Baseline Hazard Rate Research Articles

Aging Intensity for Step-Stress Accelerated Life Testing Experiments.

The aging intensity (AI), defined as the ratio of the instantaneous hazard rate and a baseline hazard rate, is a useful tool for the describing reliability properties of a random variable corresponding to a lifetime. In this work, the concept of AI is introduced in step-stress accelerated life testing (SSALT) experiments, providing new insights to the model and enabling the further clarification of the differences between the two commonly employed cumulative exposure (CE) and tampered failure rate (TFR) models. New AI-based estimators for the parameters of a SSALT model are proposed and compared to the MLEs in terms of examples and a simulation study.

Aalen’s additive Hazard model: an application in analyzing job tenure factors in private and public companies in Albania

There are several statistical methods for time to event analysis, among which is the Cox proportional hazards model, that is most commonly used. In this model, the effect of the covariates is assumed to act multiplicatively on the baseline hazard rate and the ratio of the hazards is constant over survival time. However, when the proportionality assumption for the Cox proportional hazards model is not satisfied, an additive hazard regression model may be more appropriate. In this study, we give an overview of this approach and then apply Aalen’s additive hazard regression model to a data set which contain the job tenure for Albanian employees and variables that are thought of as important factors in the job tenure. The Aalen model allows for time-varying covariate effects and is used to assess the impact that age of the employee, the age at which he started the job, salary, gender, position and years of work in front of the current position may have in job termination.

Potts-Cox survival regression

A Bayesian semi-parametric survival regression model with latent partitions is introduced. Its goal is to predict survival and to cluster survival patients within the context of building prognosis systems. In order to drive cluster formation on individuals, the Potts random partition model is chosen as a prior on the covariates space. For any given partition, the proposed model assumes an interval-wise exponential distribution for the baseline hazard rate. The number of intervals is unknown. It can be estimated with a fused-lasso type penalty given by a sequential double exponential prior. Estimation and inference are done with the aid of Markov chain Monte Carlo. To simplify the computations, the Laplace's integral approximation method is used to estimate some constants and to propose parameter updates within Markov chain Monte Carlo. The methodology is illustrated with an application to cancer survival.

On the hazard rate of α-mixture of survival functions

The α-mixture model, as a flexible family of distributions, is an effective tool for modeling heterogeneity inS population. This article investigates the hazard rate of α-mixture in terms of hazard rates of mixed baseline distributions. In particular, when the baseline hazard rate follows either additive or multiplicative models an inverse problem to obtain the baseline hazard is solved. We, also, study the α-mixture hazard rate ordering for the ordered mixing distributions in the sense of likelihood ratio order. Sufficient conditions to order two finite α-mixtures in the sense of dispersive ordering are provided. Finally, it is shown that the hazard rate of the finite α-mixture in the multiplicative model tends to the hazard rate of the strongest (weakest) population as α → + ∞ ( − ∞ ). Several examples are presented to illustrate theoretical findings.

Flexible Parametric Accelerated Hazard Model: Simulation and Application to Censored Lifetime Data with Crossing Survival Curves

This study aims to propose a flexible, fully parametric hazard-based regression model for censored time-to-event data with crossing survival curves. We call it the accelerated hazard (AH) model. The AH model can be written with or without a baseline distribution for lifetimes. The former assumption results in parametric regression models, whereas the latter results in semi-parametric regression models, which are by far the most commonly used in time-to-event analysis. However, under certain conditions, a parametric hazard-based regression model may produce more efficient estimates than a semi-parametric model. The parametric AH model, on the other hand, is inappropriate when the baseline distribution is exponential because it is constant over time; similarly, when the baseline distribution is the Weibull distribution, the AH model coincides with the accelerated failure time (AFT) and proportional hazard (PH) models. The use of a versatile parametric baseline distribution (generalized log-logistic distribution) for modeling the baseline hazard rate function is investigated. For the parameters of the proposed AH model, the classical (via maximum likelihood estimation) and Bayesian approaches using noninformative priors are discussed. A comprehensive simulation study was conducted to assess the performance of the proposed model’s estimators. A real-life right-censored gastric cancer dataset with crossover survival curves is used to demonstrate the tractability and utility of the proposed fully parametric AH model. The study concluded that the parametric AH model is effective and could be useful for assessing a variety of survival data types with crossover survival curves.

