a large sample study of cox s regression model strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in cox s regression model estimates are also derived for the underlying cumulative hazard function and survival distribution we establish the asymptotic normality of these estimates and calculate the limiting variances

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

misspecified proportional hazard models summary the properties of an estimator based on a proportional hazards model are investigated when the model is incorrect the estimator from the partial likelihood is shown to be consistent for a parameter that is defined implicitly the results are used to investigate the effects on estimation if the true model is accelerated failure time or if covariates are omitted from the proportional hazards model

sample size planning for survival prediction with focus on high dimensional data sample size planning should reflect the primary objective of a trial if the primary objective is prediction the sample size determination should focus on prediction accuracy instead of power we present formulas for the determination of training set sample size for survival prediction sample size is chosen to control the difference between optimal and expected prediction error prediction is carried out by cox proportional hazards models the general approach considers censoring as well as low dimensional and high dimensional explanatory variables for dimension reduction in the high dimensional setting a variable selection step is inserted if not all informative variables are included in the final model the effect estimates are biased towards zero the bias affects the prediction error and its magnitude is influenced by the sample size for variable selection we consider two approaches least absolute shrinkage and selection operator lasco and univariable selection for univariable selection we can calculate input parameters for the sample size formula for the lasco supportive simulations are necessary to appropriately choose the input parameters we investigate the performance of the proposed formulas with the use of simulations simulation results support the validity of the sample size formulas an application of a real data example illustrates the practical implementation of the method copyright 2012 john wiley sons ltd

eigenvalue problems for nonlinear differential equations on a measure chain abstract values of are determined for which there exist positive solutions of the second order differential equation on a measure chain u t a t f u t 0 t 0 1 satisfying either the conjugate boundary conditions u 0 u 1 0 or the right focal boundary conditions u 0 u 1 0 where a and f are positive valued and both lim x 0 f x x and lim x f x x exist

multiple positive solutions for nonlinear dynamical systems on a measure chain in this paper we consider the following dynamical system on a measure chain u 1 t f1 t u1 t u2 t 0 t a b u 2 t f2 t u1 t u2 t 0 t a b with the sturm liouville boundary value conditions ui a u i a 0 ui b ui b 0 for i 1 2 some results are obtained for the existence of three positive solutions of the above problem by using leggett williams fixed point theorem

positive solutions of the nonlinear fourth order beam equation with three parameters abstract this paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth order beam equation u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f t u 0 1 0 0 is continuous and and are parameters we show that there exists a 0 such that the above boundary value problem bvp has at least two one and no positive solutions for 0 and respectively furthermore by using the semiorder method on cones of banach space we establish a uniqueness criterion for positive solution of the bvp in particular such a positive solution u t of the bvp depends continuously on the parameter i e u t is nondecreasing in lim 0 u t 0 and lim u t for any t 0 1

the existence of countably many positive solutions for nonlinear singular m point boundary value problems on time scales in this paper by introducing a new operator improving and generating a p laplace operator for some p 1 we study the existence of countably many positive solutions for nonlinear boundary value pr

internally stable nonlinear systems with disturbances a parameterization this paper deals with the control of a nonlinear system whose output is subjected to an additive disturbance the main result is a simple parameterization of the set of all system responses that can be obtained through internally stable control of the given system the parameterization provides a clear indication of the effects of the disturbance on the response of the stabilized closed loop system the class of achievable responses is determined by the numerator of a right coprime fraction representation of the system being controlled

functional form diagnostics for cox s proportional hazards model summary we propose a new type of residual and an easily computed functional form test for the cox proportional hazards model the proposed test is a modification of the omnibus test for testing the overall fit of a parametric regression model developed by stute gonzalez manteiga and presedo quindimil 1998 journal of the american statistical association93 141 149 and is based on what we call censoring consistent residuals in addition we develop residual plots that can be used to identify the correct functional forms of covariates we compare our test with the functional form test of lin wei and ying 1993 biometrika80 557 572 in a simulation study the practical application of the proposed residuals and functional form test is illustrated using both a simulated data set and a real data set

global parametrization of feedback systems with nonlinear plants for unity feedback systems with a plant p and controller c we show that given a nonlinear stable p the class of all controller stabilizing the feedback system is globally parametrized by a stable map q we also show how q is useful in studying the effects of modeling errors on the stability

transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations abstract using a new method for dealing with the impulse term of second impulsive differential equations we are concerned with determining values of for which there exist positive solutions of the following boundary value problems in the form of x t f t x t 0 t j t t k x t k x t k c k x t k k 1 2 n a x 0 b x 0 a x 1 b x 1 0 1 h s x t d t where 0 is a positive parameter c k is a real sequence with c k 1 k 1 2 n the dependence of positive solution x t on the parameter is also studied i e lim x or lim x 0 the proof of our main result is based upon transformation techniques and fixed point theories this is probably the first time the existence of positive solutions of second order impulsive differential equations has been studied by applying this technique

