meier-based Mann–Whitney effect and a novel temporal contrast separating early and late differences. The approach provides interpretable, probability-scale effect measures and enables joint inference for global and temporal contrasts under right censoring. In simulation studies, the method outperforms the log-rank test under non-proportional hazards while maintaining nominal type-I error. A real-world application illustrates how the temporal contrast reveals clinically meaningful early treatment advantages that remain hidden in standard analyses

Survival analysis can sometimes involve individuals who will not experience the event of interest, forming what is known as the cured group. Identifying such individuals is not always possible beforehand, as they provide only right-censored data. Ignoring the presence of the cured group can introduce bias in the final model. This paper presents a method for estimating a semiparametric additive hazards model that accounts for the cured fraction. Unlike regression coefficients in a hazard ratio model, those in an additive hazard model measure hazard differences. The proposed method uses a primal-dual interior point algorithm to obtain constrained maximum penalized likelihood estimates of the model parameters, including the regression coefficients and the baseline hazard, subject to certain non-negativity constraints.

The cause-specific hazard Cox model is widely used in analyzing competing risks survival data, and the partial likelihood method is a standard approach when survival times contain only right censoring. In practice, however, interval-censored survival times often arise, and this means the partial likelihood method is not directly applicable. Two common remedies in practice are (i) to replace each censoring interval with a single value, such as the middle point; or (ii) to redefine the event of interest, such as the time to diagnosis instead of the time to recurrence of a disease. However, the mid-point approach can cause biased parameter estimates. In this article, we develop a penalized likelihood approach to fit semi-parametric cause-specific hazard Cox models, and this method is general enough to allow left, right, and interval censoring times. Penalty functions are used to regularize the baseline hazard estimates and also to make these estimates less affected by the number and location of knots used for the estimates. We will provide asymptotic properties for the estimated parameters. A simulation study is designed to compare our method with the mid-point partial likelihood approach. We apply our method to the Aspirin in Reducing Events in the Elderly (ASPREE) study, illustrating an application of our proposed method. Keywords Cause-specific Cox model , constrained optimization , penalized likelihood , Gaussian quadrature

The cause-specific hazard Cox model is widely used in analyzing competing risks survival data, and the partial likelihood method is a standard approach when survival times contain only right censoring. In practice, however, interval-censored survival times often arise, and this means the partial likelihood method is not directly applicable. Two common remedies in practice are (i) to replace each censoring interval with a single value, such as the middle point; or (ii) to redefine the event of interest, such as the time to diagnosis instead of the time to recurrence of a disease. However, the mid-point approach can cause biased parameter estimates. In this article, we develop a penalized likelihood approach to fit semi-parametric cause-specific hazard Cox models, and this method is general enough to allow left, right, and interval censoring times. Penalty functions are used to regularize the baseline hazard estimates and also to make these estimates less affected by the number and location of knots used for the estimates. We will provide asymptotic properties for the estimated parameters. A simulation study is designed to compare our method with the mid-point partial likelihood approach. We apply our method to the Aspirin in Reducing Events in the Elderly (ASPREE) study, illustrating an application of our proposed method.

This article considers the joint modeling of longitudinal covariates and partly-interval censored time-to-event data. Longitudinal time-varying covariates play a crucial role in obtaining accurate clinically relevant predictions using a survival regression model. However, these covariates are often measured at limited time points and may be subject to measurement error. Further methodological challenges arise from the fact that, in many clinical studies, the event times of interest are interval-censored. A model that simultaneously accounts for all these factors is expected to improve the accuracy of survival model estimations and predictions. In this article, we consider joint models that combine longitudinal time-varying covariates with the Cox model for time-to-event data which is subject to interval censoring. The proposed model employs a novel penalised likelihood approach for estimating all parameters, including the random effects. The covariance matrix of the estimated parameters can be obtained from the penalised log-likelihood. The performance of the model is compared to an existing method under various scenarios. The simulation results demonstrated that our new method can provide reliable inferences when dealing with interval-censored data. Data from the Anti-PD1 brain collaboration clinical trial in advanced melanoma is used to illustrate the application of the new method.

