Piecewise Constant Hazard Function Research Articles

Bayesian spatio‐temporal survival analysis for all types of censoring with application to a wildlife disease study

AbstractIn this article, we consider modeling arbitrarily censored survival data with spatio‐temporal covariates. We demonstrate that under the piecewise constant hazard function, the likelihood for uncensored or right‐censored subjects is proportional to the likelihood of multiple conditionally independent Poisson random variables. To address left‐ or interval‐censored subjects, we propose to impute the exact event times and convert them into uncensored subjects, enabling the application of the integrated nested Laplace approximation to update model parameters using the imputed data. We introduce an iterative algorithm that alternates between imputing event times for left‐ and interval‐censored subjects and re‐estimating model parameters. The proposed method is assessed through a simulation study and applied to analyze a spatio‐temporal survival dataset in a wildlife disease study investigating bovine tuberculosis in white‐tailed deer in Michigan.

Joint frailty model for recurrent events and a terminal event in the presence of cure fraction

The observations of repeated or recurrent events occur in many longitudinal studies. Furthermore, sometimes there may exist a terminal event such as death, which is strongly correlated with recurrent events. In many situations, a fraction of subjects who will never experience the event of interest during a long follow-up period is considered to be cured. In this article, we proposed a joint frailty model in the presence of cure fraction. The dependency is modeled by shared frailty that is contained in both the recurrent andterminal events hazard functions. It allows to estimate two separate sets of parameters on the recurrent, death, and cure model. We applied the maximum likelihood method under a piecewise constant hazard function for model fitting. The proposed model is evaluated by simulation studies and an application to a breast cancer data is provided.

To Wait or Not to Wait: Two-Way Functional Hazards Model for Understanding Waiting in Call Centers

ABSTRACTTelephone call centers offer a convenient communication channel between businesses and their customers. Efficient management of call centers needs accurate modeling of customer waiting behavior, which contains important information about customer patience (how long a customer is willing to wait) and service quality (how long a customer needs to wait to get served). Hazard functions offer dynamic characterization of customer waiting behavior, and provide critical inputs for agent scheduling. Motivated by this application, we develop a two-way functional hazards (tF-Hazards) model to study customer waiting behavior as a function of two timescales, waiting duration and the time of day that a customer calls in. The model stems from a two-way piecewise constant hazard function, and imposes low-rank structure and smoothness on the hazard rates to enhance interpretability. We exploit an alternating direction method of multipliers algorithm to optimize a penalized likelihood function of the model. We carefully analyze the data from a U.S. Bank call center, and provide informative insights about customer patience and service quality patterns along waiting time and across different times of a day. The findings provide primitive inputs for call center agent staffing and scheduling, as well as for call center practitioners to understand the effect of system protocols on customer waiting behavior. Supplementary materials for this article are available online.

Non‐parametric Bayesian Intensity Model: Exploring Time‐to‐Event Data on Two Time Scales

AbstractTime‐to‐event data have been extensively studied in many areas. Although multiple time scales are often observed, commonly used methods are based on a single time scale. Analysing time‐to‐event data on two time scales can offer a more extensive insight into the phenomenon. We introduce a non‐parametric Bayesian intensity model to analyse two‐dimensional point process on Lexis diagrams. After a simple discretization of the two‐dimensional process, we model the intensity by a one‐dimensional piecewise constant hazard functions parametrized by the change points and corresponding hazard levels. Our prior distribution incorporates a built‐in smoothing feature in two dimensions. We implement posterior simulation using the reversible jump Metropolis–Hastings algorithm and demonstrate the applicability of the method using both simulated and empirical survival data. Our approach outperforms commonly applied models by borrowing strength in two dimensions.

Bayes linear kinematics in a dynamic survival model

Bayes linear kinematics and Bayes linear Bayes graphical models provide an extension of Bayes linear methods so that full conditional updates may be combined with Bayes linear belief adjustment. In this paper we investigate the application of this approach to survival analysis with time-dependent covariate effects, a more complicated problem than previous applications. We use a piecewise-constant hazard function with a prior in which covariate effects are correlated over time. The need for computationally intensive methods is avoided and the relatively simple structure facilitates interpretation. Our approach eliminates the problem of non-commutativity which was observed in earlier work by Gamerman. We apply the technique to data on survival times for leukemia patients.

