Cumulative Function Research Articles

A Multi-Model Survival Analysis of Lung Cancer Using Parametric Techniques

Lung cancer remains one of the leading causes of cancer-related mortality worldwide, underscoring the critical need for effective prognostic tools. This study utilizes survival analysis to explore the factors that influence the survival outcomes of North Central Cancer Treatment data related to lung cancer. The main goal of this study is to compare and contrast various statistical models, including the Weibull, Exponential, Log-gaussian, Gumbel, and Rayleigh models. We have computed important functions, such as the survival function, the hazard function, and the cumulative hazard function, for all the considered distributions. The Anderson-Darling and Cramer Von-Mises tests, which are Goodness of fit tests, effectively compare and assess various parametric regression survival models. The Weibull survival model is interpreted to be the most effective and efficient way to study the lung cancer dataset, which is concluded upon evaluating the results of Anderson-Darling statistic 0.28745, Cramer Von-Mises statistic 0.0450, Mean Survival Probability 0.9697, Mean Cumulative Survival Probability 0.0303, Akaike Information Criterion 1,650.753 and Bayesian Information Criterion 1623.329 of the Weibull, Exponential, Log-gaussian, Gumbel, and Rayleigh parametric regression survival models.

Regression Analysis of the Generalized Accelerated Hazards Models With Interval‐Censored Data Under Case‐Cohort Studies

ABSTRACTCase‐cohort studies are typically undertaken in scenarios where disease incidence is low, or data collection on certain variables is costly. Case‐cohort researches with interval‐censored data often rely on survival models like the proportional hazards models, proportional odds models, accelerated failure time models and so on. However, these standard assumptions might not always be suitable for survival outcomes modelling in practical scenarios. To address this limitation, we provide a more adaptable class of generalized accelerated hazards models for analysing interval‐censored failure times in case‐cohort studies, for which there appears to be no well‐established approach in the literature. We develop a sieve maximum likelihood estimation method where the unknown cumulative baseline hazard function is integrated with regression parameters. The proposed estimators are proven to be consistent and normally distributed. A simulation study is carried out to assess the empirical performance of the proposed methods and suggest their practical effectiveness. Finally, we illustrate the approach using a dataset from a study on Alzheimer's disease.

Sieve Maximum Likelihood Estimation of Partially Linear Transformation Models With Interval-Censored Data.

Partially linear models provide a valuable tool for modeling failure time data with nonlinear covariate effects. Their applicability and importance in survival analysis have been widely acknowledged. To date, numerous inference methods for such models have been developed under traditional right censoring. However, the existing studies seldom target interval-censored data, which provide more coarse information and frequently occur in many scientific studies involving periodical follow-up. In this work, we propose a flexible class of partially linear transformation models to examine parametric and nonparametric covariate effects for interval-censored outcomes. We consider the sieve maximum likelihood estimation approach that approximates the cumulative baseline hazard function and nonparametric covariate effect with the monotone splines and -splines, respectively. We develop an easy-to-implement expectation-maximization algorithm coupled with three-stage data augmentation to facilitate maximization. We establish the consistency of the proposed estimators and the asymptotic distribution of parametric components based on the empirical process techniques. Numerical results from extensive simulation studies indicate that our proposed method performs satisfactorily in finite samples. An application to a study on hypobaric decompression sickness suggests that the variable TR360 exhibits a significant dynamic and nonlinear effect on the risk of developing hypobaric decompression sickness.

Cox regression model with doubly truncated and interval-censored data

Interval sampling is an efficient sampling scheme used in epidemiological studies. Doubly truncated (DT) data arise under this sampling scheme when the failure time can be observed exactly. In practice, the failure time may not be observed and might be recorded only within time intervals, leading to doubly truncated and interval censored (DTIC) data. This article considers regression analysis of DTIC data under the Cox proportional hazards (PH) model and develops the conditional maximum likelihood estimators (cMLEs) for the regression parameters and baseline cumulative hazard function of models. The cMLEs are shown to be consistent and asymptotically normal. Simulation results indicate that the cMLEs perform well for samples of moderate size.