FC084: Mineralocorticoid Receptor Antagonist Treatment in Patients with Renal Insufficiency and Associated Risk of Hyperkalemia and Death

Abstract BACKGROUND AND AIMS Mineralocorticoid receptor antagonist treatment is kidney protective but not recommended in patients with advanced renal failure due to perceived risk of hyperkalaemia and death. The aim of the present study was to examine the impact of mineralocorticoid receptor antagonist treatment in progressive chronic kidney disease on risk of hyperkalaemia and to evaluate subsequent attributable mortality. METHOD Based on data from multiple nationwide health care registers, all Danish residents >18 years with more than or equal to one recorded plasma creatinine measurement were identified in Denmark between 2008 and 2021, and followed until either incident hyperkalaemia >6 mmol/L, death or end-of follow-up (5 October 2021). Cases were identified based on incident hyperkalaemia and subsequently matched with four controls without hyperkalaemia. eGFR was computed based on the latest creatinine measurement until 30 days prior to index using the CKD-EPI equation. Patients with no recorded creatinine measurement within 2 years until 30 days prior of index were excluded. Mineralocorticoid receptor antagonist associated susceptibility for hyperkalaemia was computed across strata of estimated glomerular filtration rate (eGFR) (>60, 30–60 and <30 mL/min/1.73 m2) in a conditional logistic regression model comparing rate of hyperkalaemia with stratification of baseline hazard rate by age, gender, diabetes and hypertension. Model fitting was accomplished using a nested case–control design with 1:4 risk-set matching of cases with age-, gender-, diabetes- and hypertension-matched controls. Subsequent risk of 30-day mortality was assessed in a retrospective cohort study including patients with hyperkalemia only. Patients were followed from the date of hyperkalemia until death, 30 days or end of follow-up (5 October 2021) with comparison of MRA-attributable mortality across strata of eGFR based on the Kaplan–Meier estimator and multiple Cox regression models adjusted for age and gender. RESULTS From a population of 4 255 962 Danish citizens with recorded plasma creatinine, a total of 51 775 patients with incident hyperkalaemia were identified between 2008 and 2021. Pre-existing plasma creatinine permitting estimation of renal function was available in 34 442 (66.5%) patients. The 34 442 cases with incident hyperkalaemia were matched 1:4 with 131 410 controls. Gender distribution was 57% male, median age was 73.5 (IQR 64.0–82.0) years and median eGFR was 74 (IQR 55–89) mL/min/1.73 m2. Mineralocorticoid receptor antagonist treatment was associated with increased rates of hyperkalaemia with HRs of 8.32 (95% CI: 7.82–8.85), 4.55 (95% CI: 4.26–4.86) and 2.00 (95% CI: 1.71–2.32) in patients with eGFR > 60 mL/min/1.73 m2 eGFR 30–60 mL/min/1.73 m2 and eGFR < 30 mL/min/1.73 m2 respectively (reference = No mineralocorticoid receptor antagonist treatment). 30-day mortality in patients with hyperkalaemia stratified on mineralocorticoid receptor antagonist treatment is shown in Figures 1 and 2. Overall, mineralocorticoid receptor antagonist treatment was associated with lower subsequent mortality: HR 0.84 (95% CI: 0.78–0.90); P < 0.001, 0.67 (95% CI: 0.62–0.73); P < 0.001 and 0.98 (95% CI: 0.86–1.1); P = 0.77 for eGFR > 60 mL/min/1.73 m2 30–60 mL/min/1.73 m2 and <30 mL/min/1.73 m2 respectively (reference = mineralocorticoid receptor antagonist). CONCLUSION Although mineralocorticoid receptor antagonist treatment was associated with increased risk of hyperkalaemia for all strata of kidney impairment, subsequent 30-day mortality was lower compared with patients with corresponding kidney function without mineralocorticoid receptor antagonist treatment.