Integrated survival analysis using an event-time approach in a Bayesian framework.

Event-time or continuous-time statistical approaches have been applied throughout the biostatistical literature and have led to numerous scientific advances. However, these techniques have traditionally relied on knowing failure times. This has limited application of these analyses, particularly, within the ecological field where fates of marked animals may be unknown. To address these limitations, we developed an integrated approach within a Bayesian framework to estimate hazard rates in the face of unknown fates. We combine failure/survival times from individuals whose fates are known and times of which are interval-censored with information from those whose fates are unknown, and model the process of detecting animals with unknown fates. This provides the foundation for our integrated model and permits necessary parameter estimation. We provide the Bayesian model, its derivation, and use simulation techniques to investigate the properties and performance of our approach under several scenarios. Lastly, we apply our estimation technique using a piece-wise constant hazard function to investigate the effects of year, age, chick size and sex, sex of the tending adult, and nesting habitat on mortality hazard rates of the endangered mountain plover (Charadrius montanus) chicks. Traditional models were inappropriate for this analysis because fates of some individual chicks were unknown due to failed radio transmitters. Simulations revealed biases of posterior mean estimates were minimal (≤ 4.95%), and posterior distributions behaved as expected with RMSE of the estimates decreasing as sample sizes, detection probability, and survival increased. We determined mortality hazard rates for plover chicks were highest at <5 days old and were lower for chicks with larger birth weights and/or whose nest was within agricultural habitats. Based on its performance, our approach greatly expands the range of problems for which event-time analyses can be used by eliminating the need for having completely known fate data.

Bayesian survival trees for clustered observations, applied to tooth prognosis

AbstractTooth loss from periodontal disease or dental caries (decay) afflicts most adults over the course of their lives. Survival tree methods for correlated observations have shown potential for developing objective tooth prognosis systems; however, the current technology suffers either from prohibitive computational expense or unrealistic simplifying assumptions to overcome computational demands. In this article Bayesian tree methods are developed for correlated survival data, relying on a computationally feasible, yet flexible, frailty model with piecewise constant hazard function. Bayesian stochastic search methods, using a Laplace approximated marginal likelihood, are detailed for tree construction, and posterior ensemble averaged variable importance ranking and amalgamation procedures are developed. The proposed methods are used to assign each tooth from the Veteran Administration (VA) Dental Longitudinal Study to one of five prognosis categories and evaluate the effects of clinical factors and genetic polymorphisms in predicting tooth loss. The prognostic rules established may be used in clinical practice to optimize tooth retention and devise periodontal treatment plans.

Semiparametric Sequential Testing for Multiple Change Points in Piecewise Constant Hazard Functions with Long-Term Survivors*

In this article, we consider parameter estimation in the hazard rate with multiple change points in the presence of long-term survivors. We combine two methods: maximum likelihood based and martingale based, to estimate the change points in the hazard rate for right censored survival data that accounts for long-term survivors. A simulation study is carried out to compare the performance of estimators. The method is applied to analyze two real datasets.

Estimating the survival functions for right-censored and interval-censored data with piecewise constant hazard functions

The exponential distribution is frequently used to model the survival time of a patient population, which assumes the hazard rate to be a constant over time. This assumption is often violated as the hazard function may vary over time and exhibit one or more change points in its values. Several methods exist in the literature for detecting a single change point in a piecewise constant hazard function for right-censored data. A sequential testing approach to detecting multiple change points in the hazard function using likelihood ratio statistics and resampling is proposed, which is applicable to both right-censored and interval-censored data. Numerical results based on simulated survival data and a real example show that the proposed approach can accurately detect the change points in the hazard function for both right-censored and interval-censored data.