A modified gamma process for RUL prediction based on data with time-varying heavy-tailed distribution

Predicting remaining useful life (RUL) plays a critical role in condition-based maintenance (CBM). However, this task remains challenging as the collected data often has time-varying, non-homogenous properties (trend, variance) and exhibits non-Gaussian distributions. These aspects pose significant challenges to the classical approaches. To address these issues, we propose a modification of the gamma process (suitable for non-Gaussian data) to predict RUL for the degradation process with time-varying characteristics. Specifically, we use the Gompertz cumulative hazard function instead of the linear function used in the classical approach. The modified gamma process aims to outperform the classical one and other variants already used by exploiting its probabilistic nature to effectively manage the uncertainty in the degradation curve. In order to evaluate the effectiveness of the proposed approach, extensive experiments are performed on simulated data with both Gaussian and non-Gaussian distributions. In addition, the performance of the model is validated in real-world scenarios using two benchmark datasets. The results consistently demonstrate the effectiveness of the proposed approach for both simulated and real datasets.

Penalized Likelihood Estimation for Current Status Data with Informative Censoring

Current status data occurs when failure time of subjects in a survival study is only known to be either less or greater than the censoring time. Thus, the failure time is either left – or right – censored. Analyzing data of this structure under the Cox Proportional Hazards model with dependent censoring assumption can be challenging. To address this, a Penalized Maximum Likelihood Estimation (PMLE) approach was proposed. The unknown baseline cumulative hazard functions for both the failure time and the censoring time were estimated using splines. The advantage of penalized approach over unpenalized method is that that the desired smoothness level of the functions are controlled by their respective penalty terms. The possible dependence between the failure and censoring times was accounted for using the gamma-frailty model. An easy to implement hybrid computational algorithm is proposed to estimate the PMLEs and the Bayesian technique was employed for the estimation of the variances of the parameters. Extensive simulation studies were conducted to assess the statistical properties of the PMLEs. It was observed that the realized estimators were not only consistent, asymptotically normal and efficient, but also, were robust to the number of knots chosen, the proportion of dependent censoring used and the frailty distribution employed. The proposed PMLE method was further applied to real data obtained from tumorigenicity experiment.

MODELING THE BENEFITS OF A MARRIAGE REVERSE ANNUITY CONTRACT WITH DEPENDENCY ASSUMPTIONS USING ARCHIMEDEAN COPULA

Social security benefits may not be enough for retirement. Equity release products like marriage reverse annuities can boost retirement income for older couples. Marriage reverse annuity’s contract convert all or part of the real estate value of elderly spouses while they are living (joint life status) or after one partner dies (last survivor status). Since husband and wife face the same death risk, the chance of death between spouses is believed to be dependent for realism. Thus, copula models the future dependency model of a husband and wife. Sklar's theorem states that copulas link bivariate distribution and marginal cumulative functions. One of the most common copulas is Archimedean copula. Clayton, Gumbel, and Frank are Archimedean copula that will be used in this investigation. The Indonesian Mortality Table IV data is used to obtain the marginal distribution of the male and female which will then be used to construct copulas (Clayton, Gumbel, and Frank) that combine two marginal distributions into a joint distribution. The marginal distribution of Indonesian Mortality Table IV is uncertain, hence Canonical Maximum Likelihood parameter estimation is utilized to estimate the parameter of copulas. Multiple-state models depict the marriage reverse annuity model for joint life and last survivor status. The probability structure is based on Sklar's theorem and copula survival function. The contract benefits calculation utilizing copulas (Clayton, Gumbel, and Frank) shows that joint life status benefits are higher than last survivor status. Joint life status uses the dependence assumption with Frank's copula to calculate the smallest annual benefit value of a marriage reverse annuity contract, while last survivor status uses the independence assumption (without copula).

