cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

short communication modification of the application of a contraction mapping method on a class of fractional differential equation this letter discusses an application of a modified contraction mapping method to the proof of a global existence and uniqueness theorem of a cauchy problem for some version of differential equation of fractional order in banach space an essential condition in such applications which have previously been considered in many literatures will be dispensed with

laguerre polynomials of arbitrary fractional orders the topic of fractional calculus derivatives and integrals of arbitrary orders is enjoying growing interest not only among mathematicians but also among physicists and engineers in this paper we define the laguerre polynomials l a b x of arbitrary order a r the set of all real numbers and prove that this definition generalizes and interpolates the well known definition l n b x n 1 2 of laguerre polynomials and prove that l a b x a r is continuous as a function of a a r also the confluent hypergeometric representation of these polynomials will be given

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

on the generalized laguerre polynomials of arbitrary fractional orders and quantum mechanics the generalized laguerre polynomials l x of arbitrary order have been defined by the author el sayed 1997 math sci res hot line 1 7 14 1999 to appear in the latter reference it is proved that they are continuous as functions of and some other properties that generalize interpolate those of the classical laguerre polynomials ln x n 1 2 have been proved here we prove that l x are orthogonal in l2 0 and are particular solutions of the differential equation generalizing the one for ln x n 1 2 also some applications in quantum mechanics are discussed

reml estimation for survival models with frailty a method of estimation for generalised mixed models is applied to the estimation of regression parameters in proportional hazards models for failure times when there are repeated observations of failure on each subject the subject effect is incorporated into the model as a random frailty term best linear unbiased predictors are used as an initial step in the computation of maximum likelihood and residual maximum likelihood estimates

bivariate survival models induced by frailties abstract when two observed survival times depend via a proportional hazards model on the same unobserved variable called a frailty this common dependence induces an association between the observed times this article considers the class of bivariate survival distributions that can arise in this way extends it to allow negative association and shows that the observable bivariate distribution determines the unobserved frailty distribution up to a scale parameter a cross ratio function easily estimated even from censored data is the key to both the characterization results and inferential procedures and diagnostic plots that are introduced and illustrated by real examples the models considered are the natural counterpart for survival data of the latent variable models that have long been used in factor analysis for continuous data or in latent structure analysis for binary data the main distinguishing feature of survival data is the possibility of censoring which makes the standard methods difficu

on the perturbation of unbounded linear operators with topologically complemented ranges let x and y be normed spaces and let t d t x y be a linear transformation having a finite codimensional restriction with a continuous inverse equivalently let t be a operator for example if t is bounded below suppose that r t or r t is topologically complemented in y conditions are obtained under which r t s resp r t s is topologically complemented whenever s belongs to the class of precompact operators or to some wider class

semiparametric estimation of random effects using the cox model based on the em algorithm consider a survival experiment where individuals within a certain subset of the population share a common unobservable random frailty such a frailty could be an unobservable genetic or early environmental effect if individuals were in sibling groups or an environmental effect if individuals were grouped by households suppose that if the frailty omega is known the cox proportional hazards model for the observable covariates is valid with the consequence of the random effect being a multiplicative factor on the hazard rate assuming tht the random frailties follow a gamma distribution estimates of the fixed and random effects are obtained by using an em algorithm based on a profile likelihood construction the method developed is applied to the framingham heart study to examine the risks of smoking and cholesterol levels adjusting for potential random effects

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

multivariate survival analysis using piecewise gamma frailty in this note we propose a frailty model called piecewise gamma frailty for correlated survival data with random effects having a nested structure in frailty models a dependence function defined as a hazard ratio of one member given the failure time of another member in a unit is determined by the distributional assumptions on frailty in the piecewise gamma frailty model the nested structure of random effects or frailty allows the dependence function to vary over the time periods this model includes existing models such as the piecewise exponential model breslow 1974 biometrics 30 89 100 and the gamma frailty model clayton 1978 biometrika 65 141 151 oakes 1982 journal of the royal statistical society series b 44 414 428 as special cases a study of familial aggregation of epilepsy is used to illustrate the proposed method

on a theorem of kalton and wilansky concerning tauberian operators abstract a theorem of kalton and wilansky asserts that a bounded linear operator between banach spaces is tauberian if and only if it maps each bounded non relatively weakly compact set onto a non relatively weakly compact set in the present note this result is generalised to arbitrary linear transformations t d t x y where x and y are normed spaces this characterisation is then used to derive a theory of tauberian operators in this more general setting where only the linearity of t is assumed

two level proportional hazards models nous proposons une extension du modele de taux proportionnels a un modele a deux niveaux avec un terme constant aleatoire et des coefficients aleatoires les parametres de ce modele a plusieurs niveaux sont estimes par une combinaison des algorithmes em et de nuwton raphson meme pour des echantillons de 50 groupes la methode fournit pour les coefficients des effets fixes des estimateurs a peu pres sans biais et normalement distribues deux methodes distinctes information observee et information de la vraisemblance des profils seront utilisees pour estimer les erreurs standards ce travail a ete entrepris dans le but de comprendre les determinants des habitudes contraceptive des femmes nepaleses dans l enquete sur les familles de la vallee chitwan axinn barber et ghimire 1997 nous avons utilise ce modele de taux proportionnels a deux niveaux pour etudier la facon dont le niveau d education et les possibilites d acces a l education des enfants varient avec l initiation a un usage permanent de la contraception