The role of financial constraint factors in predicting SME default

Research background: SMEs face financial constraints in their development, which limits their access to external funds, tightens their investment possibilities, and limits their growth. Much research effort has been devoted to understanding the nature and sources of this phenomenon. In sharp contrast to this, very little has been said about the role of these factors in explaining the default probability of these types of enterprises. Understanding such interrelationships could help to adopt policies to alleviate the situation of constrained SMEs and lower their default rates.
 Purpose of the article: This study analyses the role of financial constraint factors in SME defaults. This is done by utilising the financial constraint factors in a newly derived default prediction model. A comparison of the derived model and other SME default prediction models is carried out to assess the potential of financial constraints in the discrimination power of the model.
 Methods: In this study, we use the Cox semiparametric model, while leaving the baseline hazard rate unspecified and employing macroeconomic variables as explanatory variables. The discrimination power was addressed in terms of the area under the curve (AUC), resulting in out-of-sample testing. The DeLong test was used to compare the AUC of the created and analysed models. The model was estimated on a set of over 213,731 SMEs from 28 counties, covering the period 2014?2019.
 Findings & value added: It was found that adopting the financial constraint measures can explain the default of small and medium enterprises with high accuracy; however, they do not explain the default of micro enterprises.

Impact of a non-constant baseline hazard on detection of time-dependent treatment effects: a simulation study

BackgroundNon-proportional hazards are common with time-to-event data but the majority of randomised clinical trials (RCTs) are designed and analysed using approaches which assume the treatment effect follows proportional hazards (PH). Recent advances in oncology treatments have identified two forms of non-PH of particular importance - a time lag until treatment becomes effective, and an early effect of treatment that ceases after a period of time. In sample size calculations for treatment effects on time-to-event outcomes where information is based on the number of events rather than the number of participants, there is crucial importance in correct specification of the baseline hazard rate amongst other considerations. Under PH, the shape of the baseline hazard has no effect on the resultant power and magnitude of treatment effects using standard analytical approaches. However, in a non-PH context the appropriateness of analytical approaches can depend on the shape of the underlying hazard.MethodsA simulation study was undertaken to assess the impact of clinically plausible non-constant baseline hazard rates on the power, magnitude and coverage of commonly utilized regression-based measures of treatment effect and tests of survival curve difference for these two forms of non-PH used in RCTs with time-to-event outcomes.ResultsIn the presence of even mild departures from PH, the power, average treatment effect size and coverage were adversely affected. Depending on the nature of the non-proportionality, non-constant event rates could further exacerbate or somewhat ameliorate the losses in power, treatment effect magnitude and coverage observed. No single summary measure of treatment effect was able to adequately describe the full extent of a potentially time-limited treatment benefit whilst maintaining power at nominal levels.ConclusionsOur results show the increased importance of considering plausible potentially non-constant event rates when non-proportionality of treatment effects could be anticipated. In planning clinical trials with the potential for non-PH, even modest departures from an assumed constant baseline hazard could appreciably impact the power to detect treatment effects depending on the nature of the non-PH. Comprehensive analysis plans may be required to accommodate the description of time-dependent treatment effects.

Consistent Evidence on Duration Dependence of Price Changes

We develop an estimator and tests of a discrete time mixed proportional hazard (MPH) model of duration with unobserved heterogeneity. We allow for competing risks, observable characteristics, and censoring, and we use linear GMM, making estimation and inference straightforward. With repeated spell data, our estimator is consistent and robust to the unknown shape of the frailty distribution. We apply our estimator to the duration of price spells in weekly store data from IRI. We find substantial unobserved heterogeneity, accounting for a large fraction of the decrease in the Kaplan-Meier hazard with elapsed duration. Still, we show that the estimated baseline hazard rate is decreasing and a homogeneous firm model can accurately capture the response of the economy to a monetary policy shock even if there is significant strategic complementarity in pricing. Using competing risks and spell-specific observable characteristics, we separately estimate the model for regular and temporary price changes and find that the MPH structure describes regular price changes better than temporary ones. Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Survival analysis via hierarchically dependent mixture hazards

