Specific Hazard Rates Research Articles

Exact Semiparametric Inference and Model Selection for Load-Sharing Systems

As a specific proportional hazard rates model, sequential order statistics can be used to describe the lifetimes of load-sharing systems. Inference for these systems needs to account for small sample sizes, which are prevalent in reliability applications. By exploiting the probabilistic structure of sequential order statistics, we derive exact finite sample inference procedures to test for the load-sharing parameters and for the nonparametrically specified baseline distribution, treating the respective other part as a nuisance quantity. This improves upon previous approaches for the model, which either assume a fully parametric specification or rely on asymptotic results. Simulations show that the tests derived are able to detect deviations from the null hypothesis at small sample sizes. Critical values for a prominent case are tabulated.

Changes in Colorectal Cancer 5-Year Survival Disparities in California, 1997-2014.

Colorectal cancer incidence and mortality have declined with increased screening and scientific advances in treatment. However, improvement in colorectal cancer outcomes has not been equal for all groups and disparities have persisted over time. Data from the California Cancer Registry were used to estimate changes in 5-year colorectal cancer-specific survival over three diagnostic time periods: 1997-2002, 2003-2008, and 2009-2014. Analyses included all patients in California with colorectal cancer as a first primary malignancy. Multivariable Cox proportional hazard regression models were used to evaluate the effect of race/ethnicity, insurance status, and neighborhood socioeconomic status (nSES) on 5-year colorectal cancer-specific survival. On the basis of a population-based sample of 197,060 colorectal cancer cases, racial/ethnic survival disparities decreased over time among non-Hispanic Blacks (NHB) compared with non-Hispanic Whites (NHW), after adjusting for demographic, clinical, and treatment characteristics. For cases diagnosed 1997-2002, colorectal cancer-specific hazard rates were higher for NHB [HR, 1.12; 95% confidence interval (CI), 1.06-1.19] and lower for Asians/Pacific Islanders (HR, 0.92; 95% CI, 0.87-0.96) and Hispanics (HR, 0.94; 95% CI, 0.90-0.99) compared with NHW. In 2009-2014, colorectal cancer-specific HR for NHB was not significantly different to the rate observed for NHW (HR, 1.03; 95% CI, 0.97-1.10). There were no changes in disparities in nSES, but increasing disparities by health insurance status. We found a decrease in survival disparities over time by race/ethnicity, but a persistence of disparities by neighborhood socioeconomic status and health insurance status. Further investigation into the drivers for these disparities can help direct policy and practice toward health equity for all groups.

Basic parametric analysis for a multi-state model in hospital epidemiology

BackgroundThe extended illness-death model is a useful tool to study the risks and consequences of hospital-acquired infections (HAIs). The statistical quantities of interest are the transition-specific hazard rates and the transition probabilities as well as attributable mortality (AM) and the population-attributable fraction (PAF). In the most general case calculation of these expressions is mathematically complex.MethodsWhen assuming time-constant hazards calculation of the quantities of interest is facilitated. In this situation the transition probabilities can be expressed in closed mathematical forms. The estimators for AM and PAF can be easily derived from these forms.ResultsIn this paper, we show how to explicitly calculate all the transition probabilities of an extended-illness model with constant hazards. Using a parametric model to estimate the time-constant transition specific hazard rates of a data example, the transition probabilities, AM and PAF can be directly calculated. With a publicly available data example, we show how the approach provides first insights into principle time-dynamics and data structure.ConclusionAssuming constant hazards facilitates the understanding of multi-state processes. Even in a non-constant hazards setting, the approach is a helpful first step for a comprehensive investigation of complex data.

A proportional hazards model for the analysis of doubly censored competing risks data

ABSTRACTCompeting risks data are common in medical research in which lifetime of individuals can be classified in terms of causes of failure. In survival or reliability studies, it is common that the patients (objects) are subjected to both left censoring and right censoring, which is refereed as double censoring. The analysis of doubly censored competing risks data in presence of covariates is the objective of this study. We propose a proportional hazards model for the analysis of doubly censored competing risks data, using the hazard rate functions of Gray (1988), while focusing upon one major cause of failure. We derive estimators for regression parameter vector and cumulative baseline cause specific hazard rate function. Asymptotic properties of the estimators are discussed. A simulation study is conducted to assess the finite sample behavior of the proposed estimators. We illustrate the method using a real life doubly censored competing risks data.