numerical methods for multi term fractional arbitrary orders differential equations our main concern here is to give a numerical scheme to solve a nonlinear multi term fractional arbitrary orders differential equation

proportional hazards models with frailties and random effects we discuss some of the fundamental concepts underlying the development of frailty and random effects models in survival one of these fundamental concepts was the idea of a frailty model where each subject has his or her own disposition to failure their so called frailty additional to any effects we wish to quantify via regression although the concept of individual frailty can be of value when thinking about how data arise or when interpreting parameter estimates in the context of a fitted model we argue that the concept is of limited practical value individual random effects frailties whenever detected can be made to disappear by elementary model transformation in consequence unless we are to take some model form as unassailable beyond challenge and carved in stone and if we are to understand the term frailty as referring to individual random effects then frailty models have no value random effects models on the other hand in which groups of individuals share some common effect can be used to advantage even in this case however if we are prepared to sacrifice some efficiency we can avoid complex modelling by using the considerable power already provided by the stratified proportional hazards model stratified models and random effects models can both be seen to be particular cases of partially proportional hazards models a view that gives further insight the added structure of a random effects model viewed as a stratified proportional hazards model with some added distributional constraints will for group sizes of five or more provide no more than modest efficiency gains even when the additional assumptions are exactly true on the other hand for moderate to large numbers of very small groups of sizes two or three the study of twins being a well known example the efficiency gains of the random effects model can be far from negligible for such applications the case for using random effects models rather than the stratified model is strong this is especially so in view of the good robustness properties of random effects models nonetheless the simpler analysis based upon the stratified model remains valid albeit making a less efficient use of resources

semiparametric bayesian analysis of multiple event time data abstract multiple event time data e g carcinogenic growths in different times and locations multiple attacks of cardiac arrest arise in various medical studies a bayesian analysis of such data based on proportional intensity model of multiple event time data is presented in this paper the bayesian structure is somewhat analogous to that used by kalbfleisch in a proportional hazard model an unobserved random frailty component is used in the proportional intensity model to take care of heterogeneity among the intensity processes in different subjects the monte carlo method of sampling from multivariate distributions the so called gibbs sampler is used to sample from the joint posterior distribution of the unknown parameters the methodology developed here is exemplified with the well known data set on rat tumors of gail santner and brown

linear differential equations of fractional orders abstract we are concerned here with the linear differential equations of fractional orders we will obtain some results concerning the existence uniqueness and some other properties of the solution analogous and more general to that obtained for the linear ordinary differential equations

generalized cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator abstract this paper is devoted to proving the existence and uniqueness of solutions to cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions an approach based on the equivalence of the nonlinear cauchy type problem to a nonlinear volterra integral equation of the second kind and applying a variant of the banach s fixed point theorem to prove uniqueness and existence of the solution is presented the cauchy type problems for integro differential equations of volterra type with composite fractional derivative operator which contain the generalized mittag leffler function in the kernel are considered using the method of successive approximation and the laplace transform method explicit solutions of the open problem proposed by srivastava and tomovski 2009 11 are established in terms of the multinomial mittag leffler function

mixed effect poisson log linear models for clinical and epidemiological sleep hypnogram data bayesian poisson log linear multilevel models scalable to epidemiological studies are proposed to investigate population variability in sleep state transition rates hierarchical random effects are used to account for pairings of subjects and repeated measures within those subjects as comparing diseased to non diseased subjects while minimizing bias is of importance essentially non parametric piecewise constant hazards are estimated and smoothed allowing for time varying covariates and segment of the night comparisons the bayesian poisson regression is justified through a re derivation of a classical algebraic likelihood equivalence of poisson regression with a log time offset and survival regression assuming exponentially distributed survival times such re derivation allows synthesis of two methods currently used to analyze sleep transition phenomena stratified multi state proportional hazards models and log linear models with gee for transition counts an example data set from the sleep heart health study is analyzed supplementary material includes the analyzed data set as well as the code for a reproducible analysis

a general framework for random effects survival analysis in the cox proportional hazards setting summary the use of random effects modeling in statistics has increased greatly in recent years the introduction of such modeling into event time analysis has proceeded more slowly however previously random effects models for survival data have either required assumptions regarding the form of the baseline hazard function or restrictions on the classes of models that can be fit in this paper we develop a method of random effect analysis of survival data the hierarchical cox model that is an extension of cox s original formulation in that the baseline hazard function remains unspecified this method also allows an arbitrary distribution for the random effects we accomplish this using markov chain monte carlo methods in a bayesian setting the method is illustrated with three models for a dataset with times to multiple occurrences of mammory tumors for 48 rats treated with a carcinogen and then randomized to either treatment or control this analysis is more satisfying than standard approaches such as studying the first event for each subject which does not fully use the data or assuming independence which in this case would overestimate the precision

