on the efficiency of newton and broyden numerical methods in the investigation of the regular polygon problem of n 1 bodies numerical methods of finding the roots of a system of non linear algebraic equations are treated in this paper this paper attempts to give an answer to the selection of the most efficient method in a complex problem of celestial dynamics the so called ring problem of n 1 bodies we apply newton and broyden s method to these problems and we investigate by means of their use the planar equilibrium points the five equilibrium zones which are symbolized by a 1 a 2 b c 2 and c 1 by order of appearance from the center o to the periphery of the imaginary circle on which the primaries lie t j kalvouridis a planar case of the n 1 body problem the ring problem astrophys space sci 260 3 1999 309 325 and the attracting regions of the system the efficiency of these methods is studied through a comparative process the obtained results are demonstrated in figures and are discussed

estimating medical costs with censored data incompleteness of follow up data is a common problem in estimating medical costs naive analysis using summary statistics on the collected data can result in severely misleading statistical inference this paper focuses on the problem of estimating the mean medical cost from a sample of individuals whose medical costs may be right censored a class of weighted estimators which account appropriately for censoring are introduced our estimators are shown to be consistent and asymptotically normal with easily estimated variances the efficiency of these estimators is studied with the goal of finding as efficient an estimator for the mean medical cost as is feasible extensive simulation studies are used to show that our estimators perform well in finite samples even with heavily censored data for a variety of circumstances the methods are applied to a set of cost data from a cardiology trial conducted by the duke university medical center extensions to other censored data problems are also discussed

temporal process regression summary we consider regression for response and covariates which are temporal processes observed over intervals a functional generalised linear model is proposed which includes extensions of standard models in multi state survival analysis simple nonparametric esti mators of time indexed parameters are developed using working independence estimating equations and are shown to be uniformly consistent and to converge weakly to gaussian processes the procedure does not require smoothing or a markov assumption unlike approaches based on transition intensities the usual definition of optimal estimating equations for parametric models is then generalised to the functional model and the optimum is identified in a class of functional generalised estimating equations simulations demonstrate large efficiency gains relative to working independence at times where censoring is heavy the estimators are the basis for new tests of the covariate effects and for the estimation of models in which greater structure is imposed on the parameters providing novel goodness of fit tests the methodology s practical utility is illustrated in a data analysis

association measures for clustered competing risks we propose a semiparameteric model for multivariate clustered competing risks data when the cause specific failure times and the occurrence of competing risk events among subjects within the same cluster are of interest the cause specific hazard functions are assumed to follow cox proportional hazard models and the associations between failure times given the same or different cause events and the associations between occurrences of competing risk events within the same cluster are investigated through copula models a cross odds ratio measure is explored under our proposed models two stage estimation procedure is proposed in which the marginal models are estimated in the first stage and the dependence parameters are estimated via an expectation maximization algorithm in the second stage the proposed estimators are shown to yield consistent and asymptotically normal under mild regularity conditions simulation studies are conducted to assess finite sample performance of the proposed method the proposed technique is demonstrated through an application to a multicenter bone marrow transplantation dataset

absolute risk regression for competing risks interpretation link functions and prediction in survival analysis with competing risks the transformation model allows different functions between the outcome and explanatory variables however the model s prediction accuracy and the interpretation of parameters may be sensitive to the choice of link function we review the practical implications of different link functions for regression of the absolute risk or cumulative incidence of an event specifically we consider models in which the regression coefficients have the following interpretation the probability of dying from cause d during the next t years changes with a factor exp for a one unit change of the corresponding predictor variable given fixed values for the other predictor variables the models have a direct interpretation for the predictive ability of the risk factors we propose some tools to justify the models in comparison with traditional approaches that combine a series of cause specific cox regression models or use the fine gray model we illustrate the methods with the use of bone marrow transplant data

an algebraic method to solve the radial schrodinger equation we propose a method of numerical integration of differential equations of the type x 2y f x y 0 by approximating its solution with solutions of equations of the type x 2y ax 2 bx c y 0 this approximation is performed by segmentary approximation on an interval we apply the method to obtain approximate solutions of the radial schrodinger equation on a given interval and test it for two different potentials we conclude that our method gives a similar accuracy than the taylor method of higher order

global optimal solutions for proximal fuzzy contractions abstract best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution x that is optimal in the sense that the error d x t x assumes the global minimum value d a b in the present paper we initiate some new classes of proximal contraction mappings and obtain best proximity point theorems for such fuzzy mappings in a non archimedean fuzzy metric space as outcomes of these theorems we conclude evident new best proximity and fixed point theorems in non archimedean fuzzy metric spaces with partial order furthermore we provide an example to elaborate the usability of the established results