A novel cross-entropy-based importance sampling method for cumulative time-dependent failure probability function

Cumulative time-dependent failure probability function (C-T-FPF) can reflect the effects of distribution parameters of random inputs and the upper bound of service time interval, which vary in their design domains, on time-dependent failure probability. However, solving C-T-FPF involves a time-consuming triple-layer framework. Thus, we propose an efficient method by combining cross-entropy-based importance sampling (IS) with adaptive Kriging model (CE-IS-AK). The innovations of CE-IS-AK include two aspects. Firstly, we construct a space augmented by both distribution parameters and the upper bound of service time interval, on which the triple-layer framework is decoupled to a single-layer one. And in the augmented space, an optimal IS density is proposed to reduce the required candidate sample size for estimating C-T-FPF. Secondly, we employ Gaussian mixture model (GMM) to approximate the optimal IS density, and the parameters in GMM are determined by minimizing the cross entropy of GMM and the optimal IS density. Moreover, to reduce the model evaluations in the proposed method, Kriging model is adaptively embedded to replace the actual model, and a first failure instant based learning function is proposed to train Kriging model adaptively. Due to the proposed single-layer framework and IS strategy assisted by the Kriging model, the efficiency is greatly improved for estimating C-T-FPF, which is validated by several examples.

Frequency-of-seeing curves (psychometric functions) for perimetric stimuli in age-related macular degeneration.

Frequency-of-seeing (FoS) curves (psychometric functions) for perimetric stimuli have been widely used in computer simulations of new visual field test procedures. FoS curves for age-related macular degeneration (AMD) are not available in the literature and are needed for the development of improved microperimetry test procedures, which are of particular interest for use as clinical trial endpoints. Data were refitted from a previous study to generate FoS curves for 20 participants with AMD, each tested at nine locations within the central 10°. Stimulus parameters, background luminance and dB scale were matched to the MAIA-2 microperimeter, and stimuli were presented in a method of constant stimuli to build up FoS curves over multiple runs. FoS curves were fitted with a modified cumulative Gaussian function. The relationship between sensitivity and slope of fitted FoS curves was modelled by robust linear regression, producing models both with and without an eccentricity parameter. FoS curves were satisfactorily fitted to data from 174 visual field locations in 20 participants (age 65-83 years, 11 female). Each curve was made up of a median of 243 (range 177-297) stimulus presentations over a median of 12 (range 9-32) levels. Median sensitivity was 25.5 dB (range 3.8-31.4 dB). The median slope (SD of fitted function) was 1.6 dB (range 0.5-8.5 dB). As in previous studies of other conditions, the slope of fitted FoS curves increased as sensitivity decreased (p < 0.001). FoS are provided for participants with AMD, as well as models of the relationship between sensitivity and slope. These fitted models and data may be useful for computer simulation studies of microperimetry procedures. Full details of the fitted curves are provided as supporting information.

The Time Course of Injury Risk After Return-to-Play in Professional Football (Soccer).

Injury risk in professional football (soccer) is increased in the weeks following return-to-play (RTP). However, the time course of injury risk after RTP (the hazard curve) as well as its influencing factors are largely unknown. This knowledge gap, which is arguably due to the volatility of instantaneous risk when calculated for short time intervals, impedes on informed RTP decision making and post-RTP player management. This study aimed to characterize the hazard curve for non-contact time-loss injuries after RTP in male professional football and to investigate the influence of the severity of the index injury and playing position. Media-based injury records from the first German football league were collected over four seasons as previously published. Time-to-event analysis was employed for non-contact time-loss injury after RTP. The Kaplan-Meier survival function was used to calculate the cumulative hazard function, from which the continuous hazard function was retrieved by derivation. There were 1623 observed and 1520 censored events from 646 players analyzed. The overall shape of the hazard curve was compatible with an exponential decline of injury risk, from an approximately two-fold level shortly after RTP towards baseline, with a half-time of about 4weeks. Interestingly, the peak of the hazard curve was slightly delayed for moderate and more clearly for severe index injuries. The time course of injury risk after RTP (the hazard curve) can be characterized based on the Kaplan-Meier model. The shape of the hazard curve and its influencing factors are of practical as well as methodological relevance and warrant further investigation.