fractional order evolutionary integral equations the topic of fractional calculus derivatives and integrals of arbitrary orders is enjoying growing interest not only among mathematicians but also among physicists and engineers also the evolutionary integral equations have many applications in applied mathematics in this work we are concerned with a class of evolutionary integral equations of arbitrary fractional orders the existence uniqueness and some other properties of the solutions will be proved also the continuation of the problems will be established

a class of bayesian shared gamma frailty models with multivariate failure time data nous proposons ici dans le cadre des donnees de survie une nouvelle classe de modeles a fragilite gamma partagee en appliquant la transformation de box cox a la fonction de hasard ainsi qu au produit du risque de base et de la fragilite cette nouvelle classe de modeles offre un large eventail de formes et relations entre la fonction de hasard et le risque de base et inclut en tant que cas particuliers le modele de cox usuel a fragilite gamma ainsi qu un nouveau modele additif a fragilite gamma de par la contrainte que le risque soit non negatif ce modele a fragilite gamma partagee presente un defi calculatoire dans le paradigme bayesien les distributions conjointes a priori sont definies par une distribution conditionnelle marginale ou la distribution conditionnelle est univariee cela permet d integrer les contraintes non lineaires sur les parametres la partie marginale de la distribution a priori quant a elle est libre de toute contrainte les distributions a priori permettent de programmer facilement l integralite des distributions conditionnelles requises pour l echantillonnage de gibbs tout en integrant les contraintes un jeu reel de donnees permet d illustrer cette classe de modeles a fragilite gamma partagee

set valued integral equations of fractional orders the topic of fractional calculus derivative and integral of arbitrary orders is enjoying growing interest not only among mathematicians but also among physicists and engineers see 6 16 18 20 23 25 the set valued integral equations integral inclusions arises in the study of control system see 21 22 26 in this paper we prove the existence of locally bounded variation solution of a volterra type set valued integral equation of arbitrary not necessarily integer order the proof will be based on the measure of weak noncompactness and the existence of caratheodory selectors as a consequence we study the initial value problem for some set valued differential and integro differential equations the corresponding single valued problems will be firstly considered

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

conditional akaike information under generalized linear and proportional hazards mixed models we study model selection for clustered data when the focus is on cluster specific inference such data are often modelled using random effects and conditional akaike information was proposed in vaida blanchard 2005 and used to derive an information criterion under linear mixed models here we extend the approach to generalized linear and proportional hazards mixed models outside the normal linear mixed models exact calculations are not available and we resort to asymptotic approximations in the presence of nuisance parameters a profile conditional akaike information is proposed bootstrap methods are considered for their potential advantage in finite samples simulations show that the performance of the bootstrap and the analytic criteria are comparable with bootstrap demonstrating some advantages for larger cluster sizes the proposed criteria are applied to two cancer datasets to select models when the cluster specific inference is of interest

hierarchical proportional hazards regression models for highly stratified data summary in clinical trials conducted over several data collection centers the most common statistically defensible analytic method a stratified cox model analysis suffers from two important defects first identification of units that are outlying with respect to the baseline hazard is awkward since this hazard is implicit rather than explicit in the cox partial likelihood second and more seriously identification of modest treatment effects is often difficult since the model fails to acknowledge any similarity across the strata we consider a number of hierarchical modeling approaches that preserve the integrity of the stratified design while offering a middle ground between traditional stratified and unstratified analyses we investigate both fully parametric weibull and semiparametric models the latter based not on the cox model but on an extension of an idea by gelfand and mallick 1995 biometrics51 843 852 which models the integrated baseline hazard as a mixture of monotone functions we illustrate the methods using data from a recent multicenter aids clinical trial comparing their ease of use interpretation and degree of robustness with respect to estimates of both the unit specific baseline hazards and the treatment effect

bell polynomials of arbitrary fractional orders we define the generalized bell polynomials of arbitrary order email protected r the set of all real numbers and study some properties of these polynomials the continuation to the integer order polynomials will be established also we study some properties of the laguerre polynomials of fractional orders and its generalization which were expressed by the bell polynomials

bayesian semiparametric dynamic frailty models for multiple event time data many biomedical studies collect data on times of occurrence for a health event that can occur repeatedly such as infection hospitalization recurrence of disease or tumor onset to analyze such data it is necessary to account for within subject dependency in the multiple event times motivated by data from studies of palpable tumors this article proposes a dynamic frailty model and bayesian semiparametric approach to inference the widely used shared frailty proportional hazards model is generalized to allow subject specific frailties to change dynamically with age while also accommodating nonproportional hazards parametric assumptions on the frailty distribution are avoided by using dirichlet process priors for a shared frailty and for multiplicative innovations on this frailty by centering the semiparametric model on a conditionally conjugate dynamic gamma model we facilitate posterior computation and lack of fit assessments of the parametric model our proposed method is demonstrated using data from a cancer chemoprevention study

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures