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

statistical estimation in the proportional hazards model with risk set sampling thomas partial likelihood estimator of regression parameters is widely used in the analysis of nested case control data with cox s model this paper proposes a new estimator of the regression parameters which is consistent and asymptotically normal its asymptotic variance is smaller than that of thomas estimator away from the null unlike some other existing estimators the proposed estimator does not rely on any more data than strictly necessary for thomas estimator and is easily computable from a closed form estimating equation with a unique solution the variance estimation is obtained as minus the inverse of the derivative of the estimating function and therefore the inference is easily available a numerical example is provided in support of the theory

computation of error bounds for p matrix linear complementarity problems we give new error bounds for the linear complementarity problem where the involved matrix is a p matrix computation of rigorous error bounds can be turned into a p matrix linear interval system moreover for the involved matrix being an h matrix with positive diagonals an error bound can be found by solving a linear system of equations which is sharper than the mathias pang error bound preliminary numerical results show that the proposed error bound is efficient for verifying accuracy of approximate solutions

improved error bounds based on α m for the linear complementarity problem abstract in this paper the error bounds of the approximate solution and the bounds of the true solution to the linear complementarity problem lcp m q are presented based on the function m which is introduced by mathias and pang in 23 in order to show the accuracy and the effectiveness of our results we perform some numerical experiments to reveal our results are better than other results in the literature

error bounds for the linear complementarity problem with a p matrix abstract we introduce a fundamental quantity associated with a p matrix and show how this quantity is useful in deriving error bounds for the linear complementarity problem of the p type we also obtain upper and lower bounds for the quantity introduced

error bounds for monotone linear complementarity problems we give a bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered when the point satisfies the linear inequalities of the complementarity problem the residual consists of the complementarity condition plus its square root this latter term is essential and without it the error bound cannot hold we also show that another natural residual that has been employed to bound errors for strictly monotone linear complementarity problems fails to bound errors for the monotone case considered here

approximate case influence for the proportional hazards regression model with censored data a method is presented for approximating the influence of individual cases upon regression coefficient estimates obtained from the cox proportional hazards model observations can thus be identified which may greatly influence statistical inferences regarding the effects of prognostic factors upon survival time an example from a cancer clinical trial is given

information bounds for cox regression models with missing data we derive information bounds for the regression parameters in cox models when data are missing at random these calculations are of interest for understanding the behavior of efficient estimation in case cohort designs a type of two phase design often used in cohort studies the derivations make use of key lemmas appearing in robins rotnitzky and zhao j amer statist assoc 89 1994 846 866 and robins hsieh and newey j roy statist soc ser b 57 1995 409 424 but in a form suited for our purposes here we begin by summarizing the results of robins rotnitzky and zhao in a form that leads directly to the projection method which will be of use for our model of interest we then proceed to derive new information bounds for the regression parameters of the cox model with data missing at random mar in the final section we exemplify our calculations with several models of interest in cohort studies including an i i d version of the classical case cohort design of prentice biometrika 73 1986 1 11 and self and prentice ann statist 16 1988 64 81

new improved error bounds for the linear complementarity problem new local and global error bounds are given for both nonmonotone and monotone linear complementarity problems comparisons of various residuals used in these error bounds are given a possible candidate for a best error bound emerges from our comparisons as the sum of two natural residuals

on estimation of partially linear transformation models we study a general class of partially linear transformation models which extend linear transformation models by incorporating nonlinear covariate effects in survival data analysis a new martingale based estimating equation approach consisting of both global and kernel weighted local estimation equations is developed for estimating the parametric and nonparametric covariate effects in a unified manner we show that with a proper choice of the kernel bandwidth parameter one can obtain the consistent and asymptotically normal parameter estimates for the linear effects asymptotic properties of the estimated nonlinear effects are established as well we further suggest a simple resampling method to estimate the asymptotic variance of the linear estimates and show its effectiveness to facilitate the implementation of the new procedure an iterative algorithm is developed numerical examples are given to illustrate the finite sample performance of the procedure supplementary materials are available online

