Cumulative Hazard Function Research Articles

A Comprehensive and Systematic Analysis of Minimal Residual Disease (MRD) Monitoring in Follicular Lymphoma: Results from the Fondazione Italiana Linfomi (FIL) FOLL12 Trial

A Comprehensive and Systematic Analysis of Minimal Residual Disease (MRD) Monitoring in Follicular Lymphoma: Results from the Fondazione Italiana Linfomi (FIL) FOLL12 Trial

Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints

Modern systems are operating under dynamic environments and the components therein often exhibit positively correlated lifetimes. Moreover, due to various practical reasons such as the load sharing mechanism, it is not uncommon that lifetimes of some components dominate the others within the same system. In this study, we first propose a statistical model for system reliability evaluation by jointly considering the correlated component lifetimes and the lifetime ordering constraints. In specific, the effects of the dynamic environments are incorporated by modelling the cumulative hazard function as an exponential dispersion process and the lifetime ordering constraints are modelled by truncating the support of the joint lifetime distribution. We then discuss the statistical inference based on the proposed model. The point estimates of the model parameters as well as the lifetime quantiles are obtained by the maximum likelihood method, and the confidence intervals are constructed by using the generalized pivots. Extensive simulation show that the proposed interval estimation procedures can achieve accurate coverage even under small sample sizes. Two real examples are used to illustrate the proposed modelling and estimation framework.

The Maxwell – Exponential Distribution: Theory and Application to Lifetime Data

Two parameters Maxwell – Exponential distribution was proposed using the Maxwell generalized family of distribution. The probability density function, cumulative distribution function, survival function, hazard function, quantile function, and statistical properties of the proposed distribution are discussed. The parameters of the proposed distribution have been estimated using the maximum likelihood estimation method. The potentiality of the estimators was shown using a simulation study. The overall assessment of the performance of Maxwell - Exponential distribution was determined by using two real-life datasets. Our findings reveal that the Maxwell – Exponential distribution is more flexible compared to other competing distributions as it has the least value of information criteria.

Normal-Power Function Distribution with Logistic Quantile Function: Properties and Application

Developing compound probability distributions is very important in the field of probability and statistics because there are different datasets from different fields with different features. These features range from high skewness, peakedness (kurtosis), bimodality, highly dispersed, and so on. Existing distributions might not easily fit well to these emerging data of interest. So, there is a need to develop more robust and flexible distributions that are positively skewed, negatively skewed, and bathup shape, to handle some of these features in the emerging data of interest. This paper, therefore, proposed a new four-parameter distribution called the Normal-Power{logistic} distribution. The proposed distribution was characterized by its density, distribution, survival, hazard, cumulative hazard, reversed hazard, and quantile functions. Properties such as the r-th moment, heavy tail property, stochastic ordering, mean inactive time were obtained. A useful transformation of the proposed distribution to normal distribution was shown to help generate its quantiles. The method of Maximum Likelihood Estimation (MLE) was used to estimate the model parameters. A simulation study was carried out to test the consistency of the maximum likelihood parameter estimates. The result of the simulation shows that the biases reduce as the sample size increases for different parameter values. The importance of the new distribution was proved empirically using a real-life dataset of gauge lengths of 10mm. The proposed distribution was compared with five other competing distributions, and the results show that the proposed Normal-Power{logistic} distribution (NPLD) performed favourably than the other five distributions using the AIC, CAIC, BIC, HQIC criteria.

Non-parametric estimation of reciprocity and triadic effects in relational event networks

Relational event networks arise naturally from social interactions: interactions can be considered time-stamped edges between vertices of social actors. By studying the factors that influence the interaction inter-arrival times we can get insight in the drivers of interaction dynamics. Relational event models provide a succinct way to analyse a broad family of interactional patterns and their influence on network dynamics. These models can easily incorporate a wide range of mechanisms involving both endogenous and exogenous network effects. It has been typical for quantitive models to assume that endogenous network mechanisms, such as reciprocity or triadic effects, remain stable over time. However, a number of studies argued that reciprocity has a strong tendency to decay over time.This paper proposes to model the dynamic structure of reciprocal and triadic effects in relational event networks via stratified baseline hazards. This avoids having to define arbitrary temporal windows for what can be considered reciprocity or triadic closure. A two-step estimation framework is used, based on the stratified Cox proportional hazards model, followed by non-parametric estimation of the stratified cumulative hazard functions. This framework is illustrated by a simulation study and its application to two studies involving email communication and classroom interactions.