Nested case–control sampling without replacement

Nested case–control design (NCC) is a cost-effective outcome-dependent design in epidemiology that collects all cases and a fixed number of controls at the time of case diagnosis from a large cohort. Due to inefficiency relative to full cohort studies, previous research developed various estimation methodologies but changing designs in the formulation of risk sets was considered only in view of potential bias in the partial likelihood estimation. In this paper, we study a modified design that excludes previously selected controls from risk sets in view of efficiency improvement as well as bias. To this end, we extend the inverse probability weighting method of Samuelsen which was shown to outperform the partial likelihood estimator in the standard setting. We develop its asymptotic theory and a variance estimation of both regression coefficients and the cumulative baseline hazard function that takes account of the complex feature of the modified sampling design. In addition to good finite sample performance of variance estimation, simulation studies show that the modified design with the proposed estimator is more efficient than the standard design. Examples are provided using data from NIH-AARP Diet and Health Cohort Study.

A novel mathematical method to estimate rice phenological parameters across spatial scales for the ORYZA model

While crop models are increasingly applied to large-scale areas, inadequate observation data make it difficult to calibrate the model’s phenological parameters at the regional scale. The present study proposed a simple mathematical method for estimating rice phenological parameters across spatial scales for the ORYZA model. The method establishes the cumulative function of the phenological parameters (CFPP). By fitting the CFPP with the so-called growth curve, the 4 phenological parameters of the ORYZA model were transformed into 3 fitted parameters in the equation of CFPP. Functions between the fitted parameters and several meteorological and field management factors were established. These established functions were substituted back into CFPP to construct a modified CFPP. Due to the inter-translational relationship between CFPP and the original phenological parameters, the values of phenological parameters could be estimated by the modified CFPP based on meteorological and field management factors. The newly proposed mathematical method was applied in the Yangtze River Basin (YRB), China. The results indicated that the multi-station average of the absolute value of relative errors for the rice panicle initiation, flowering, and maturity dates within the YRB were 12.3 %, 10.5 %, and 8.7 %, respectively, which were at most 4.8 % larger than that simulated using parameters calibrated using each station’s phenological data. The phenological parameters estimated by the novel mathematical method had close performance to those calibrated directly based on observed data at most stations in terms of rice phenology simulation. The present study provided a new solution for phenological parameter calibration for crop models when applied in a large-scale area.

How Electrical Storms Recur Over Time in Patients With Implantable Cardioverter Defibrillators - Subanalysis of the Nippon Storm Study.

Electrical storms (E-storms), defined as multiple fatal ventricular arrhythmias over a short period, negatively affect the prognosis of patients receiving an implantable cardioverter defibrillator or cardiac resynchronization therapy with a defibrillator (ICD/CRT-D). However, the prognostic impact of recurrent E-storms has not been well elucidated. We analyzed the association between E-storm recurrences and mortality using data from 1,274 participants in the Nippon Storm Study, a prospective observational study conducted at 48 ICD/CRT-D centers in Japan. Differences in E-storm recurrences by patient characteristics were evaluated using the mean cumulative function (MCF), which is the cumulative number of E-storm episodes per patient as a function of time. Patients with multiple E-storms had a 3.39-fold higher mortality risk than those without E-storms (95% confidence interval 1.82-6.28; P<0.01). However, there was no significant difference in mortality risk between patients with a single E-storm and those without E-storms. The MCF curve exhibited a slower ascent in patients who received primary prevention ICD/CRT-D than in those who received secondary prevention ICD/CRT-D. However, when analyzing only patients with E-storms, the MCF curves demonstrated comparable trajectories in both groups. E-storm recurrences may have a negative impact on prognosis. Once patients with primary prevention experience an E-storm episode, they face a similar risk of subsequent recurrent E-storms as patients with secondary prevention.