information and asymptotic efficiency in parametric nonparametric models asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems for regular estimates and asymptotic minimax bounds the methods used involve i the notion of a hellinger differentiable root density where part of the differentiation is with respect to the nonparametric part of the model to obtain appropriate scores and ii calculation of the effective score for the real or vector finite dimensional parameter of interest as that component of the score function orthogonal to all nuisance parameter scores perhaps infinite dimensional the resulting asymptotic information for estimation of the parametric component of the model is just 4 times the squared l 2 norm of the effective score a corollary of these results is a simple necessary condition for adaptive estimation adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model examples considered include the one sample location model with and without symmetry mixture models the two sample shift model and cox s proportional hazards model

estimation of the survival function for gray s piecewise constant time varying coefficients model gray s extension of cox s proportional hazards ph model for right censored survival data allows for a departure from the ph assumption via introduction of time varying regression coefficients tvc using penalized splines gray s work focused on estimation inference and residual analyses but no estimator for the survival function has been proposed we derive a survival function estimator for one important member of the class of tvc models a piecewise constant time varying coefficients pc tvc model we also derive an estimate for the confidence limits of the survival function accuracy in estimating underlying survival times and survival quantiles is assessed for both cox s and gray s pc tvc model using a simulation study featuring scenarios violating the ph assumption finally an example of the estimated survival functions and the corresponding confidence limits derived from cox s ph and gray s pc tvc model respectively is presented for a liver transplant data set in the second part of the thesis we examine the effect of model misspecification for two classes of regression models for right censored survival data additive and multiplicative models for the conditional hazard rate a particular attention is given to data exhibiting time varying regression coefficients the class of multiplicative models is represented by cox ph model and gray s tvc model respectively and for additive models we use aalen s linear model both gray s tvc model and aalen s linear model incorporate time varying coefficients a simulation study is performed to cross analyze survival data which follows either a multiplicative or an additive model for the conditional hazard rate the effect of misspecifying the true model for the conditional hazard rate is assessed by looking at the power of the individual models to detect an existing effect bias and mean square error observed for each conditional model based estimator of survival we also show that aalen s model formulae is a first order taylor series approximation of that of gray s model which explains the comparably higher flexibility on part of the aalen s model as compared to the cox ph when the gray s tvc model for the data is misspecified

chaotic iterative methods for the linear complementarity problems a class of parallel multisplitting chaotic relaxation methods is established for the large sparse linear complementarity problems and the global and monotone convergence is proved for the h matrix and the l matrix classes respectively moreover comparison theorem is given which describes the influences of the parameters and the multiple splittings upon the monotone convergence rates of the new methods

maximum weighted partial likelihood estimators for the cox model new estimators for cox regression that maximize a weighted version of the partial likelihood are considered their asymptotic properties on the model are sketched estimators with bounded influence curves are identified and shown to have good asymptotic relative efficiency the results of a small simulation study are included the estimators are applied to a well known data set the possible influence of an outlying observation on the cox estimator can be of any magnitude individuals who survive longest have a high potential of being influential in analogy with weighted linear rank tests we introduce estimators of the regression coefficients that maximize a weighted partial likelihood for instance by using weights proportional to the number at risk one obtains a wilcoxon type estimator this estimator has bounded influence provided the covariate space is compact and in a wide variety of situations its asymptotic relative efficiency is at least as good as that of the wilcoxon test asymptotic prop

a nonparametric test to compare survival distributions with covariate adjustment summary the analysis of covariance is a technique that is used to improve the power of a k sample test by adjusting for concomitant variables if the end point is the time of survival and some observations are right censored the score statistic from the cox proportional hazards model is the method that is most commonly used to test the equality of conditional hazard functions in many situations however the proportional hazards model assumptions are not satisfied specifically the relative risk function is not time invariant or represented as a log linear function of the covariates we propose an asymptotically valid k sample test statistic to compare conditional hazard functions which does not require the assumption of proportional hazards a parametric specification of the relative risk function or randomization of group assignment simulation results indicate that the performance of this statistic is satisfactory the methodology is demonstrated on a data set in prostate cancer