Time-to-event data in medical studies may involve some patients who are cured and will never experience the event of interest. In practice, those cured patients are right censored. However, when data contain a cured fraction, standard survival methods such as Cox proportional hazards models can produce biased results and therefore misleading interpretations. In addition, for some outcomes, the exact time of an event is not known; instead an interval of time in which the event occurred is recorded. This article proposes a new computational approach that can deal with both the cured fraction issues and the interval censoring challenge. To do so, we extend the traditional mixture cure Cox model to accommodate data with partly interval censoring for the observed event times. The traditional method for estimation of the model parameters is based on the expectation-maximization (EM) algorithm, where the log-likelihood is maximized through an indirect complete data log-likelihood function. We propose in this article an alternative algorithm that directly optimizes the log-likelihood function. Extensive Monte Carlo simulations are conducted to demonstrate the performance of the new method over the EM algorithm. The main advantage of the new algorithm is the generation of asymptotic variance matrices for all the estimated parameters. The new method is applied to a thin melanoma dataset to predict melanoma recurrence. Various inferences, including survival and hazard function plots with point-wise confidence intervals, are presented. An R package is now available at Github and will be uploaded to R CRAN.

This paper introduces the R package ordinalCont, which implements an ordinal regression framework for response variables which are recorded on a visual analogue scale (VAS). This scale is used when recording subjects' perception of an intangible quantity such as pain, anxiety or quality of life, and consists of a mark made on a linear scale. We implement continuous ordinal regression models for VAS as the appropriate method of analysis for such responses, and introduce smoothing terms and random effects in the linear predictor. The model parameters are estimated using constrained optimization of the penalized likelihood and the penalty parameters are automatically selected via maximization of their marginal likelihood. The estimation algorithm is shown to perform well, in a simulation study. Two examples of application are given: the first involves the analysis of pain outcomes in a clinical trial for laser treatment for chronic neck pain; the second is an analysis of quality of life outcomes in a clinical trial for chemotherapy for the treatment of breast cancer.

Credit-granting institutions need to estimate the probability of loan default, which represents the chance a customer fails to make repayments as promised. Critically this estimation is intertwined with the competing risk a customer fully repays their loan while also having key predictive drivers with values that change over time. A conventional model in this setting is a competing risks Cox Model with time-varying covariates. However partial likelihood estimation of this model has two shortcomings: (1) the baseline hazard is not estimated, so calculating probabilities requires a further estimation step; and (2) a covariance matrix for both regression coefficients and the baseline hazard is not produced. This paper caters for these shortcomings by devising a maximum likelihood technique to jointly estimate regression coefficients and the cause-specific baseline hazards using constrained optimisation to ensure the latter’s non-negativity. We show via simulation our technique produces regression coefficients estimates with lower bias in small samples with heavy censoring. When applied to a real-world credit risk dataset consisting of home loan data our Maximum Likelihood approach produces a smoother estimate of the cause-specific baseline hazards for default and redemption than those obtained using the Partial Likelihood and Breslow approach. This provides better clarity of the shape of these functions through both a less volatile central estimate as well as quantifying the error of this central estimate. We implement our method in R.

In survival analysis, the semiparametric accelerated failure time model is an important alternative to the widely used Cox proportional hazard model. The existing methods for accelerated failure time models include least-squares, log rank-based estimating equations and approximations to the nonparametric error distribution. In this paper, we propose another fitting method for the accelerated failure time model, formulated from the hazard function of the exponential error term. Our method can handle partly interval-censored data which contains event time, as well as left, right and interval censoring time. We adopt the maximum penalized likelihood method to estimate all the parameters in the model, including the nonparametric component. The penalty function is used to regularize the nonparametric component of the accelerated failure time model. Asymptotic properties of the penalized likelihood estimate are developed. A simulation study is conducted to investigate the performance of the proposed method and an application of this method to an AIDS study is presented as an example.

考虑不修、最小维修、换件维修和多中间维修水平,提出了一种基于粒子群优化（PSO）算法和多员维修的复杂系统选择性维修模型,将组件维修前状态、组件有效役龄和维修费用等因素引入不完全维修模型,更符合工程实际。提出了一种基于多员维修的系统组件维修分配算法,解决了如何将多维修任务分配给多维修人员,使得系统维修时间最小的问题,并将所提算法引入到PSO算法中,求解考虑多维修人员和不完全维修条件的复杂系统选择性维修模型。案例表明:所提模型和求解算法有效,能够为复杂系统提供切实有效的维修决策方案。

Existing likelihood methods for the additive hazards model with interval censored survival data are limited and often ignore the non-negative constraints on hazards. This paper proposes a maximum penalized likelihood method to fit additive hazards models with interval censoring. Our method firstly models the baseline hazard using a finite number of non-negative basis functions, and then regression coefficients and baseline hazard are estimated simultaneously by maximizing a penalized log-likelihood function, where a penalty function is introduced to regularize the baseline hazard estimate. In the estimation procedure, non-negative constraints are imposed on both the baseline hazard and the hazard of each subject. A primal–dual interior-point algorithm is applied to solve the constrained optimization problem. Asymptotic properties are obtained and a simulation study is conducted for assessment of the proposed method.