Maximum Product of Spacing Parameter Estimation of Gompertz Rayleigh Distribution and Application to Rainfall Datasets

Gompertz Rayleigh (GomR) distribution was introduced in an earlier study with few statistical properties derived and parameters estimated using only the most common traditional method, Maximum Likelihood Estimation (MLE). This paper aimed at deriving more statistical properties of the GomR distribution, estimating the three unknown parameters via a competitive method, Maximum Product of Spacing (MPS) and evaluating goodness of fit using rainfall data sets from Nigeria, Malaysia and Argentina.
 Properties of statistical distributions including distribution of smallest and largest order statistics, cumulative or integrated hazard function, odds function, rth non-central moments, moment generating function, mean, variance and entropy measures for GomR distribution were explicitly derived. The fitted data sets reveal the flexibility of GomR distribution over other distributions been compared with. Simulation study was used to evaluate the consistency, accuracy and unbiasedness of the GomR distribution parameter estimates obtained from the method of MPS. The study found that GomR distribution could not provide a better fit for Argentine rainfall data but it was the best distribution for the rainfall data sets from Nigeria and Malaysia in comparison with the distributions; Generalized Weibull Rayleigh (GWR), Exponentiated Weibull Rayleigh (EWR), Type (II) Topp Leone Generalized Inverse Rayleigh (TIITLGIR), Kumarawamy Exponential Inverse Raylrigh (KEIR), Negative Binomial Marshall-Olkin Rayleigh (NBMOR) and Exponentiated Weibull (EW). Furthermore, the estimates from MPSE were consistent as the sample size increases but not as efficient as those from MLE.

Exposure assessment for Cox proportional hazards cure models with interval-censored survival data.

Mixture cure models have been developed as an effective tool to analyze failure time data with a cure fraction. Used in conjunction with the logistic regression model, this model allows covariate-adjusted inference of an exposure effect on the cured probability and the hazard of failure for the uncured subjects. However, the covariate-adjusted inference for the overall exposure effect is not directly provided. In this paper, we describe a Cox proportional hazards cure model to analyze interval-censored survival data in the presence of a cured fraction and then apply a post-estimation approach by using model-predicted estimates difference to assess the overall exposure effect on the restricted mean survival time scale. For baseline hazard/survival function estimation, simple parametric models as fractional polynomials or restricted cubic splines are utilized to approximate the baseline logarithm cumulative hazard function, or, alternatively, the full likelihood is specified through a piecewise linear approximation for the cumulative baseline hazard function. Simulation studies were conducted to demonstrate the unbiasedness of both estimation methods for the overall exposure effect estimates over various baseline hazard distribution shapes. The methods are applied to analyze the interval-censored relapse time data from a smoking cessation study.

Ventriculoperitoneal shunt failure in pediatric patients: an analysis of a national hospitalization database in Thailand.

Shunt failure is common among patients undergoing ventriculoperitoneal shunting for treatment of hydrocephalus. The present study examined long-term shunt failure and associated risk factors in pediatric patients by using a national hospitalization database of Thailand. Patients 17 years or younger who had been admitted to 71 public hospitals in 2012-2017 for first-time ventriculoperitoneal shunting for diseases with known etiology and discharged alive were followed through 2019 to ascertain shunt failure. Shunt survivals were calculated using Kaplan-Meier estimates and time to failure was analyzed to identify risk factors for the first failure by using Cox proportional hazards regression. Differences in risks of subsequent failures with respect to place in the order of failures (i.e., first, second, third) were determined using a cumulative hazard function. Over a median follow-up of 29.9 months, shunt failure occurred in 33.7% of 2072 patients (median age 8.8 months), with a higher proportion in patients < 1 year than in patients 1-17 years (37.8% vs 28.9%, p < 0.001), and ranged from 26.1% of those having posttraumatic hydrocephalus to 35.9% of those having infectious diseases. The shunt failure rates at 3, 6, and 12 months were 11.5%, 19.0%, and 25.2%, respectively. Patients < 1 year had a higher risk of the first failure than patients 1-17 years (hazard ratio 1.45, 95% CI 1.20-1.76). Among those with shunt failure, 35.8% had multiple failures and 52.9% failed within 180 days after the index shunting. The cumulative hazard of subsequent failure was consistently higher than that of an earlier failure regardless of age and etiology, and the cumulative hazard of the second failure in the patients with 180-day failure was higher than that in the patients in whom shunts failed beyond 180 days. Shunt failure occurred more frequently in younger pediatric patients. Much attention should be placed on the initial shunt operation so as to mitigate the failure risk. Close follow-up was crucial once patients had developed the failure, because the risk of subsequent failure was more likely than an earlier one among those with multiple failures.