censored quantile regression with partially functional effects quantile regression offers a flexible approach to analyzing survival data allowing each covariate effect to vary with quantiles in practice constancy is often found to be adequate for some covariates in this paper we study censored quantile regression tailored to the partially functional effect setting with a mixture of varying and constant effects such a model can offer a simpler view regarding covariate survival association and moreover can enable improvement in estimation efficiency we propose profile estimating equations and present an iterative algorithm that can be readily and stably implemented asymptotic properties of the resultant estimators are established a simple resampling based inference procedure is developed and justified extensive simulation studies demonstrate efficiency gains of the proposed method over a naive two stage procedure the proposed method is illustrated via an application to a recent renal dialysis study

a class of k sample tests for comparing the cumulative incidence of a competing risk in this paper for right censored competing risks data a class of tests developed for comparing the cumulative incidence of a particular type of failure among different groups the tests are based on comparing weighted averages of the hazards of the subdistribution for the failure type of interest asymptotic results are derived by expressing the statistics in terms of counting processes and using martingale central limit theory it is proposed that weight functions very similar to those for the g p tests from ordinary survival analysis be used simulation results indicate that the asymptotic distributions provide adequate approximations in moderate sized samples

analysing competing risks data with transformation models we present a flexible class of marginal models for the cumulative incidence function the semiparametric transformation model is utilized in a decomposition for the marginal failure probabilities which extends previous work on farewell s cure model novel estimation inference and prediction procedures are developed with large sample properties derived from the theory of martingales and u statistics a small simulation study demonstrates that the methods are appropriate for practical use the methods are illustrated with a thorough analysis of a prostate cancer clinical trial simple graphical displays are used to check for the goodness of fit

a two step explicit p stable method of high phase lag order for linear periodic ivps in this paper we present a nonlinear two step explicit p stable method of fourth algebraic order and 12th phase lag order for solving one dimensional second order linear periodic initial value problems ivps of ordinary differential equations based on a special vector arithmetic with respect to an analytic function the method can be extended to be vector applicable for multi dimensional problems directly some numerical results are reported to illustrate the efficiency of the method

maximum likelihood estimation of semiparametric mixture component models for competing risks data summary in the analysis of competing risks data the cumulative incidence function is a useful quantity to characterize the crude risk of failure from a specific event type in this article we consider an efficient semiparametric analysis of mixture component models on cumulative incidence functions under the proposed mixture model latency survival regressions given the event type are performed through a class of semiparametric models that encompasses the proportional hazards model and the proportional odds model allowing for time dependent covariates the marginal proportions of the occurrences of cause specific events are assessed by a multinomial logistic model our mixture modeling approach is advantageous in that it makes a joint estimation of model parameters associated with all competing risks under consideration satisfying the constraint that the cumulative probability of failing from any cause adds up to one given any covariates we develop a novel maximum likelihood scheme based on semiparametric regression analysis that facilitates efficient and reliable estimation statistical inferences can be conveniently made from the inverse of the observed information matrix we establish the consistency and asymptotic normality of the proposed estimators we validate small sample properties with simulations and demonstrate the methodology with a data set from a study of follicular lymphoma

generalised linear models for correlated pseudo observations with applications to multi state models will be in a given state at some time is a complex nonlinear function of the intensity regression coefficients we present a technique which models the state probabilities directly this method is based on the pseudo values from a jackknife statistic constructed from simple summary statistic estimates of the state probabilities these pseudo values are then used in a generalised estimating equation to obtain estimates of the model parameters we illustrate how this technique works by studying examples of common regression problems we apply the technique to model acute graft versus host disease in bone marrow transplants

a two step explicit p stable method of high phase lag order for second order ivps this paper presents a two step explicit p stable method of high phase lag order for solving second order ordinary differential equations basing on a special vector operation with respect to a smooth function the method can be extended to be vector applicable in multi dimensional space

estimating subject specific dependent competing risk profile with censored event time observations in a longitudinal study suppose that the primary endpoint is the time to a specific event this response variable however may be censored by an independent censoring variable or by the occurrence of one of several dependent competing events for each study subject a set of baseline covariates is collected the question is how to construct a reliable prediction rule for the future subject s profile of all competing risks of interest at a specific time point for risk benefit decision makings in this paper we propose a two stage procedure to make inferences about such subject specific profiles for the first step we use a parametric model to obtain a univariate risk index score system we then estimate consistently the average competing risks for subjects which have the same parametric index score via a nonparametric function estimation procedure we illustrate this new proposal with the data from a randomized clinical trial for evaluating the efficacy of a treatment for prostate cancer the primary endpoint for this study was the time to prostate cancer death but had two types of dependent competing events one from cardiovascular death and the other from death of other causes