Interval censored survival data occur if the onset time for an event of interest cannot be observed exactly, rather, one only observes a time interval where the event time is located. Interval censoring may happen, for example, in medical or health studies where periodic follow-ups are used to collect survival data. In fact many clinical trials and longitudinal studies adopt this type of design. If a survival data set involves only right censoring then the standard approach for fitting a proportional hazard model is Cox's maximum partial likelihood (Cox, 1972), which treats the baseline hazard as a nuisance parameter and only estimates the regression coefficients. For interval censored data, however, partial likelihood is not available in closed form. Also counting process methods for right censored survival data (e.g. Andersen & Gill, 1982; Andersen et al., 1993) are difficult to extend to interval censoring. In contrast, likelihood based methods are feasible since the likelihood function can be established and manipulated for the proportional hazard model with interval censored data. This book, edited by Chen, Sun and Peace, focuses on recent developments in the analysis of interval censored survival data where estimation is effected using likelihood functions or loss functions. It naturally forms supplementary reading material to a number of existing books on basic interval censoring methods, for example Sun (2006). The book starts by providing a literature review of recent developments in survival analysis with interval censored data in Chapter 1. Different types of interval censoring and their associated likelihood functions are discussed in Chapter 2. Chapters 3–5 are devoted to a special form of interval censoring, namely current status data which specify only that the event of interest occurs either before or after the observation time. Chapter 3 summarizes the likelihood based methods for survival function estimation and comparison of survival functions. Chapter 4 explains how to fit the semiparametric proportional hazard regression model in the context of current status data. Chapter 5 illustrates how to handle dependent censoring in current status data using the frailty method. Chapters 6 and 7 deal with Bayesian methods for semiparametric regression with general interval censored data. Specifically, Chapter 6 describes a Bayesian method for several semiparametric models where a nonparametric monotone function (such as the cumulative hazard function) is approximated by an expansion involving monotone basis functions, and Chapter 7 discusses Bayesian inference for proportional hazard models with time-varying regression coefficients. Then Chapters 8 and 9 shift to different topics. Chapter 8 reviews an approach for estimating the causal effect of a baseline treatment on the endpoint, from longitudinal data containing a combination of interval and right censoring. Chapter 9 discusses an important issue usually associated with fitting semiparametric models, namely estimation of the variance of regression coefficient estimates. Here the difficulty lies in the fact that for the nonparametric component of the model, which is approximated by a finite number of basis functions, the Hessian matrix can be nearly singular and consequently will produce unreliable variance estimates. The proposed method adopts a least-squares approach that provide a stable estimate of the semiparametric information matrix, this estimate being asymptotically consistent. Finally, Chapters 10–14 form a group of application and computing chapters. Chapters 10 and 11 study the bias and mean-squared-error of the likelihood based interval censoring method as compared with some ad hoc (mainly single imputation) methods. The comparisons indicate that the likelihood method consistently provides smaller biases than the imputation methods. Chapter 12 discusses an application of a parametric model for studying acute ischemic stroke data where the event time is subject to interval censoring. Chapter 13 deals with log-rank tests in the context of interval censored events in clinical trials and Chapter 14 reviews an R package for general log-rank tests. In conclusion, this book furnishes a nice summary of interval censored survival data analysis and in addition describes some recent advances in this area. It is suitable for researchers and postgraduate students who require skills in survival analysis with interval censored data, and furthermore can be used as supplementary reading to some existing books and book chapters on interval censoring.

This paper considers simultaneous estimation of the regression coefficients and baseline hazard in proportional hazard models using the maximum penalized likelihood (MPL) method where a penalty function is used to smooth the baseline hazard estimate. Although MPL methods exist to fit proportional hazard models, they suffer from the following deficiencies: (i) the positivity constraint on the baseline hazard estimate is either avoided or poorly treated leading to efficiency loss, (ii) the asymptotic properties of the MPL estimator are lacking, and (iii) simulation studies comparing the performance of MPL to that of the partial likelihood have not been conducted. In this paper we propose a new approach and aim to address these issues. We first model baseline hazard using basis functions, then estimate this approximate baseline hazard and the regression coefficients simultaneously. The penalty function included in the likelihood is quite general but typically assumes prior knowledge about the smoothness of the baseline hazard. A new iterative optimization algorithm, which combines Newton's method and a multiplicative iterative algorithm, is developed and its convergence properties studied. We show that if the smoothing parameter tends to zero sufficiently fast, the new estimator is consistent, asymptotically normal and retains full efficiency under independent censoring. A simulation study reveals that this method can be more efficient than the partial likelihood method, particularly for small to moderate samples. In addition, our simulation shows that the new estimator is substantially less biased under informative censoring.