A Semiparametric Bayesian Approach for the Analysis of Competing Risks Data

In survival studies, the failure (death) of an individual may be classified into one of k (k > 1) mutually exclusive classes, usually causes of failure. Competing risks models are used to analyze such situations, where the eventual failure (death) of an individual may due to any one of the k possible causes. In the present study, we introduce a semiparametric Bayesian method for the analysis of competing risks data. We assume that each cumulative baseline cause specific hazard rate function has a gamma prior distribution. A marginal likelihood function based on data and the prior parameter values is proposed for the estimation of regression parameters, by considering cumulative baseline cause specific hazard rate functions as a nuisance parameter. We derive posterior distributions of cumulative baseline cause specific hazard rate functions and then propose a Bayes estimator for the same under squared error loss function. A simulation study is carried out to assess the finite sample performance of the proposed method. We illustrate the practical utility of the method using a real lifetime data set.

Excess mortality and morbidity in patients surviving infective endocarditis

Mortality and morbidity associated with infective endocarditis may extend beyond successful treatment. The primary objective was to analyze rates, temporal changes, and predictors of excess mortality in patients surviving the acute phase of endocarditis. The secondary objective was to determine the rate of recurrence and the need for late cardiac surgery. An observational cohort study was conducted at a university-affiliated tertiary medical center, among 328 patients who survived the active phase of endocarditis. We used age-, sex-, and calendar year-specific mortality hazard rates of the Bouches-du-Rhone French district population to calculate expected survival and excess mortality. The risk of recurrence and late valve surgery was also assessed. Compared with expected survival, patients surviving a first episode of endocarditis had significantly worse outcomes (P = .001). The relative survival rates at 1, 3, and 5 years were 92% (95% CI, 88%-95%), 86% (95% CI, 77%-92%), and 82% (95% CI, 59%-91%), respectively. This excess mortality was observed during the entire follow-up period but was the highest during the first year after hospital discharge. Most of the recurrences and late cardiac surgeries also occurred during this period. Women exhibited a higher risk of age-adjusted excess mortality (adjusted excess hazard ratio, 2.0; 95% CI, 1.05-3.82; P = .03). Comorbidity index, recurrence of endocarditis, and history of an aortic valve endocarditis in women were independent predictors of excess mortality. These results justify close monitoring of patients after successful treatment of endocarditis, at least during the first year. Special attention should be paid to women with aortic valve damage.

Additive hazards models for gap time data with multiple causes

Recurrent event data with multiple causes are often observed in biomedical studies. The additive hazards model describes a different aspect of the association between covariates and the failure time than does the proportional hazards model. In this paper, we introduce additive hazards models for the analysis of gap time data of recurrent events with multiple causes. We estimate the regression parameter vector and cumulative baseline cause specific hazard rate function using counting process approach. Asymptotic properties of the estimators are studied. The proposed model is applied to the kidney dialysis data given in Lawless (2003). A simulation study is carried out to assess the performance of the estimates.