positive solutions of singular p laplacian dynamic equations with sign changing nonlinearity abstract let t be a time scale such that 0 t t by the schauder fixed point theorem and the upper and lower solution method we present some existence criteria of the positive solution of m point singular p laplacian dynamic equation p u t q t f t u t 0 t 0 t t with boundary conditions u 0 0 i 1 m 1 i u i u t 0 m 2 where p s s p 2 s with p 1 i r r is continuous for i 1 2 m 1 and nonincreasing if m 3 0 1 2 m 2 m 1 t the nonlinear term may be singular in its dependent variable and is allowed to change sign our results are new even for the corresponding differential t r and difference equations t z as an application an example is given to illustrate our result

the existence of countably many positive solutions for nonlinear singular m point boundary value problems on the half line in this paper by introducing a new operator improving and generating a p laplace operator for some p 1 we study the existence of countably many positive solutions for nonlinear boundary value problems on the half line f u a t f u t 0 0r is the increasing homeomorphism and positive homomorphism and f 0 0 we show the sufficient conditions for the existence of countably many positive solutions by using the fixed point index theory and a new fixed point theorem in cones

non linear systems stability and rationality the problem of when a non linear system can be represented as a quotient of two stable non linear systems is considered attention is mainly directed toward nonlinear discrete time recursive systems where recursive means that the relationship between an input sequence and the corresponding output sequence can be expressed in terms of a finite number of recursive equations necessary and sufficient conditions are derived for the existence of a fraction representation of a recursive system where the numerator and the denominator are stable recursive systems the explicit construction of such a fraction representation is described

eigenvalue problems for second order nonlinear dynamic equations on time scales abstract this paper is concerned with the existence and nonexistence of positive solutions of the second order nonlinear dynamic equation u t a t f u t 0 t 0 1 satisfying either the conjugate boundary conditions u 0 u 1 0 or the right focal boundary conditions u 0 u 1 0 where a and f are positive we show that there exists a 0 such that the above boundary value problem has at least two one and no positive solutions for 0 and respectively furthermore by using the semiorder method on cones of the banach space we establish an existence and uniqueness criterion for positive solution of the problem in particular such a positive solution u t of the problem depends continuously on the parameter i e u t is nondecreasing in lim 0 u 0 and lim u

the robust inference for the cox proportional hazards model abstract we derive the asymptotic distribution of the maximum partial likelihood estimator for the vector of regression coefficients under a possibly misspecified cox proportional hazards model as in the parametric setting this estimator converges to a well defined constant vector in addition the random vector n 1 2 is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated the newly proposed robust covariance matrix estimator is similar to the so called sandwich variance estimators that have been extensively studied for parametric cases for many misspecified cox models the asymptotic limit or part of it can be interpreted meaningfully in those circumstances valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of and the related results for the score statistics extensive studies demonstrate that the proposed robust tests and interval estimation procedures

uniqueness implies existence on time scales abstract the uniqueness of solutions of initial value boundary value problems along with some uniqueness conditions on two point boundary value problems imply the existence of solutions for the same boundary value problems for the second order nonlinear dynamic equation y f t y y on a time scale this existence is established using shooting methods

existence of three positive solutions to a second order boundary value problem on a measure chain growth conditions are imposed on f such that the boundary value problem x t f x t t t1 t2 x t1 x t1 0 and x t2 x t2 0 where t1 t2 from a measure chain t has at least three positive solutions by way of the five functionals fixed point theorem

positive solutions for nonlinear three point boundary value problems on time scales abstract let t be a time scale such that 0 t t consider the following three point boundary value problem on time scales u t a t f t u t 0 t 0 t u 0 u 0 0 u u t where 0 0 0 t 0 t and d t 1 0 by using fixed point theorems in cones some new and general results are obtained for the existence of single and multiple positive solutions of the above problem in particular our criteria generalize and improve some known results

sturm liouville eigenvalue problems on time scales for sturm liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish rayleigh s principle our results not only unifly the corresponding theories for differential and difference equations but are also new in the discrete case

triple positive pseudo symmetric solutions of three point bvps for p laplacian dynamic equations on time scales abstract let t be a time scale such that 0 t t we consider the three point boundary value problem for p laplacian dynamic equations on time scales t of the form p u t h t f t u t 0 for t 0 t t with boundary conditions u 0 0 u u t where t is symmetric in t t and p u u p 2 u with p 1 by using a pseudo symmetric technique and the five functionals fixed point theorem we prove that the boundary value problem has at least three positive pseudo symmetric solutions under some assumptions as an application an example is given to illustrate the result