confidence bands for cumulative incidence curves under the additive risk model in the context of competing risks the cumulative incidence function is often used to summarize the cause specific failure time data as an alternative to the proportional hazards model the additive risk model is used to investigate covariate effects by specifying that the subject specific hazard function is the sum of a baseline hazard function and a regression function of covariates based on such a formulation we present an approach to constructing simultaneous confidence intervals for the cause specific cumulative incidence function of patients with given risk factors a melanoma data set is used for the purpose of illustration

prediction of cumulative incidence function under the proportional hazards model in the presence of dependent competing risks in survival analysis the cox model can be utilized to examine the covariate effects on the cause specific hazard function for the failure type of interest for this situation the cumulative incidence function provides an intuitively appealing summary curve for marginal probabilities of this particular event in this paper we show how to construct confidence intervals and bands for such a function under the cox model for future patients with certain covariates our proposals are illustrated with data from a prostate cancer trial

the proportional odds cumulative incidence model for competing risks we suggest an estimator for the proportional odds cumulative incidence model for competing risks data the key advantage of this model is that the regression parameters have the simple and useful odds ratio interpretation the model has been considered by many authors but it is rarely used in practice due to the lack of reliable estimation procedures we suggest such procedures and show that their performance improve considerably on existing methods we also suggest a goodness of fit test for the proportional odds assumption we derive the large sample properties and provide estimators of the asymptotic variance the method is illustrated by an application in a bone marrow transplant study and the finite sample properties are assessed by simulations

a family of numerov type exponentially fitted predictor corrector methods for the numerical integration of the radial schro dinger equation a family of predictor corrector exponential numerov type methods is developed for the numerical integration of the one dimensional schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions the new methods are very simple and integrate more exponential functions than both the well known fourth order numerov type exponentially fitted methods and the sixth algebraic order runge kutta type methods numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above

a two step method for the numerical solution of the radial schrodinger equation abstract a two step method is developed for the numerical solution of the radial schrodinger equation numerical results obtained for the integration of resonance problem and eigenvalue problem show that this new method is better than other similar methods

best proximity points for cyclic meir keeler contractions abstract we introduce a notion of cyclic meir keeler contractions and prove a theorem which assures the existence and uniqueness of a best proximity point for cyclic meir keeler contractions this theorem is a generalization of a recent result due to eldred and veeramani

best proximity point theorems an exploration of a common solution to approximation and optimization problems abstract given a non self mapping t a b in the setting of a metric space this work concentrates on the resolution of the non linear programming problem of globally minimizing the real valued function x d x tx thereby yielding an optimal approximate solution to the equation tx x an iterative algorithm is also presented to compute a solution of such problems as a sequel it is possible to compute an optimal approximate solution to some non linear equations

checking the cox model with cumulative sums of martingale based residuals summary this paper presents a new class of graphical and numerical methods for checking the adequacy of the cox regression model the procedures are derived from cumulative sums of martingale based residuals over follow up time and or covariate values the distributions of these stochastic processes under the assumed model can be approximated by zero mean gaussian processes each observed process can then be compared both visually and analytically with a number of simulated realizations from the approximate null distribution these comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are special attention is given to checking the functional form of a covariate the form of the link function and the validity of the proportional hazards assumption an omnibus test consistent against any model misspecification is also studied the proposed techniques are illustrated with two real data sets

best proximity points convergence and existence theorems for p cyclic mappings abstract we introduce a new class of mappings called p cyclic contractions which contains the p cyclic contraction mappings as a subclass then convergence and existence results of best proximity points for p cyclic contraction mappings are obtained moreover we prove results of the existence of best proximity points in a reflexive banach space these results are generalizations of the results of al thagafi and shahzad 2009 8

instrumental variable with competing risk model in this paper we discuss causal inference on the efficacy of a treatment or medication on a time to event outcome with competing risks although the treatment group can be randomized there can be confoundings between the compliance and the outcome unmeasured confoundings may exist even after adjustment for measured covariates instrumental variable methods are commonly used to yield consistent estimations of causal parameters in the presence of unmeasured confoundings on the basis of a semiparametric additive hazard model for the subdistribution hazard we propose an instrumental variable estimator to yield consistent estimation of efficacy in the presence of unmeasured confoundings for competing risk settings we derived the asymptotic properties for the proposed estimator the estimator is shown to be well performed under finite sample size according to simulation results we applied our method to a real transplant data example and showed that the unmeasured confoundings lead to significant bias in the estimation of the effect about 50 attenuated copyright 2017 john wiley sons ltd

a discrete eigenfunctions method for computing mixed hyperbolic problems based on an implicit difference scheme in this paper numerical solutions of mixed hyperbolic problems are computed using a discrete eigenfunctions method combined with an implicit difference scheme this new numerical technique preserves the qualitative properties of the analytic solution due to the sturm liouville structure of the underlying discrete linear boundary value problem and has computational stability advantages vs other methods illustrative examples are included

high order closed newton cotes trigonometrically fitted formulae for the numerical solution of the schrodinger equation abstract in this paper we investigate the connection between closed newton cotes formulae trigonometrically fitted methods symplectic integrators and efficient integration of the schrodinger equation the study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry see the relevant literature and the references here in this paper we study the closed newton cotes formulae and we write them as symplectic multilayer structures based on the closed newton cotes formulae we also develop trigonometrically fitted symplectic methods an error analysis for the one dimensional schrodinger equation of the new developed methods and a comparison with previous developed methods is also given we apply the new symplectic schemes to the well known radial schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems

weighted npmle for the subdistribution of a competing risk direct regression modeling of the subdistribution has become popular for analyzing data with multiple competing event types all general approaches so far are based on nonlikelihood based procedur

estimating and comparing time dependent areas under receiver operating characteristic curves for censored event times with competing risks the area under the time dependent roc curve auc may be used to quantify the ability of a marker to predict the onset of a clinical outcome in the future for survival analysis with competing risks two alternative definitions of the specificity may be proposed depending of the way to deal with subjects who undergo the competing events in this work we propose nonparametric inverse probability of censoring weighting estimators of the auc corresponding to these two definitions and we study their asymptotic properties we derive confidence intervals and test statistics for the equality of the aucs obtained with two markers measured on the same subjects a simulation study is performed to investigate the finite sample behaviour of the test and the confidence intervals the method is applied to the french cohort paquid to compare the abilities of two psychometric tests to predict dementia onset in the elderly accounting for death without dementia competing risk the timeroc r package is provided to make the methodology easily usable

estimating twin concordance for bivariate competing risks twin data for twin time to event data we consider different concordance probabilities such as the casewise concordance that are routinely computed as a measure of the lifetime dependence correlation for specific diseases the concordance probability here is the probability that both twins have experienced the event of interest under the assumption that both twins are censored at the same time we show how to estimate this probability in the presence of right censoring and as a consequence we can then estimate the casewise twin concordance in addition we can model the magnitude of within pair dependence over time and covariates may be further influential on the marginal risk and dependence structure we establish the estimators large sample properties and suggest various tests for example for inferring familial influence the method is demonstrated and motivated by specific twin data on cancer events with the competing risk death we thus aim to quantify the degree of dependence through the casewise concordance function and show a significant genetic component

regression modeling of competing risks data based on pseudovalues of the cumulative incidence function typically regression models for competing risks outcomes are based on proportional hazards models for the crude hazard rates these estimates often do not agree with impressions drawn from plots of cumulative incidence functions for each level of a risk factor we present a technique which models the cumulative incidence functions directly the method is based on the pseudovalues from a jackknife statistic constructed from the cumulative incidence curve these pseudovalues are used in a generalized estimating equation to obtain estimates of model parameters we study the properties of this estimator and apply the technique to a study of the effect of alternative donors on relapse for patients given a bone marrow transplant for leukemia

an extended numerov type method for the numerical solution of the schrodinger equation abstract a new method which is based on the new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases 1 i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x 2 ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new method is more efficient compared with previously derived methods

shooting methods for the schrodinger equation a simple stable shooting method for the schrodinger equation is described and is shown to work with various integrators including the numerov one quantum mechanical techniques are extended to apply to problems for which e is not a traditional eigenvalue and are shown to permit the extraction of useful information even from an unstable shooting method

four step methods for y f x y abstract a systematic investigation is undertaken here on the possible practical consequences of the fact that the linear four step methods for y f x y form a family with a certain structure the mentioned property is found to be surprisingly rich in consequences and three of them are of particular importance i drastic reduction in the number of iterations at each step ii increased flexibility with respect to the start of integration and to the modification of the step size during the integration process iii possibility of direct and simple appraisal of the local truncation error

regression analysis with randomly right censored data this paper proposes a new estimator of the parameter vector in a linear regression model when the observations are randomly censored on the right and when the error distribution is unknown this estimator is explicitly defined and easily computable the paper contains sufficient conditions under which this estimator is mean square consistent and asymptotically normal a numerical example is given

summarizing differences in cumulative incidence functions the cumulative incidence function is widely reported in competing risks studies with group differences assessed by an extension of the log rank test however simple interpretable summaries of group differences are not available an adaptation of the proportional hazards model to the cumulative incidence function is often employed but the interpretation of the hazard ratio may be somewhat awkward unlike the usual survival set up we propose nonparametric inferences for general summary measures which may be time varying and for time averaged versions of the measures theoretical justification is provided using counting process techniques a real data example illustrates the practical utility of the methods

predicting cumulative incidence probability by direct binomial regression we suggest a new simple approach for estimation and assessment of covariate effects for the cumulative incidence curve in the competing risks model we consider a semiparametric regression model where some effects may be time varying and some may be constant over time our estimator can be implemented by standard software our simulation study shows that the estimator works well and has finite sample properties comparable with the subdistribution approach we apply the method to bone marrow transplant data and estimate the cumulative incidence of death in complete remission following a bone marrow transplantation here death in complete remission and relapse are two competing events