directionally continuous selection in banach spaces soit e un espace de banach m 0 et on considere le cone m dans r e m t x r e x mt on dit qu une application f r e e est m continue si pour tout t 0 x 0 et e 0 il existe 0 tel que f t x f t 0 x 0 quand t 0 t t 0 x x 0 m t t 0 etant donnee une multifonction semicontinue inferieurement g r e e a valeurs fermees non vides on etudie l existence d une selection m continue f t x g t x

influence functions for proportional hazards regression summary influence functions for the regression parameters in the proportional hazards model are presented it is suggested that empirical influence functions computed for each observation and each covariate can be useful in an informal way to identify influential observations this is illustrated on the stanford heart transplant data and two other examples

a diagnostic for cox regression and general conditional likelihoods abstract diagnostics for changes in the maximum likelihood estimate of due to deletion of single observations in linear and logistic regression models are equivalent to diagnostics obtained from fitting augmented regression models for the cox proportional hazards model and in conditional models for arbitrarily matched or stratified data diagnostics derived from augmented regression models are again useful and easy to compute these diagnostics are applicable to different formulations for the multiplicative covariable effects over time or over strata an exact or approximate treatment of tied failure times or multiple case matching and time dependent covariates

asymptotic distribution theory for cox type regression models with general relative risk form the theory and application of the cox 1972 failure time regression model has almost without exception assumed an exponential form for the dependence of the hazard function on regression variables other regression forms may be more natural or descriptive in some applications for example a linear relative risk regression model provides a convenient framework for studying epidemiologic risk factor interactions this note uses the counting process formulation of andersen and gill 1982 to develop asymptotic distribution theory for a class of intensity function regression models in which the usual exponential regression form is relaxed to an arbitrary non negative twice differentiable form some stability and regularity conditions beyond those of andersen and gill are required to show the consistency of the observed information matrix which in general need not be positive semidefinite

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

the estimation of average hazard ratios by weighted cox regression often the effect of at least one of the prognostic factors in a cox regression model changes over time which violates the proportional hazards assumption of this model as a consequence the average hazard ratio for such a prognostic factor is under or overestimated while there are several methods to appropriately cope with non proportional hazards in particular by including parameters for time dependent effects weighted estimation in cox regression is a parsimonious alternative without additional parameters the methodology which extends the weighted k sample logrank tests of the tarone ware scheme to models with multiple binary and continuous covariates has been introduced in the nineties of the last century and is further developed and re evaluated in this contribution the notion of an average hazard ratio is defined and its connection to the effect size measure p x y is emphasized the suggested approach accomplishes estimation of intuitively interpretable average hazard ratios and provides tools for inference a monte carlo study confirms the satisfactory performance advantages of the approach are exemplified by comparing standard and weighted analyses of an international lung cancer study sas and r programs facilitate application

resampling procedures to compare two survival distributions in the presence of right censored data summary in this simulation study the accuracy of the size of four classes of test procedures asymptotic normal test permutation test bootstrap hypothesis test and bootstrap interval test were compared using the logrank and prentice wilcoxon two sample test statistics sample sizes of 20 and 50 per group were used twenty observations per group yielded differences in the test procedures when the prentice wilcoxon test statistic was employed the asymptotic test procedure was the most accurate resampling procedures did not improve upon its accuracy using the logrank test statistic the permutation or bootstrap hypothesis testing procedures were closest to the nominal significance level the more popular asymptotic normal logrank test was consistently anticonservative when the number of observations in each group was increased to 50 the test procedures were comparable power comparisons were also examined the test with the greatest power was dependent upon the two survival patterns and the choice of test statistic

generalized martingale residual processes for goodness of fit inference in cox s type regression models in the paper a general class of stochastic processes based on the sums of weighted martingale transform residuals for goodness of fit inference in general cox s type regression models is studied their form makes the inference robust to covariate outliers a weak convergence result for such processes is obtained giving the possibility of establishing the randomness of their graphs together with the construction of the formal x 2 type goodness of fit tests by using the khmaladze innovation approach a modified version of the initial class of processes is also defined weak convergence results for the processes are derived this leads to the main application which concerns the formal construction of the kolmogorov smirnov and cramer von mises type goodness of fit tests this is done within the general situation considered