ON THE PROPERTIES AND APPLICATIONS OF A NEW EXTENSION OF EXPONENTIATED RAYLEIGH DISTRIBUTION

Statistical distributions already in existence are not the most appropriate model that adequately describes real-life data such as those obtained from experimental investigations. Therefore, there are needs to come up with their extended forms to give substitutive adaptable models. By adopting the method of Transformed-Transformer family of distributions, an extension of Exponentiated Rayleigh distribution titled Gompertz- Exponentiated Rayleigh (GOM-ER) distribution was proposed and proved to be valid. Some properties of the new distribution including random number generator, quartiles, distribution of smallest and largest order statistics, reliability function, hazard rate function, cumulative or integrated hazard function, odds function, non-central moments, moment generating function, mean, variance and entropy measures were derived. Using the methods of maximum likelihood and maximum product of spacing, the four unknown parameters were estimated. Shapes of the hazard function depicts that GOM-ER is a distribution that is strictly increasing while those of the PDF depicts that GOM-ER can be skewed or symmetrical. Two datasets were fitted to determine the flexibility of GOM-ER. Simulation study evaluates the consistency, accuracy and unbiasedness of the GOM-ER parameter estimates obtained from the two frequentist estimation methods adopted.

New methods for the additive hazards model with the informatively interval-censored failure time data.

The additive hazards model is one of the most commonly used models for regression analysis of failure time data and many inference procedures have been developed for it under various situations. In particular, Wang etal. (2018a, Computational Statistics and Data Analysis, 125, 1-9) discussed the situation where one observes informatively interval-censored data and proposed a likelihood estimation approach. However , it involves estimation of the unknown baseline cumulative hazard function and thus may be time-consuming . Corresponding to this, we propose two new procedures, an estimating equation-based one and an empirical likelihood-based one, and both do not need estimation of the cumulative hazard function and can be easily implemented. The asymptotic properties of the proposed methods are established and an extensive simulation study suggests that they work well in practical situations. An application is alsoprovided.

Study of factors caused diabetes mellitus and stroke diseases toward gender

Gender consists of female and male. Diabetes mellitus is a chronic disease characterized by high levels of sugar in the blood. Glucose is controlled by the hormone insulin which is produced by the pancreas. In diabetics, the pancreas is unable to produce insulin according to the body’s needs. Blood sugar tests can be done to find out someone has diabetes. This method by a time blood sugar test and fasting blood sugar test. Stroke is a condition that occurs when the blood supply to the brain is interrupted or reduced due to a blockage (ischemic stroke) or rupture of a blood vessel (hemorrhagic stroke). We can use the proportional hazards regression and the stratified proportional hazards model to determine the factors causing diabetes mellitus and stroke. The significant covariate could be seen as a factor causing disease. Significant covariate also yields hazard ratio. If the event time assumed has loglogistics distribution. Some of the functions discussed in this paper, namely the cumulative distribution function, survival function, hazard rate function, and cumulative hazard function. These functions are useful for prediction tools provided the event time is known. The purpose of this study was to find the factors that cause diabetes mellitus and stroke in the gender strata and making functions of random variables with the logistic distribution. The results obtained from this study are the factors that influence a person at risk of stroke namely the type of stroke, hypertension, systolic blood pressure, diastolic blood pressure, and diabetes mellitus. The factors that affect a person at risk of stroke according to gender are the type of stroke, hypertension, systolic blood pressure. If the sample includes all women who have had a stroke, it is primarily affected by the type of stroke. If the sample consists of all men who have had a stroke, it is concluded that the significant covariates are type of stroke, hypertension, systolic blood pressure, and diabetes mellitus. Men have a higher rate of stroke than women.