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

a p stable exponentially fitted method for the numerical integration of the schrodinger equation a p stable exponentially fitted method is developed in this paper for the numerical integration of the schrodinger equation an application to the resonance problem of the radial schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentially fitted methods of the same kind

a noumerov type method with minimal phase lag for the integration of second order periodic initial value abstract in a recent paper 2 we gave a noumerov type method with minimal phase lag for the integration of second order initial value problems y f t y y t0 y0 y t0 y 0 however the method given there is implicit we show here the interesting result that if the noumerov type methods of 2 are made explicit with the help of the classical second order method then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size 1 40320 h 6 and also a slightly larger interval of periodicity of size 2 75 than the phase lag of size 1 12096 h 6 and interval of periodicity of size 2 71 for the implicit method 2 more interestingly it turns out that noumerov made explicit of chawla 3 also has less frequency distortion than the implicit noumerov method we shall assume a familiarity with the notation and discussion given in 2

on estimands arising from misspecified semiparametric rate based analysis of recurrent episodic conditions marginal rate based analyses are widely used for the analysis of recurrent events in clinical trials in many areas of application the events are not instantaneous but rather signal the onset of a symptomatic episode representing a recurrent infection respiratory exacerbation or bout of acute depression in rate based analyses it is unclear how to best handle the time during which individuals are experiencing symptoms and hence are not at risk we derive the limiting value of the nelson aalen estimator and estimators of the regression coefficients under a semiparametric rate based model in terms of an underlying two state process we investigate the impact of the distribution of the episode durations heterogeneity and dependence on the asymptotic and finite sample properties of standard estimators we also consider the impact of these features on power in trials designed to test intervention effects on rate functions an application to a trial of individuals with herpes simplex virus is given for illustration

a sixth order exponentially fitted method for the numerical solution of the radial abstract a new sixth order method of runge kutta type is developed for the numerical integration of the single channel radial schrodinger equation the formula derived contains certain free parameters which allows it to be fitted automatically to exponential functions extensive numerical testing on the resonance problem and on the bound states problem indicates that this new method is generally more efficient than other previously developed finite difference methods

a generator of high order embedded p stable methods for the numerical solution of the schro dinger equation a generator of new embedded p stable methods of order 2n 2 where n is the number of layers used by the embedded methods for the approximate numerical integration of the one dimensional schrodinger equation is developed in this paper these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for one dimensional differential equations of the schrodinger type show the validity of the developed theory

explicit exponentially fitted methods for the numerical solution of the schrodinger equation we present here some accurate explicit methods for the numerical integration of the radial schrodinger equation the formulae considered contain free parameters which are defined in order to integrate exponential functions numerical results also indicate that the new methods are much more accurate than other explicit numerov type wellknown methods

a family of four step exponential fitted methods for the numerical integration of the radial schrodinger equation abstract a family of exponential four step methods is developed for the numerical integration of the one dimensional schrodinger equation the formula developed contains certain free parameters which allows it to be fitted automatically to exponential functions the new methods integrate more exponential functions and are very simple compared with the well known sixth algebraic order runge kutta type methods numerical results indicate that the new methods are much more accurate than other exponentially fitted methods

exponentially fitted methods for solving time fractional nonlinear reaction diffusion equation abstract in this article we will develop a new numerical scheme with the second order in time and a class of fourth order or sixth order in space based on the exponential fitting techniques to approximate the nonlinear time fractional reaction diffusion equation with fixed order and distributed order derivatives these techniques depend on a parameter which will be used to annihilate the error and increase the order of accuracy the proposed methods are proved to be unconditionally stable and convergent by fourier analysis also the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical test problems and a comparison with other methods is presented

competing risks regression for stratified data for competing risks data the fine gray proportional hazards model for subdistribution has gained popularity for its convenience in directly assessing the effect of covariates on the cumulative incidence function however in many important applications proportional hazards may not be satisfied including multicenter clinical trials where the baseline subdistribution hazards may not be common due to varying patient populations in this article we consider a stratified competing risks regression to allow the baseline hazard to vary across levels of the stratification covariate according to the relative size of the number of strata and strata sizes two stratification regimes are considered using partial likelihood and weighting techniques we obtain consistent estimators of regression parameters the corresponding asymptotic properties and resulting inferences are provided for the two regimes separately data from a breast cancer clinical trial and from a bone marrow transplantation registry illustrate the potential utility of the stratified fine gray model

a two step explicit p stable method for solving second order initial value problems the conceptions of periodicity interval and p stability were introduced by lambert and watson j inst math appl 18 1976 189 in connection with multistep methods for second order initial value problems this paper presents a new two step explicit p stable method of order two for solving initial value problems of second order ordinary differential equations based on a special vector operation the method can be extended to be vector applicable