Electronic structure evolution upon lithiation: A Li K-edge study of silicon oxide anode through X-ray Raman spectroscopy

The comprehensive understanding of the local structural changes surrounding lithium in lithium silicate (LixSiOy) and silicide (LixSi) within Li/SiOx batteries during the reversible structural transformations has been hindered by the limitations of current methodologies. In this work, the evolution of electronic structure at various lithiation stages has been addressed well by examining the Li K-edge spectra through X-ray Raman spectroscopy (XRS). The features observed in the Li K-edge XRS spectra provide insights into the development and alteration of LixSiOy, which emerges in the initial phases and may be accompanied by a reduction in the ionicity of Li–O bonding during lithiation. These features also agree well with the accompanying FDMNES code simulation. The correlation between electrochemical mechanisms and spectral characteristics is further explored by applying pseudo-Voigt peaks and cumulative pseudo-Voigt functions for fitting purposes. The absence of a significant edge shift indicates a similarity in the electronic structure of LixSi throughout lithiation, and no evidence of Li2O formation has also been observed. The Li K-edge XRS spectra exhibit strong agreement with the electrochemical behavior, establishing it as a valuable tool for investigating the evolution of electronic structure in Li/SiOx batteries.

On the scaling of river network biogeochemical function

AbstractRiver networks play a fundamental biogeochemical role in the Earth system by transporting and processing materials from terrestrial to ocean ecosystems. The cumulative biogeochemical function of a watershed of area can broadly be referred to as the total processing rate of material performed by its river network. An important recent research, conducted through network simulations, has revealed that the biogeochemical function of rivers can scale superlinearly with the area under certain scenarios. This finding has significant implications for the role of river networks in regional and global biogeochemical cycles. Here, we demonstrate how such scaling can be derived analytically by combining the power law distribution of drainage area, the universal fractal signature of river networks and the scaling of channel hydraulic geometry, utilising the theory of finite‐size scaling. The results enable the discrimination between linear and superlinear behaviours, as well as the calculation of the exact exponent based on parameters that define how the biogeochemical function and the river width change with river drainage area. Furthermore, we investigate the difference between the scaling of the biogeochemical function with the area of the watershed and with the area of a region drained by multiple river networks, emphasising the implications for upscaling efforts.

Joint Probability Densities on Riemannian Manifolds are Symmetric Tensor Densities

This paper presents the tensor properties of joint probability densities on a Riemannian manifold. Initially, we develop a binary data matrix to record the values of a large number of particles confining in a closed system at a certain time in order to retrieve the joint probability densities of related variables. By introducing the particle-oriented coordinate and the generalized inner product as a multi-linear operation on the basis of this coordinate, we extract the set of joint probabilities and prove them to meet covariant tensor properties on a general Riemannian space of variables. Based on the Taylor expansion of scalar fields in Riemannian manifolds, it has been shown that the symmetrized iterative covariant derivatives of the cumulative probability function defined on Riemannian manifolds also give the set of related joint probability densities equivalent to the aforementioned multi-linear method. We show these covariant tensors reduce to classical ordinary partial derivatives in ordinary Euclidean space with Cartesian coordinates and give the formal definition of joint probabilities by partial derivatives of the cumulative distribution function. The equivalence between the symmetrized covariant derivative and the generalized inner product has been concluded. Some examples of well-known physical tensors clarify that many deterministic physical variables are presented as tensor densities with an interpretation similar to probability densities.

Compositional nutrient analysis of greenhouse grown ryegrass (Lolium multiforum) fertilized with raw and hydrothermal-treated igneous rock as diagnostic tool for balanced fertilization in tropical environments