Estimate survival function of the Topp-Leone exponential distribution with application

This is a new lifetime Exponential using the Topp-Leone generated family of distributions proposed by Rezaei et al. The new distribution is called the Topp-Leone Exponential (TLE) distribution. What is done in this paper is an estimation of the two parameters for Topp-Leone Exponential distribution model by using the maximum likelihood estimator method to get the derivation of the point estimators for all unlabeled parameters according to iterative techniques as Newton $-$ Raphson method, then to derive Ordinary least squares estimator method. Applying all two methods to estimate related probability functions; death density function, cumulative distribution function, survival function and hazard function (rate function). When examining the numerical results for probability survival function by employing mean squares error measure and mean absolute percentage measure, this may lead to work on the best method in modeling a set of real data.

Assessing the adequacy of the gompertz regression model in the presence of right censored data

The Gompertz distribution is often used to model human mortality and establish actuarial tables. Cox-Snell residual is the most general practice to evaluate the model adequacy. However, standard statistical procedures are not amenable to handle the censored observations. Therefore, this research investigates the adequacy of the two parameter Gompertz parametric survival model that was extended to incorporate with fixed covariate in the presence of right censored data. Performance of the model is assessed and compared at various combinations of sample sizes and censoring proportions via simulation study. The newly proposed modifications to the Cox-Snell residuals based on the geometric mean and harmonic mean as well as the jackknife means were compared with Cox-Snell and Modified Cox-Snell residuals via simulation studies at different settings. The log-cumulative hazard plot of residuals is obtained by plotting the proposed residuals against the cumulative hazard function to assess the model’s fit. Based on the model adequacy study, we found out that jackknife geometric mean and jackknife harmonic mean outperform CS, MCS, GMCS and HMCS residuals.

Generalized Rank Mapped Transmuted Distributions with Properties and Application: A Review

Generalizing probability distributions is a very common practice in the theory of statistics. Researchers have proposed several generalized classes of distributions which are very flexible and convenient to study various statistical properties of the distribution and its ability to fit the real-life data. Several methods are available in the literature to generalize new family of distributions. The Quadratic Rank Transmutation Map (QRTM) is a tool for the construction of new families of non-Gaussian distributions and to modulate a given base distribution for modifying the moments like the skewness and kurtosis with the ability to explore its tail properties and improve the adequacy of the distribution. Recently, a new family of transmutation map, defined as Cubic Rank Transmutation (CRT) has been used by several authors to develop new distributions with application to real-life data. In this article, we have done a review work on the existing generalized rank mapped transmuted probability distributions, available in the literature with various statistical properties such as the reliability, hazard rate and cumulative hazard functions, moments, mean, variance, moment-generating function, order statistics, generalized entropy and quantile function along with its applications. Some future works have also been discussed for generalized rank mapped transmuted distributions.

Marginal posterior distributions for regression parameters in the Cox model using Dirichlet and gamma process priors

A Bayesian treatment of the proportional hazard (PH) model is revisited using Dirichlet process and gamma process priors for the baseline survival and cumulative hazard functions respectively. Such priors, due to their discrete support, conflict with the absolutely continuous nature of survival responses expressed through the hazard function structure that defines the PH model. We resolve this conflict through the use of a proposed ε-grid likelihood approach which we apply to the PH model to consider marginal inference for the regression parameter β using expansions as ε↓0. Using Dirichlet process priors, the ε-grid likelihood approach leads to a new explicit marginal posterior density expansion for β which accommodates the most general setting with arbitrary ties and right censoring. In the previously considered case of gamma process priors, the approach extends the results of Kalbfleisch (1978) to deal with arbitrary arrangements of ties and also provides a rigorous justification for his posterior expressions. As in Kalbfleisch (1978), we show that the leading terms of both expansions approximate Cox’s partial likelihood when there are no ties and the process priors are diffuse. The ε-grid likelihood approach is similar in concept to the grouped data likelihood approach of Sinha et al. (2003) but differs in the likelihood approximation. Whereas our expansions are Poincaré expansions (with relative error O(ε) as ε↓0), those of Sinha et al. (2003) are stochastic expansions and not proper Poincaré expansion (as we show), so that they lack relative error O(ε). Using a flat improper prior for β, the two marginal posterior expressions for β are shown to be integrable for general arrangements of ties and censoring under weak conditions on the design and failures.