on the existence of solutions to differential inclusions with nonconvex right hand sides we study the existence of solutions of differential inclusions with upper semicontinuous right hand sides the investigation was prompted by the well known filippov examples we define a new concept colliding on a set in the case when the admissible velocities do not collide on the set of discontinuities of the right hand side we expect that at least one trajectory emanates from every point if the velocities do collide on the set of discontinuities of the right hand side the existence of solutions is not guaranteed as is seen from one of filippov s examples in this case we impose an additional condition in order to prove the existence of a solution starting at a point of the discontinuity set for the right hand sides under consideration we assume the following whenever the velocities collide on a set s there exist tangent velocities belonging to the clarke tangent cone to s on a dense subset of s then we prove the existence of an varepsilon solution for every varepsilon 0 under additional assumptions we can pass to the limit as varepsilon to 0 and obtain a solution of the considered differential inclusion

some new estimators for cox regression new estimators for cox regression are considered their asymptotic properties both on and off the model are established corollaries include conditions under which the maximum partial likelihood estimator defines a parameter in the population and the asymptotics of the case cohort estimator robust estimators that minimize the asymptotic variance subject to a bound on the maximal bias on infinitesimal neighborhoods are discussed the estimators are illustrated with medical data

goodness of fit analysis for the cox regression model based on a class of parameter estimators abstract in this article we propose a class of estimation functions for the vector of regression parameters in the cox proportional hazards model with possibly time dependent covariates by incorporating the weight functions commonly used in weighted log rank tests into the partial likelihood score function the resulting estimators behave much like the conventional maximum partial likelihood estimator in that they are consistent and asymptotically normal when the cox model is inappropriate however the estimators with different weight functions generally converge to nonidentical constant vectors for example the magnitude of the parameter estimator using the kaplan meier survival estimator as the weight function will be stochastically larger than that of the maximum partial likelihood estimator if covariate effects diminish over time such facts motivate us to develop goodness of fit methods for the cox regression model by comparing parameter estimators with different weight functions under the assume

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

information bounds for the conditional hazard ratio in a nested family of regression models information bounds are obtained for a parametric treatment effect in a nested family of models for the conditional distribution of survival given confounding variables the family extends from the fully parametric weibull model to a completely nonparametric model for right censored data and includes the cox model and the stratified version of it it is shown that in certain special cases there is no loss of information about the treatment effect when going from a fully parametric to a nonparametric specification of the conditional hazard function

a comparison of error bounds for linear complementarity problems of h matrices we give new error bounds for the linear complementarity problem when the involved matrix is an h matrix with positive diagonals we find classes of h matrices for which the new bounds improve considerably other previous bounds we also show advantages of these new bounds with respect the computational cost a new perturbation bound of h matrices linear complementarity problems is also presented

analysis of competing risks survival data when some failure types are missing we propose a method to analyse competing risks survival data when failure types are missing for some individuals our approach is based on a standard proportional hazards structure for each of the failure types and involves the solution to estimating equations we present consistent and asymptotically normal estimators of the regression coefficients and related score tests an appealing feature is that individuals with known failure types make the same contributions as they would to a standard proportional hazards analysis contributions of individuals with unknown failure types are weighted according to the probability that they failed from the cause of interest efficiency and robustness are discussed results are illustrated with data from a breast cancer trial

maximum likelihood estimation in semiparametric regression models with censored data summary semiparametric regression models play a central role in formulating the effects of covariates on potentially censored failure times and in the joint modelling of incomplete repeated measures and failure times in longitudinal studies the presence of infinite dimensional parameters poses considerable theoretical and computational challenges in the statistical analysis of such models we present several classes of semiparametric regression models which extend the existing models in important directions we construct appropriate likelihood functions involving both finite dimensional and infinite dimensional parameters the maximum likelihood estimators are consistent and asymptotically normal with efficient variances we develop simple and stable numerical techniques to implement the corresponding inference procedures extensive simulation experiments demonstrate that the inferential and computational methods proposed perform well in practical settings applications to three medical studies yield important new insights we conclude that there is no reason theoretical or numerical not to use maximum likelihood estimation for semiparametric regression models we discuss several areas that need further research