Hierarchical nonparametric processes are popular tools for defining priors on collections of probability distributions, which induce dependence across multiple samples. In survival analysis problems, one is typically interested in modeling the hazard rates, rather than the probability distributions themselves, and the currently available methodologies are not applicable. Here, we fill this gap by introducing a novel, and analytically tractable, class of multivariate mixtures whose distribution acts as a prior for the vector of sample-specific baseline hazard rates. The dependence is induced through a hierarchical specification of the mixing random measures that ultimately corresponds to a composition of random discrete combinatorial structures. Our theoretical results allow to develop a full Bayesian analysis for this class of models, which can also account for right-censored survival data and covariates, and we also show posterior consistency. In particular, we emphasize that the posterior characterization we achieve is the key for devising both marginal and conditional algorithms for evaluating Bayesian inferences of interest. The effectiveness of our proposal is illustrated through some synthetic and real data examples.

Abstract MP7: Multicenter Study of Intracranial Hemorrhage and Death Risk and Predictors in the Untreated Course of Brain Arteriovenous Malformation Patients

Background: Accurate and reliable models of intracranial hemorrhage (ICH) risk in the untreated course of brain arteriovenous malformation (BAVM) patients are needed to help weigh the risk-benefit of treatment. We present preliminary estimates of rates and risk factors for ICH or death in the Multicenter AVM Research Study (MARS), which is the largest individual patient data meta-analysis (IPDMA) of cohort studies of unruptured BAVM. Methods: Longitudinal data from 9 cohorts comprising 2,839 unruptured BAVM at time of diagnosis were included: UCSF (n=557), Macquarie (n=462), Kaiser Permanente Northern California (n=354), Columbia (n=335), Barrow Neurological Institute (n=324), Tiantan Hospital (n=303), Mayo Clinic-Rochester (n=244), Scottish Intracranial Vascular Malformation Study (n=194), and Toronto Western Hospital (n=66). Clinical and angiographic data were collected using standardized definitions across cohorts. Cox proportional hazards analysis of time-to-event (ICH or death) in the untreated course after diagnosis was performed, censoring at first treatment or last visit. Baseline hazard rates were stratified by cohort to account for different rates and follow-up times. Results: The combined cohort was 49% female, 72% white race, and 17% Hispanic ethnicity; mean age at diagnosis was 37±17 years and mean follow-up was 3.4±7.0 years. The most common presenting symptom was seizures (43%). Mean AVM size was 3.5±1.9 cm and 46% were Spetzler-Martin grade I-II. Overall, 123 ICH and 131 deaths occurred over 9,608 person-years (PY) for an event rate of 2.64 per 100 PY [95% CI: 2.34-2.99]. Cox regression analyses adjusting for age at diagnosis (HR=1.67 [1.54, 1.82], P<0.001) identified cerebellar location (HR=1.73 [1.03-2.93], P=0.040) and exclusively deep venous drainage (HR=1.83 [1.09-3.09], P=0.022) as predictors. Conclusion: Our current model identified increasing age, exclusively deep venous drainage, and cerebellar location as risk factors for ICH or death in the untreated course of unruptured bAVM patients. Review of imaging data, multiple imputation of missing angioarchitectural data, and additional data from pending cohorts will provide greater power to search for additional risk factors in this ongoing IPDMA.