Hazard of Prostate Cancer Specific Mortality After Radical Prostatectomy

Hazard is defined as an event rate at a certain time that is conditional on survival until that time. For most patients with localized malignancies the mortality hazard decreases with time after an initial period of high failure risk. We assessed prostate cancer specific mortality hazard changes with time in men treated with radical prostatectomy. The cohort included 127,236 men from the SEER (Surveillance, Epidemiology and End Results) database who were treated with RP between 1988 and 2003. Pathological stage was organ confined in 38,684 men (30%), nonorgan confined in 41,806 (33%) and unstaged in 46,746 (37%). Gleason score 7 or less was present in 100,816 men (79%) and Gleason score 8 or greater in 26,420 (21%). Patients were stratified into groups, including group 1—71,106 (59%) with Gleason score 7 or less, organ confined, group 2—23,063 (19%) with Gleason score 7 or less, nonorgan confined, group 3—13,660 (12%) with Gleason score 8 or greater, organ confined and group 4—12,158 (10%) with Gleason score 8 or greater, nonorgan confined tumors. Median followup was 7.2 years (range 0 to 19). Hazard was estimated from a Cox regression model including patient age, race, stage and grade. The overall annual prostate cancer specific mortality hazard rate was 0.4%, 0.7% and 1% 5, 10 and 15 years after radical prostatectomy, respectively. Between 5 and 15 years after radical prostatectomy the hazard increased annually from 0.2% to 0.5% in group 1, from 0.5% to 1.2% in group 2, from 0.7% to 1.6% in group 3 and from 1.5% to 3.7% in group 4. In contrast to other prevalent cancers, the hazard of prostate cancer specific mortality shows a modest, constant increase for at least 15 years after radical prostatectomy.

A Phase III Extension Trial With a 1-Arm Crossover From Leuprolide to Degarelix: Comparison of Gonadotropin-Releasing Hormone Agonist and Antagonist Effect on Prostate Cancer

We investigated the efficacy and safety of degarelix treatment and the effects of switching from leuprolide to degarelix in an ongoing extension study with a median 27.5-month followup of a pivotal 1-year prostate cancer trial. Patients who completed a 1-year pivotal phase III trial continued on the same monthly degarelix maintenance dose (160 or 80 mg in 125 each), or were re-randomized from leuprolide 7.5 mg to degarelix 240/80 mg (69) or 240/160 mg (65). Data are shown on the approved degarelix 240/80 mg dose. The primary end point was safety/tolerability and the secondary end points were testosterone, prostate specific antigen, luteinizing hormone and follicle-stimulating hormone responses, and prostate specific antigen failure and progression-free survival. During followup testosterone and prostate specific antigen suppression were similar to those in the 1-year trial in patients who continued on degarelix or switched from leuprolide. The prostate specific antigen progression-free survival hazard rate was decreased significantly after the switch in the leuprolide/degarelix group while the rate in those who continued on degarelix was consistent with the rate in treatment year 1. The same hazard rate change pattern occurred in the group with baseline prostate specific antigen greater than 20 ng/ml. Adverse event frequency was similar between the groups and decreased with time. Data support the statistically significant prostate specific antigen progression-free survival benefit for degarelix over leuprolide seen during year 1 and the use of degarelix as first line androgen deprivation therapy as an alternative to a gonadotropin-releasing hormone agonist.

Recurrence dynamics does not depend on the recurrence site

IntroductionThe dynamics of breast cancer recurrence and death, indicating a bimodal hazard rate pattern, has been confirmed in various databases. A few explanations have been suggested to help interpret this finding, assuming that each peak is generated by clustering of similar recurrences and different peaks result from distinct categories of recurrence.MethodsThe recurrence dynamics was analysed in a series of 1526 patients undergoing conservative surgery at the National Cancer Institute of Milan, Italy, for whom the site of first recurrence was recorded. The study was focused on the first clinically relevant event occurring during the follow up (ie, local recurrence, distant metastasis, contralateral breast cancer, second primary tumour), the dynamics of which was studied by estimating the specific hazard rate.ResultsThe hazard rate for any recurrence (including both local and distant disease relapses) displayed a bimodal pattern with a first surge peaking at about 24 months and a second peak at almost 60 months. The same pattern was observed when the whole recurrence risk was split into the risk of local recurrence and the risk of distant metastasis. However, the hazard rate curves for both contralateral breast tumours and second primary tumours revealed a uniform course at an almost constant level. When patients with distant metastases were grouped by site of recurrence (soft tissue, bone, lung or liver or central nervous system), the corresponding hazard rate curves displayed the typical bimodal pattern with a first peak at about 24 months and a later peak at about 60 months.ConclusionsThe bimodal dynamics for early stage breast cancer recurrence is again confirmed, providing support to the proposed tumour-dormancy-based model. The recurrence dynamics does not depend on the site of metastasis indicating that the timing of recurrences is generated by factors influencing the metastatic development regardless of the seeded organ. This finding supports the view that the disease course after surgical removal of the primary tumour follows a common pathway with well-defined steps and that the recurrence risk pattern results from inherent features of the metastasis development process, which are apparently attributable to tumour cells.