time dependent predictive accuracy in the presence of competing risks competing risks arise naturally in time to event studies in this article we propose time dependent accuracy measures for a marker when we have censored survival times and competing risks time dependent versions of sensitivity or true positive tp fraction naturally correspond to consideration of either cumulative or prevalent cases that accrue over a fixed time period or alternatively to incident cases that are observed among event free subjects at any select time time dependent dynamic specificity 1 false positive fp can be based on the marker distribution among event free subjects we extend these definitions to incorporate cause of failure for competing risks outcomes the proposed estimation for cause specific cumulative tp dynamic fp is based on the nearest neighbor estimation of bivariate distribution function of the marker and the event time on the other hand incident tp dynamic fp can be estimated using a possibly nonproportional hazards cox model for the cause specific hazards and riskset reweighting of the marker distribution the proposed methods extend the time dependent predictive accuracy measures of heagerty lumley and pepe 2000 biometrics 56 337 344 and heagerty and zheng 2005 biometrics 61 92 105

inference for means and covariances of point processes through estimating functions summary liang zeger 1986 introduced methodology for the analysis of longitudinal data that provides an alternative to likelihood based inference they considered modelling the marginal means of the response follow up measures and proposed the use of unbiased estimating functions to handle inference here we wish to do the same for point or jump processes we consider parametric models for the marginal means and possibly the covariance structures of processes that allow covariates inference is performed with unbiased estimating functions and robust variance estimates are provided the optimal linear estimating function is presented in general the special case of mixed poisson processes is discussed in further detail with an asymptotic efficiency study and simulations

p stable exponentially fitted methods for the numerical integration of the schrodinger equation a p stable exponentially fitted method is developed in this paper for the numerical integration of the schrodinger equation an application to the bound states problem we solve the radial schrodinger equation in order to find eigenvalues for which the wavefunction and its derivative are continuous and the boundary conditions are satisfied and the resonance problem the point of a resonance is that phase changes rapidly through of the radial schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentially fitted methods of the same kind the method can be applied to any problem of physics and chemistry which can be expressed as system of coupled second order differential equations which have oscillatory or periodic solutions this is because it has the property of the p stability i e the interval of periodic stability of the proposed method is equal to 0 which allow is to integrate successful problems with high oscillatory or periodic solution

a new variable step method for the numerical integration of the one dimensional schro dinger equation the numerical integration of the one dimensional schrodinger equation is considered a new variable step method is presented within the canonical functions scheme characterized namely by i the replacement of the schrodinger equation by the equivalent integral equation ii the representation of the wavefunction by a superposition of simple integrals iii the definition of taylor series like difference equations at each interval an estimate of the step size h is determined in order to obtain the desired local accuracy for the practical case of a given potential function with polynomial interpolations the step size h is directly given by a simple formula depending on i the potential function in the interval ii the initial values of the wavefunction at the origin of the interval this variable step method is applied to the morse potential function used for the recent raptis and cash method comput phys commun 36 113 1985 for this application the present method presents some advantages over the previous one

a finite difference method for the numerical solution of the schro dinger equation a new approach which is based on a new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x ii the general case of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient compared to previously derived methods

the strong coupling expansion for anharmonic oscillators we accurately calculate the strong coupling expansion for anharmonic oscillators by means of a robust and stable numerical algorithm the method applies to any state and to any anharmonicity degree by means of the perturbation coefficients we estimate the location of the branch points that determine the convergence radius of the strong coupling expansion

a semiparametric random effects model for multivariate competing risks data we propose a semiparametric random effects model for multivariate competing risks data when the failures of a particular type are of interest under this model the marginal cumulative incidence functions follow a generalized semiparametric additive model the associations between the cause specific failure times can be studied through dependence parameters of copula functions that are allowed to depend on cluster level covariates a cross odds ratio type measure is proposed to describe the associations between cause specific failure times and its relationship to the dependence parameters is explored we develop a two stage estimation procedure where the marginal models are estimated in the first stage and the dependence parameters are estimated in the second stage the large sample properties of the proposed estimators are derived the proposed procedures are applied to danish twin data to model the cumulative incidence for the age of natural menopause and to investigate the association in the onset of natural menopause between monozygotic and dizygotic twins

a variable mesh c splage method of accuracy o k2hl 1 khl hl3 for 1d nonlinear parabolic equations a new high accuracy two level implicit cubic spline formulation for the solution of nonlinear parabolic equation u x email protected x t u u x u t is presented the strategy of the cubic spline alternating group explicit c splage method is developed the proposed c splage method requires only 3 spatial grid points and applicable to singular problems the convergence analysis of the proposed c splage method is given the obtained c splage method is compared to the corresponding successive over relaxation sor method both in terms of stability and performance