Further Results on Piecewise Constant Hazard Functions in Aalen's Additive Risk Model

The modifications suggested in Uhm et al. (2011) are studied using a partly parametric version of Aalen's additive risk model. A follow-up time period is partitioned into intervals, and hazard functions are estimated as a piecewise constant in each interval. A maximum likelihood estimator by iteratively reweighted least squares and variance estimates are suggested based on the model as well as evaluated by simulations using mean square error and a coverage probability, respectively. In conclusion the modifications are needed when there are a small number of uncensored deaths in an interval to estimate the piecewise constant hazard function.

Additive Risk Model Using Piecewise Constant Hazard Function

We study a weighted least squares estimator for Aalen's additive risk model with right-censored survival data which allows for a very flexible handling of covariates. We divide the follow-up period into intervals and assume a constant hazard rate in each interval. The model is motivated as a piecewise approximation of a hazard function composed of three parts: arbitrary nonparametric functions for some covariate effects, smoothly varying functions for others, and known (or constant) functions for yet others. The proposed estimator is an extension of the grouped data version of the Huffer and McKeague (1991) estimator. For our model, since the number of parameters is finite (although large), conventional approaches (such as maximum likelihood) are easy to formulate and implement. The approach is illustrated by simulations, and is compared to the previous studies. The method is also applied to the Framingham study data.

Full and conditional likelihood approaches for hazard change-point estimation with truncated and censored data

Hazard function plays an important role in reliability and survival analysis. In some real life applications, abrupt changes in the hazard function may be observed and it is of interest to detect the location and the size of the change. Hazard models with a change-point are considered when the observations are subject to random left truncation and right censoring. For a piecewise constant hazard function with a single change-point, two estimation methods based on the maximum likelihood ideas are considered. The first method assumes parametric families of distributions for the censoring and truncation variables, whereas the second one is based on conditional likelihood approaches. A simulation study is carried out to illustrate the performances of the proposed estimators. The results indicate that the fully parametric method performs better especially for estimating the size of the change, however the difference between the two methods vanish as the sample size increases. It is also observed that the full likelihood approach is not robust to model misspecification.

Detecting multiple change points in piecewise constant hazard functions

The National Cancer Institute (NCI) suggests a sudden reduction in prostate cancer mortality rates, likely due to highly successful treatments and screening methods for early diagnosis. We are interested in understanding the impact of medical breakthroughs, treatments, or interventions, on the survival experience for a population. For this purpose, estimating the underlying hazard function, with possible time change points, would be of substantial interest, as it will provide a general picture of the survival trend and when this trend is disrupted. Increasing attention has been given to testing the assumption of a constant failure rate against a failure rate that changes at a single point in time. We expand the set of alternatives to allow for the consideration of multiple change-points, and propose a model selection algorithm using sequential testing for the piecewise constant hazard model. These methods are data driven and allow us to estimate not only the number of change points in the hazard function but where those changes occur. Such an analysis allows for better understanding of how changing medical practice affects the survival experience for a patient population. We test for change points in prostate cancer mortality rates using the NCI Surveillance, Epidemiology, and End Results dataset.

Estimation of a change point in a hazard function based on censored data.

The hazard function plays an important role in reliability or survival studies since it describes the instantaneous risk of failure of items at a time point, given that they have not failed before. In some real life applications, abrupt changes in the hazard function are observed due to overhauls, major operations or specific maintenance activities. In such situations it is of interest to detect the location where such a change occurs and estimate the size of the change. In this paper we consider the problem of estimating a single change point in a piecewise constant hazard function when the observed variables are subject to random censoring. We suggest an estimation procedure that is based on certain structural properties and on least squares ideas. A simulation study is carried out to compare the performance of this estimator with two estimators available in the literature: an estimator based on a functional of the Nelson-Aalen estimator and a maximum likelihood estimator. The proposed least squares estimator tums out to be less biased than the other two estimators, but has a larger variance. We illustrate the estimation method on some real data sets.

Estimation of the Jump-Point in a Hazard Function

We consider a piecewise constant hazard function with exactly one jump point, say τ . It uniquely determines an Exponential distribution whose density features a discontinuity of the first kind at the change point τ . Assuming that τ is the unknown parameter of interest, the maximum likelihood estimator is shown to be strongly consistent for τ . Its computation is very simple, because it requires merely a finite number of comparisons. Some graphics and calculations illustrate our results.