Due to the escalating demand for conventional potassium sources, and their unavailability in many African countries, there is a growing interest in hydrothermally treated igneous rocks as viable alternatives. These materials exhibit promising potential in enhancing potassium release, comparable to traditional sources like muriate of potash (M.O.P.). However, it remains uncertain whether they can adequately fulfill the nutrient requirements of plants. In this study, we evaluated nutrient balance in ryegrass (Lolium multiflorum) using raw and hydrothermally treated K-bearing silicate rocks as fertilizers. The Compositional Nutrient Diagnostic (CND) approach was used to assess the plants’ nutrient status, involving identification of high-yield subpopulations, and setting a yield cutoff. We derived a theoretical threshold for nutrient imbalance (CNDr2) using statistical methods like the chi-square distribution function and low-yield subpopulation proportion. CNDr2 was validated via Cate–Nelson partition and sum of squared individual nutrient indexes. Results showed a yield cutoff of 37.759 kg ha−1, distinguishing low-yield and high-yield subpopulations based on cumulative variance ratio functions from survey data. The theoretical threshold derived using the chi-square function was 10.1, subsequently validated by the Cate–Nelson method. Critical CND nutrient indices were symmetrical around zero, but their sum indicated foliar nutrient imbalance, notably zinc, potassium, and copper excess, attributed to 20 interactions identified (4 synergetic and 16 antagonistic). This study underscores hydrothermal treatment’s efficacy in improving nutrient availability and achieving a balanced nutrient profile compared to raw materials. It offers a promising solution for enhancing agricultural productivity, especially in tropical regions like Africa, where traditional sources are scarce.

Dynamic Analysis of the Nonlinear Relationship Between Key Metal Prices and New Energy Vehicle Market

Up against the global energy transformation, the new energy vehicle industry has been in a stage of rapid development. Key metals represented by nickel are indispensable raw materials supporting the new energy vehicle industry, with their price fluctuations influencing its development, thereby affecting the global energy transformation. Studying the dynamic nonlinear relationship between the price fluctuation of the nickel as the key metal and the new energy vehicle market will provide a new perspective for understanding the dynamic development of the new energy vehicle market. Based on nickel price data, new energy vehicle market index, combined with Markov vector autoregressive model and cumulative impulse response function, this paper conducts research on the impact of nickel price fluctuations on the new energy vehicle market. According to the empirical results, the impact of nickel price fluctuations on the new energy vehicle market is different under various regimes. During the low-speed and high-speed development periods, nickel price fluctuations have a negative impact on the new energy vehicle market. During the stable development period, nickel price fluctuations mainly have a positive impact on the new energy vehicle market. Based on the above conclusions, this paper proposes policy suggestions for the relevant subjects.

The Centennial Allograft: Cumulative Kidney and Liver Function for More Than 100 Years

The maximum cumulative life span of kidneys and livers first in donors and then in transplant recipients has not been established. The purpose of this study was to determine if cumulative organ function for more than 90 years is possible for transplanted kidneys and livers. This study included kidney and liver transplants from living or deceased donors ≥55 years. Cumulative organ function (COF) = Organ Age at Donation [Years] + Tx Allograft Function [Years]. Univariate and multivariable methods were used to describe characteristics and outcomes. Between 1987 and 2022, a total of 81,807 kidney and 37,099 liver transplants were included in this study. Of all kidney grafts 2.7% but 16.6% of all liver grafts reached the 90-year COF mark. There were only 2 living donor kidneys that surpassed the 100-year mark versus 29 deceased liver grafts. The longest kidney function was 104 years and longest liver function 108 years. Multivariate analysis showed that optimal donor and recipient selection and management are predictors for allograft longevity. COF in organs exceeding 100 physiologic years is possible. Extended organ longevity was 5 times more common for livers than kidneys. These analyses support that age alone should not exclude older kidney and liver donors from consideration for transplantation.

Unit information Dirichlet process prior.

Prior distributions, which represent one's belief in the distributions of unknown parameters before observing the data, impact Bayesian inference in a critical and fundamental way. With the ability to incorporate external information from expert opinions or historical datasets, the priors, if specified appropriately, can improve the statistical efficiency of Bayesian inference. In survival analysis, based on the concept of unit information (UI) under parametric models, we propose the unit information Dirichlet process (UIDP) as a new class of nonparametric priors for the underlying distribution of time-to-event data. By deriving the Fisher information in terms of the differential of the cumulative hazard function, the UIDP prior is formulated to match its prior UI with the weighted average of UI in historical datasets and thus can utilize both parametric and nonparametric information provided by historical datasets. With a Markov chain Monte Carlo algorithm, simulations and real data analysis demonstrate that the UIDP prior can adaptively borrow historical information and improve statistical efficiency in survival analysis.