Consistent Evidence on Duration Dependence of Price Changes

We develop an estimator and tests of a discrete time mixed proportional hazard (MPH) model of duration with unobserved heterogeneity. We allow for competing risks, observable characteristics, and censoring, and we use linear GMM, making estimation and inference straightforward. With repeated spell data, our estimator is consistent and robust to the unknown shape of the frailty distribution. We apply our estimator to the duration of price spells in weekly store data from IRI. We find substantial unobserved heterogeneity, accounting for a large fraction of the decrease in the Kaplan-Meier hazard with elapsed duration. Still, we show that the estimated baseline hazard rate is decreasing and a homogeneous firm model can accurately capture the response of the economy to a monetary policy shock even if there is significant strategic complementarity in pricing. Using competing risks and spell-specific observable characteristics, we separately estimate the model for regular and temporary price changes and find that the MPH structure describes regular price changes better than temporary ones.

Time-to-event modeling of hypertension reveals the nonexistence of true controls.

Given a lifetime risk of ~90% by the ninth decade of life, it is unknown if there are true controls for hypertension in epidemiological and genetic studies. Here, we compared Bayesian logistic and time-to-event approaches to modeling hypertension. The median age at hypertension was approximately a decade earlier in African Americans than in European Americans or Mexican Americans. The probability of being free of hypertension at 85 years of age in African Americans was less than half that in European Americans or Mexican Americans. In all groups, baseline hazard rates increased until nearly 60 years of age and then decreased but did not reach zero. Taken together, modeling of the baseline hazard function of hypertension suggests that there are no true controls and that controls in logistic regression are cases with a late age of onset.

The Lambert-F Distributions Class: An Alternative Family for Positive Data Analysis

In this article, we introduce a new probability distribution generator called the Lambert-F generator. For any continuous baseline distribution F, with positive support, the corresponding Lambert-F version is generated by using the new generator. The result is a new class of distributions with one extra parameter that generalizes the baseline distribution and whose quantile function can be expressed in closed form in terms of the Lambert W function. The hazard rate function of a Lambert-F distribution corresponds to a modification of the baseline hazard rate function, greatly increasing or decreasing the baseline hazard rate for earlier times. Herein, we study the main structural properties of the new class of distributions. Special attention is given to two particular cases that can be understood as two-parameter extensions of the well-known exponential and Rayleigh distributions. We discuss parameter estimation for the proposed models considering the moments and maximum likelihood methods. Finally, two applications were developed to illustrate the usefulness of the proposed distributions in the analysis of data from different real settings.

Analysis of joint modeling of longitudinal zero-inflated power series and zero-inflated time to event data

ABSTRACT In longitudinal studies measurements are often collected on different types of responses for each individual. These may contain several longitudinally measured responses (such as the CD4 count) and the time at which an event occurs (e.g., HIV, death, or dropout from the study). These outcomes are often separately analyzed. Compared to separate modeling, joint modeling and simultaneous analysis allows for more coherent, robust analysis and may produce a better insight into the process under study. However, there has always been difficulty to the analyst that finding a proper multi-variable joint distribution for linking responses. In this article, we survey the zero-inflated property for longitudinal count and time to event data. We apply a member of the family of power series distributions (PSDs) and the Cox proportional hazard regression model (Cox PH) with Weibull baseline hazard rate, respectively, for these correlated responses. Also we consider both right and left censoring mechanisms in time to event process. This modeling strategy leads to expand the class of joint models and presents some new joint models which, as far as we know, have not yet been investigated by other researchers. The parameters in the joint model are estimated by using likelihood techniques.

Condition-Based Maintenance Planning for Systems Subject to Dependent Soft and Hard Failures

Most systems can fail in multiple ways, and the failure modes are usually positively correlated. This phenomenon complicates the reliability analysis and makes the corresponding maintenance planning challenging. This article proposes a condition-based maintenance policy for systems that are subject to both degradation-induced soft failure and sudden hard failure, where a higher degradation level leads to a higher hazard rate of the hard failure. The Wiener process is adopted for the degradation process, and the Weibull model is used to describe the baseline hazard rate of the hard failure. The degradation level is then treated as a time-varying covariate that affects the hazard rate of the hard failure, and the closed-form of the reliability function is derived by using the Brownian bridge theory. An inspection/replacement maintenance policy is employed, and the long-run cost rate is formulated based on the semiregenerative property of the system state. The optimal inspection interval and the preventive replacement threshold are then jointly determined by minimizing the long-run cost rate. A numerical study on a hydraulic sliding spool system is conducted to validate the derived reliability function and the maintenance policy.

A Bayesian inference for the penalized spline joint models of longitudinal and time-to-event data: A prior sensitivity analysis

Abstract Bayesian approaches have been used in the literature to estimate the parameters for joint models of longitudinal and time-to-event data. The main aim of this paper is to analyze the impact of prior distributions on estimating parameters in a proposed fully Bayesian analysis setting for the penalized spline joint models. To achieve this aim, the joint posterior distribution of parameters in survival and longitudinal submodels is presented. The Markov chain Monte Carlo (MCMC) algorithm is then proposed, which consists of the Gibbs sampler (GS) and Metropolis Hastings (MH) algorithms to sample for the target conditional posterior distributions. The prior sensitivity analysis for the baseline hazard rate and association parameters is performed through simulation studies and a case study.

Regression Models and Multivariate Life Tables

Semiparametric, multiplicative-form regression models are specified for marginal single and double failure hazard rates for the regression analysis of multivariate failure time data. Cox-type estimating functions are specified for single and double failure hazard ratio parameter estimation, and corresponding Aalen–Breslow estimators are specified for baseline hazard rates. Generalization to allow classification of failure times into a smaller set of failure types, with failures of the same type having common baseline hazard functions, is also included. Asymptotic distribution theory arises by generalization of the marginal single failure hazard rate estimation results of Lin et al. The Péano series representation for the bivariate survival function in terms of corresponding marginal single and double failure hazard rates leads to novel estimators for pairwise bivariate survival functions and pairwise dependency functions, at specified covariate history. Related asymptotic distribution theory follows from that for the marginal single and double failure hazard rates and the continuity, compact differentiability of the Péano series transformation and bootstrap applicability. Simulation evaluation of the proposed estimation procedures is presented, and an application to multiple clinical outcomes in the Women’s Health Initiative Dietary Modification Trial is provided. Higher dimensional marginal hazard rate regression modeling is briefly mentioned. Supplementary materials for this article are available online.

In-Sample Hazard Forecasting Based on Survival Models with Operational Time

We introduce a generalization of the one-dimensional accelerated failure time model allowing the covariate effect to be any positive function of the covariate. This function and the baseline hazard rate are estimated nonparametrically via an iterative algorithm. In an application in non-life reserving, the survival time models the settlement delay of a claim and the covariate effect is often called operational time. The accident date of a claim serves as covariate. The estimated hazard rate is a nonparametric continuous-time alternative to chain-ladder development factors in reserving and is used to forecast outstanding liabilities. Hence, we provide an extension of the chain-ladder framework for claim numbers without the assumption of independence between settlement delay and accident date. Our proposed algorithm is an unsupervised learning approach to reserving that detects operational time in the data and adjusts for it in the estimation process. Advantages of the new estimation method are illustrated in a data set consisting of paid claims from a motor insurance business line on which we forecast the number of outstanding claims.

Testing Linear and Nonlinear Hypotheses in a Cox Proportional Hazards Model with Errors in Covariates

We investigate linear and nonlinear hypotheses testing in a Cox proportional hazards model for right-censored survival data when the covariates are subject to measurement errors. In Kukush and Chernova (2018) [Theor. Probability and Math. Statist. 96, 101–110], a consistent simultaneous estimator is introduced for the baseline hazard rate and the vector of regression parameters. Therein the baseline hazard rate belongs to an unbounded set of nonnegative Lipschitz functions, with fixed constant, and the vector of regression parameters belongs to a compact parameter set. Based on the estimator, we develop two procedures to test nonlinear and linear hypotheses about the vector of regression parameters: Wald-type and score-type tests. The latter is based on an unbiased estimating equation. The consistency of the tests is shown.