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

existence multiplicity and dependence on a parameter for a periodic boundary value problem the authors consider the boundary value problem y 2 y g t f y 0 0 t 2 y 0 y 2 y 0 y 2

effect of misspecification of regression models in the analysis of survival data the accelerated life and proportional hazards families of models are widely used in the analysis of survival data we examine the question if interest lies in the qualitative effect on failure of various explanatory variables how critical is the choice of model family in assessing the relative importance of the explanatory variables we consider small effects when censoring is independent of the explanatory variables to first order the regression parameters are proportional under the alternative models this is not true when the censoring varies appreciably with the explanatory variables some higherorder theory is examined for the special case of the two sample problem

model misspecification in proportional hazards regression the proportional hazards model is frequently used to evaluate the effect of treatment on failure time events in randomised clinical trials concomitant variables are usually available and may be considered for use in the primary analyses under the assumption that incorporating them may reduce bias or improve efficiency in this paper we consider two approaches to including covariate information regression modelling and stratification we focus on the setting where covariate effects are nonproportional and we compare the bias efficiency and coverage properties of these approaches these results indicate that our intuition based on linear model analysis of covariance is misleading covariate adjustment in proportional hazards models has little effect on the variance but may significantly improve the accuracy of the treatment effect estimator

twin solutions of boundary value problems for differential equations on measure chains a new twin fixed point theorem is applied to obtain the existence of at least two positive solutions for the right focal boundary value problem on a measure chain t y t f y t 0 t 0 1 t y 0 y 1 0

estimating cumulative treatment effects in the presence of nonproportional hazards often in medical studies of time to an event the treatment effect is not constant over time in the context of cox regression modeling the most frequent solution is to apply a model that assumes the treatment effect is either piecewise constant or varies smoothly over time i e the cox nonproportional hazards model this approach has at least two major limitations first it is generally difficult to assess whether the parametric form chosen for the treatment effect is correct second in the presence of nonproportional hazards investigators are usually more interested in the cumulative than the instantaneous treatment effect e g determining if and when the survival functions cross therefore we propose an estimator for the aggregate treatment effect in the presence of nonproportional hazards our estimator is based on the treatment specific baseline cumulative hazards estimated under a stratified cox model no functional form for the nonproportionality need be assumed asymptotic properties of the proposed estimators are derived and the finite sample properties are assessed in simulation studies pointwise and simultaneous confidence bands of the estimator can be computed the proposed method is applied to data from a national organ failure registry

existence of a positive solution for one dimensional singular p laplacian problems and its parameter dependence article history we consider one dimensional singular p laplacian problems of the form q 1 and may be singular at t 0a nd ort 1 using fixed point techniques combined with the partially ordered structure of banach space we establish some new and more general existence multiplicity and nonexistence results we also study the dependence of the positive solution u t on the parameter t hat is

triple positive solutions of m point bvps for p laplacian dynamic equations on time scales abstract this paper is concerned with the existence of positive solutions of p laplacian dynamic equation p u t a 1 t f u t 0 subject to boundary conditions u 0 b 0 i 1 m 2 a i u i 0 u t 0 or u 0 0 u t b 1 i 1 m 2 b i u i 0 where p v v p 2 v with p 1 by using the five functionals fixed point theorem we prove that the boundary value problem has at least three positive solutions as an application an example is given to illustrate the result

on functional misspecification of covariates in the cox regression model model misspecification is discussed for the analysis of censored survival data with the cox regression model the maximum partial likelihood estimator for covariate effects has an asymptotic normal distribution under model misspecification lin wei 1989 situations where the assumed functional form for covariates is wrong are considered in a general framework in which we also derive the limit of breslow s 1972 estimator furthermore we give explicit expressions for the asymptotic variance of the maximum partial likelihood estimator numerical values for the errors of the maximum partial likelihood estimator and of the usual variance estimator are obtained for functional misspecification of a single relevant covariate and looking at the effect of a first factor representing treatment or something similar for functional wrong incorporation of auxiliary covariate measurements in the latter situations we analyse the impact of dependence between the covariates

biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates summary certain important nonlinear regression models lead to biased estimates of treatment effect even in randomized experiments if needed covariates are omitted the asymptotic bias is determined both for estimates based on the method of moments and for maximum likelihood estimates the asymptotic bias from omitting covariates is shown to be zero if the regression of the response variable on treatment and covariates is linear or exponential and in regular cases this is a necessary condition for zero bias many commonly used models do have such exponential regressions thus randomization ensures unbiased treatment estimates in a large number of important nonlinear models for moderately censored exponential survival data analysis with the exponential survival model yields less biased estimates of treatment effect than analysis with the proportional hazards model of cox if needed covariates are omitted simulations confirm that calculations of asymptotic bias are in excellent agreement with the bias observed in experiments of modest size

effects of covariate omission and categorization when analysing randomized trials with the cox model when analysing the survival of patients in comparative randomized clinical trials using the cox proportional hazards model important prognostic factors may be included for the adjustment of the treatment effect in this paper we examine two of the most common misspecifications of the model i an important prognostic factor is omitted from the analysis and ii an important prognostic factor originally present on continuous scale is included in categorized form both situations may emerge from the occurrence of missing values we investigate the properties of the maximum partial likelihood estimator of the treatment effect under this kind of misspecification the estimate of the treatment effect is found to be asymptotically biased toward zero for its asymptotic variance we obtain a quantity with the so called sandwich structure thus variance estimation by the inverse of the second order derivative of the likelihood is not consistent the magnitude of overestimation or underestimation is evaluated numerically for specific settings the precision of the treatment effect estimate under covariate omission or categorization is compared with the precision of the estimate in the correct and not misspecified model it turns out that correct adjustment does not lead to a higher precision of the treatment effect estimate but due to the resulting underestimation covariate omission or categorization lead to loss of power of the test of no treatment effect 1997 by john wiley sons ltd