Comparison of failure probabilities in the presence of competing risks

Typically, differences in the effect of treatment on competing risks are compared by a weighted log-rank test. This test compares the cause specific hazard rates between the groups. Often the test does not agree with the impressions gained from plots of the cumulative incidence functions. Here we discuss several K-sample tests allowing us to directly compare cumulative incidence functions. These include tests based on the weighted integrated difference between the subdistribution hazards or cumulative incidence functions, Kolmogorov-Smirnov type test, and Renyi type test. In addition to unadjusted comparison techniques, tests based on the regression modeling of the cumulative incidence functions are considered. A simulation study is used to compare the various tests and to assess their power against different alternatives. The methods are illustrated using real data examples.

A Credit Risk Model with Dynamic Frailties for Default Intensity Estimation

In the spirit of reduced-form models, default is treated as an unpredictable jump with an exogenously specific hazard rate process. Most researchers using reduced-form models assume doubly stochastic properties. In practice, this assumption is annulled in the presence of frailty factors, which are unobservable explanatory variables correlated across firms. However, few if any methods are used to estimate a credit risk model with frailty factors. This paper proposes a maximization likelihood estimation method using a proportional hazard model and an algorithm for implementation. The proportional hazard model is a popular method for analyzing the effect of covariates on the hazard rate. An application to real data is then presented, with the results showing that the model with frailty is more accurate than the model without it.

Conditional tests in a competing risks model

Testing for equality of competing risks based on their cumulative incidence functions (CIFs) or their cause specific hazard rates (CSHRs) has been considered by many authors. The finite sample distributions of the existing test statistics are in general complicated and the use of their asymptotic distributions can lead to conservative tests. In this paper we show how to perform some of these tests using the conditional distributions of their corresponding test statistics instead (conditional on the observed data). The resulting conditional tests are initially developed for the case of k = 2 and are then extended to k > 2 by performing a sequence of two sample tests and by combining several risks into one. A simulation study to compare the powers of several tests based on their conditional and asymptotic distributions shows that using conditional tests leads to a gain in power. A real life example is also discussed to show how to implement such conditional tests.

Inference for the Dependent Competing Risks Model with Masked Causes of Failure

The competing risks model is useful in settings in which individuals/units may die/fail for different reasons. The cause specific hazard rates are taken to be piecewise constant functions. A complication arises when some of the failures are masked within a group of possible causes. Traditionally, statistical inference is performed under the assumption that the failure causes act independently on each item. In this paper we propose an EM-based approach which allows for dependent competing risks and produces estimators for the sub-distribution functions. We also discuss identifiability of parameters if none of the masked items have their cause of failure clarified in a second stage analysis (e.g. autopsy). The procedures proposed are illustrated with two datasets.

Nonparametric tests for cause specific hazard rates with censored data for competing risks among several groups

For the competing risk setting where the lifetime data are due to one of several distinct and exclusive causes, comparison of cause-specific hazard rates is of interest to researchers. In this paper we survey existing methods for related tests and provide a new test for a common overall rate for all causes and groups. Tests given in the literature for checking for a common rate for causes and a common rate for the groups are shown to be in the same framework as the proposed test for common overall rate. Asymptotics are shown to follow a common theme for each test. An extensive numerical and graphical investigation and an example are presented to substantiate the proposed methods.

Consistent estimation of distributions with type II bias with applications in competing risks problems

A random variable X is symmetric about 0 if X and -X have the same distribution. There is a large literature on the estimation of a distribution function (DF) under the symmetry restriction and tests for checking this symmetry assumption. Often the alternative describes some notion of skewness or one-sided bias. Various notions can be described by an ordering of the distributions of X and -X. One such important ordering is that $P(0<X<X\le x)-P(-x\le X<0)$ is increasing in $x<0$. The distribution of X is said to have a Type II positive bias in this case. If X has a density f, then this corresponds to the density ordering $f(-x)\le f(x)$ for $x<0$. It is known that the nonparametric maximum likelihood estimator (NPMLE) of the DF under this restriction is inconsistent. We provide a projection-type estimator that is similar to a consistent estimator of two DFs under uniform stochastic ordering, where the NPMLE also fails to be consistent. The weak convergence of the estimator has been derived which can be used for testing the null hypothesis of symmetry against this one-sided alternative. It also turns out that the same procedure can be used to estimate two cumulative incidence functions in a competing risks problem under the restriction that the cause specific hazard rates are ordered. We also provide some real life examples.

Generalized supremum tests for the equality of cause specific hazard rates.

In this paper we propose two new classes of asymptotically distribution-free Renyi-type tests for testing the equality of two risks in a competing risk model with possible censoring. This work extends the work of Aly, Kochar and McKeague [1994, Journal of American Statistical Association, 89, 994-999] and many of the existing tests for this problem belong to these newly proposed classes. The asymptotic properties of the proposed tests are investigated. Simulation studies are done to compare the performance with existing tests. A competing risks data set is analyzed to demonstrate the usefulness of the procedure.

Likelihood Based Inference for Cause Specific Hazard Rates under Order Restrictions

We consider the competing risks model with grouped data or with discrete Failure times where a unit is exposed to several risks, but its eventual failure is due to exactly one of the causes. Nonparametric maximum likelihood estimates of the cause specific hazard rates are obtained under the restriction that these risks are uniformly ordered. We allow for random censoring. Unlike most papers on this topic, no assumption is made about the independence of the various risks, although we do assume that the censoring mechanism is acting independently of the life distribution. We derive the likelihood ratio statistic for testing the null hypothesis of equality of cause specific hazard rates against ordered alternatives. The asymptotic null distribution of the test statistic is seen to be of the chi-bar squared ( χ 2) type. The procedures developed here are illustrated with the help of an example involving survival rates for mice exposed to radiation.

Survival analyses of radiated animals incorporating competing risks and covariates

AbstractThe United States Air Force has been interested in studying the effect of different types of radiation encountered by its personnel in space. A study was, therefore, conducted at the USAF School of Aerospace Medicine, Brooks Air Force, Texas with rhesus monkeys as experimental subjects. These subjects were exposed to different types of radiation such as: (1) electromagnetic radiation; (2) electrons; (3) protons; and (4) nuclei of elements of higher numbers with different amount of radiation. These subjects were followed over a period of 338 months. In this paper an interesting problem related to health and radiation has been addressed. The effects of radiation, taking into account the cause of death (cancer or heart disease) along with the covariates such as sex, age, type of exposure, dose, are examined. A general log‐linear hazard model approach is studied. The model estimates the cause specific hazard rates, assuming piecewise exponential distribution, and exhibits the survival function for each of the covariate groups and the probability of death due to each cause. A data set called ‘Delayed Bio‐Effects Colony’, of radiated animals, is analysed and some conclusions are drawn. Overall the study has brought out the effect of high and low doses of radiation on both the male and female groups. The procedure presented in the paper distinguishes between time varying hazards. Thus the methodology may be useful for other survival related environmental problems, be this with regard to animal or human survival. Thus, the paper could contribute to its more widespread use.

Les déterminants individuels de la durée du chômage

Individual determinants of unemployment durations, by Liliane Bonnal, Denis Fougère. From a statistical standpoint, this paper analyzes the effects of individual characteristics on average unemployment durations and the types of duration with competing risks. The results show that time-profiles of specific hazard rates or exit rates from unemployment depend on the destination state [the two options basically being: finding a job, and being removed from the ANPE (National Agency for Employment) lists for not checking in]. Furthermore, the individual factors determining these rates differ from one destination state to another. Variables which influence the instantaneous probability to be hired are demographic (sex, age, citizenship, marital status) and socio-economic (education, job qualifications, socio-professional categories) individual characteristics.