A family of Gamma-generated distributions: Statistical properties and applications.

In this paper, we concentrate on the statistical properties of Gamma-X family of distributions. A special case of this family is the Gamma-Weibull distribution. Therefore, the statistical properties of Gamma-Weibull distribution as a sub-model of Gamma-X family are discussed such as moments, variance, skewness, kurtosis and Rényi entropy. Also, the parameters of the Gamma-Weibull distribution are estimated by the method of maximum likelihood. Some sub-models of the Gamma-X are investigated, including the cumulative distribution, probability density, survival and hazard functions. The Monte Carlo simulation study is conducted to assess the performances of these estimators. Finally, the adequacy of Gamma-Weibull distribution in data modeling is verified by the two clinical real data sets.Mathematics Subject Classification: 62E99; 62E15.

Semiparametric analysis of zero-inflated recurrent events with a terminal event.

Recurrent event data frequently arise in longitudinal studies and observations on recurrent events could be terminated by a major failure event such as death. In many situations, there exist a large fraction of subjects without any recurrent events of interest. Among these subjects, some are unsusceptible to recurrent events, while others are susceptible but have no recurrent events being observed due to censoring. In this article, we propose a zero-inflated generalized joint frailty model and a sieve maximum likelihood approach to analyze zero-inflated recurrent events with a terminal event. The model provides a considerable flexibility in formulating the effects of covariates on both recurrent events and the terminal event by specifying various transformation functions. In addition, Bernstein polynomials are employed to approximate the unknown cumulative baseline hazard (intensity) function. The estimation procedure can be easily implemented and is computationally fast. Extensive simulation studies are conducted and demonstrate that our proposed method works well for practical situations. Finally, we apply the method to analyze myocardial infarction recurrences in the presence of death in a clinical trial with cardiovascular outcomes.

Generalized accelerated hazards mixture cure models with interval-censored data

Existing semiparametric mixture cure models with interval-censored data often assume a survival model, such as the Cox proportional hazards model, proportional odds model, accelerated failure time model, or their transformations for the susceptible subjects. There are cases in practice that such conventional assumptions may be inappropriate for modeling survival outcomes of susceptible subjects. We propose a more flexible class of generalized accelerated hazards mixture cure models for analysis of interval-censored failure times in the presence of a cure fraction. We develop a sieve maximum likelihood estimation in which the unknown cumulative baseline hazard function is approximated by means of B-splines and bundled with regression parameters. The proposed estimator possesses the properties of consistency and asymptotic normality, and can achieve the optimal global convergence rate under some conditions. Simulation results demonstrate that the proposed estimator performs satisfactorily in finite samples. The application of the proposed method is illustrated by the analysis of smoking cessation data from a lung health study.

OptBand: optimization-based confidence bands for functions to characterize time-to-event distributions

Classical simultaneous confidence bands for survival functions (i.e., Hall–Wellner, equal precision, and empirical likelihood bands) are derived from transformations of the asymptotic Brownian nature of the Nelson–Aalen or Kaplan–Meier estimators. Due to the properties of Brownian motion, a theoretical derivation of the highest confidence density region cannot be obtained in closed form. Instead, we provide confidence bands derived from a related optimization problem with local time processes. These bands can be applied to the one-sample problem regarding both cumulative hazard and survival functions. In addition, we present a solution to the two-sample problem for testing differences in cumulative hazard functions. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies. The proposed bands are applied to clinical trial data to assess survival times for primary biliary cirrhosis patients treated with D-penicillamine.

Using Cauchy Distribution To Estimate Survival Function

This paper intends to estimate the unlabeled two parameters for Cauchy distribution model depend on employing the maximum likelihood estimator method to obtain the derivation of the point estimators for all unlabeled parameters depending on iterative techniques , as Newton – Raphson method , then to derive “Lindley approximation estimator method and then to derive Ordinary least squares estimator method. Applying all these methods to estimate related probability functions; death density function, cumulative distribution function, survival function and hazard function (rate function)”.&#x0D; “When examining the numerical results for probability survival function by employing mean squares error measure and mean absolute percentage measure, this may lead to work on the best method in modeling a set of real data”