embedded methods for the numerical solution of the schrodinger equation abstract new methods for the approximate numerical integration of the one dimensional schrodinger equation are developed in this paper complete phase lag analysis of the new methods is included these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for a one dimensional schrodinger equation show the validity of the developed theory

direct modeling of regression effects for transition probabilities in the progressive illness death model in this work we present direct regression analysis for the transition probabilities in the possibly non markov progressive illness death model the method is based on binomial regression where the response is the indicator of the occupancy for the given state along time randomly weighted score equations that are able to remove the bias due to censoring are introduced by solving these equations one can estimate the possibly time varying regression coefficients which have an immediate interpretation as covariate effects on the transition probabilities the performance of the proposed estimator is investigated through simulations we apply the method to data from the registry of systematic lupus erythematosus relesser a multicenter registry created by the spanish society of rheumatology specifically we investigate the effect of age at lupus diagnosis sex and ethnicity on the probability of damage and death along time copyright 2017 john wiley sons ltd

inference for events with dependent risks in multiple endpoint studies abstract kaplan meier and cumulative incidence functions are not sufficient descriptive devices for studies that have multiple time to event endpoints for example in cancer treatment research the probability of tumor recurrence conditional on not having died from treatment related toxicities and the prevalence of graft versus host disease among leukemia free patients surviving a bone marrow transplant are of interest these quantities can be estimated nonparametrically using simple functions of several kaplan meier and cumulative incidence estimates for events with possibly dependent risks we derive asymptotic distribution theory for such functions by representing kaplan meier cumulative incidence and cumulative hazard estimators as sums of iid random variables variance estimation also follows directly from this representation two sample test statistics with asymptotic null distribution theory are presented several examples illustrate the utility of these results

an exponentially fitted eighth order method for the numerical solution of the schrodinger equation an eighth algebraic order exponentially fitted method is developed for the numerical integration of the schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions an comparative error analysis is also given numerical and theoretical results indicate that the new method is much more efficient than other classical and exponentially fitted methods

a partly parametric additive risk model summary aalen s additive risk model allows the influence of each covariate to vary separately over time although allowing greater flexibility of temporal structure than a cox model aalen s model is more limited in the number of covariates it can handle we introduce a partly parametric version of aalen s model in which the influence of only a few covariates varies nonparametrically over time and that of the remaining covariates is constant efficient procedures for fitting this new model are developed and studied the approach is applied to data from the medical research council s myelomatosis trials it is the first step of a taylor series expansion of a general hazard function about the zero of the covariate vector however in estimating the unknown functions in such a general model there is a variance bias trade off that may be critical in small and medium samples also after fitting the model one does not have parameters or formulae that are easily reported we propose a model that takes the additive structure of aalen s model and imposes parametric constraints to obtain a semiparametric submodel which may be more appropriate in some applications the model will be illustrated with data from clinical trials on myelomatosis covariates include treatment sex and four age strata which will be treated parametrically together with serum levels of haemoglobin and f32 microglobulin whose effects will be investigated nonparametrically the additive form can be interpreted loosely in terms of unobserved competing risks since the hazard function for the minimum of independent random vari ables is the sum of the hazard functions for the individual variables microglobulin levels are related to kidney function and tumour mass whereas haemoglobin is unaffected by kidney function hence one might anticipate that the hazard function associated with each covariate represents a different cause of death

the existence of best proximity points in metric spaces with the property uc abstract eldred and veeramani in a a eldred p veeramani existence and convergence of best proximity points j math anal appl 323 2006 1001 1006 mr2260159 proved a theorem which ensures the existence of a best proximity point of cyclic contractions in the framework of uniformly convex banach spaces in this paper we introduce a notion of the property uc and extend the eldred and veeramani theorem to metric spaces with the property uc

estimates of clinically useful measures in competing risks survival analysis competing risks occur frequently in follow up clinical studies to assess treatment or covariate effects measures of clinical impact based on crude cumulative incidence should be considered such as relative risks or the absolute risk reduction in this work transformation models through suitable link functions provide a straightforward approach to obtain point and interval estimates of such measures an extension of the klein and andersen proposal based on pseudo values is considered non additive effects were tested by interactions between baseline spline function on time and covariates the methods are applied to the evaluation of the impact of axillary lymph node nanometastases on metastatic relapse of breast cancer patients further a literature data set on prostate cancer was used for illustration

on computing eigenvalues of the schrodinger equation for symmetrical potentials the problem of the numerical determination of eigenvalues of the one dimensional schrodinger equation with symmetrical potential v x is considered the problem is reduced to the definition of an eigenvalue function g e well defined for the given potential v it is theoretically proved that the even parity eigenvalues are given by g e 0 and the odd parity eigenvalues are given by g e infinity the method is canonical in the sense that it is independent of the eigenfunction yet along with the eigenvalue it allows the determination of the eigenfunction initial values this method is applied to the potentials v x2 v x6 bx2 v x2 lambda x2 1 gx2 where exact eigenvalues are available the numerical results compared with the exact ones and with those of previous confirmed methods show that the present method is accurate and efficient

convergence and existence results for best proximity points we provide a positive answer to a question raised by eldred and veeramani a a eldred p veeramani existence and convergence of best proximity points j math anal appl 323 2006 1001 1006 about the existence of a best proximity point for a cyclic contraction map in a reflexive banach space moreover we introduce a new class of maps called cyclic contractions which contains the cyclic contraction maps as a subclass convergence and existence results of best proximity points for cyclic contraction maps are also obtained 2008 elsevier ltd all rights reserved

a four step method for the numerical solution of the schro dinger equation abstract a new four step exponentially fitted method is developed in this paper the expressions for the coefficients of the method are found such as to ensure the optimal approximation to the eigenvalue schrodinger equation i e equivalent to positive energy

frequency evaluation in exponential fitting multistep algorithms for odes we consider the linear multistep algorithms for first order odes and examine the problem of how the frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms we find out that the key of the answer consists in analysing the behaviour of the error on further investigating the simple case of two step bdf algorithms we produce formulae for the optimal s and show that if the optimal s are used the order of the method is increased by one unit the reported numerical illustrations suggest that further investigations along these lines deserve a real attention

marginal regression models for multivariate failure time data abstract in this article we propose a general cox type regression model to formulate the marginal distributions of multivariate failure time data this model has a nested structure in that it allows different baseline hazard functions among distinct failure types and imposes a common baseline hazard function on the failure times of the same type we prove that the maximum quasi partial likelihood estimator for the vector of regression parameters under the independence working assumption is consistent and asymptotically normal with a covariance matrix for which a consistent estimator is provided furthermore we establish the uniform consistency and joint weak convergence of the aalen breslow type estimators for the cumulative baseline hazard functions and develop a resampling technique to approximate the joint distribution of these processes which enables one to make simultaneous inference about the survival functions over the time axis and across failure types finally we assess the small sample pro

existence and convergence of best proximity points abstract consider a self map t defined on the union of two subsets a and b of a metric space and satisfying t a b and t b a we give some contraction type existence results for a best proximity point that is a point x such that d x t x dist a b we also give an algorithm to find a best proximity point for the map t in the setting of a uniformly convex banach space

numerical methods for solving radial schro dinger equations abstract an algorithm previously introduced by brown et al 1963 for solving radial schrodinger equations is revisited and implemented in a more accurate way the method is firstly applied to equations where potentials are present which are finite at the origin and which have an asymptotic behaviour v r 0 as r typical examples of potentials belonging to that class are the woods saxon and the morse potential furthermore the method is also used for coulomb like potentials such as the hulthen and the hellmann potential a comparison between the approximated numerical values and other available numerical and exact bound state energies is made

marginal models for clustered time to event data with competing risks using pseudovalues many time to event studies are complicated by the presence of competing risks and by nesting of individuals within a cluster such as patients in the same center in a multi center study several methods have been proposed for modeling the cumulative incidence function with independent observations however when subjects are clustered one needs to account for the presence of a cluster effect either through frailty modeling of the hazard or subdistribution hazard or by adjusting for the within cluster correlation in a marginal model we propose a method for modeling the marginal cumulative incidence function directly we compute leave one out pseudo observations from the cumulative incidence function at several time points these are used in a generalized estimating equation to model the marginal cumulative incidence curve and obtain consistent estimates of the model parameters a sandwich variance estimator is derived to adjust for the within cluster correlation the method is easy to implement using standard software once the pseudo values are obtained and is a generalization of several existing models simulation studies show that the method works well to adjust the standard error for the within cluster correlation we illustrate the method on a dataset looking at outcomes after bone marrow transplantation

assessing cumulative incidence functions under the semiparametric additive risk model in analyzing competing risks data a quantity of considerable interest is the cumulative incidence function often the effect of covariates on the cumulative incidence function is modeled via the proportional hazards model for the cause specific hazard function as the proportionality assumption may be too restrictive in practice we consider an alternative more flexible semiparametric additive hazards model of biometrika 1994 81 501 514 for the cause specific hazard this model specifies the effect of covariates on the cause specific hazard to be additive as well as allows the effect of some covariates to be fixed and that of others to be time varying we present an approach for constructing confidence intervals as well as confidence bands for the cause specific cumulative incidence function of subjects with given values of the covariates furthermore we also present an approach for constructing confidence intervals and confidence bands for comparing two cumulative incidence functions given values of the covariates the finite sample property of the proposed estimators is investigated through simulations we conclude our paper with an analysis of the well known malignant melanoma data using our method