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

proportional hazards estimate of the conditional survival function summary we introduce a new estimator of the conditional survival function given some subset of the covariate values under a proportional hazards regression the new estimate does not require estimating the base line cumulative hazard function an estimate of the variance is given and is easy to compute involving only those quantities that are routinely calculated in a cox model analysis the asymptotic normality of the new estimate is shown by using a central limit theorem for kaplan meier integrals we indicate the straightforward extension of the estimation procedure under models with multiplicative relative risks including non proportional hazards and to stratified and frailty models the estimator is applied to a gastric cancer study where it is of interest to predict patients survival based only on measurements obtained before surgery the time at which the most important prognostic variable stage becomes known

tree structured proportional hazards regression modeling a method for fitting piecewise proportional hazards models to censored survival data is described stratification is performed recursively using a combination of statistical tests and residual analysis the bootstrap is employed to keep the probability of a type i error the error of discovering two or more strata when there is only one of the method close to a predetermined value the proposed method can thus also serve as a formal goodness of fit test for the proportional hazards model real and simulated data are used for illustration

on classification of n 2 supersymmetric theories we find a relation between the spectrum of solitons of massiven 2 quantum field theories ind 2 and the scaling dimensions of chiral fields at the conformal point the condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetricn 2 conformal theories and their massive deformations in terms of a suitable generalization of dynkin diagrams which coincides with the a d e dynkin diagrams for minimal models the landau ginzburg theories are a proper subset of this classification in the particular case of lg theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of picard lefschetz theory

assessing time by covariate interactions in proportional hazards regression models using cubic spline functions proportional hazards or cox regression is a popular method for modelling the effects of prognostic factors on survival use of cubic spline functions to model time by covariate interactions in cox regression allows investigation of the shape of a possible covariate time dependence without having to specify a specific functional form cubic spline functions allow one to graph such time by covariate interactions to test formally for the proportional hazards assumption and also to test for non linearity of the time by covariate interaction the functions can be fitted with existing software using relatively few parameters the regression coefficients are estimated using standard maximum likelihood methods

a regression survival model for testing the proportional hazards hypothesis a semi parametric generalization of the proportional hazards regression model is defined whereby the hazard functions can cross for different values of the covariates in the two sample comparison it includes in particular the case of two weibull distributions differing in scale and shape parameters a global test of the proportional hazards assumption is proposed against such defined alternatives its power in the two sample case is compared to that of previously described tests by using simulation experiments survival data of patients with breast carcinoma including several prognostic factors are presented as an illustration

tests for comparing estimated survival functions we describe a class of statistical tests for the comparison of two or more survival curves typically estimated using the kaplan meier method the class is based on the construction of o quigley 2003 and some special cases are of particular interest underlying the inferential development are the arguments of efron hinkley 1978 leading to a theoretical sampling model that is in some sense closer to the observed data the log rank and weighted log rank tests arise as special members of the class in practice the log rank test will often be a suboptimal even poor test due to the presence of non proportional hazards the proposed test maintains good power and in all the cases considered has greater power than the log rank test under non proportional hazards the power will depend on the alternatives being considered and under reasonable assumptions on the alternatives we conclude that the proposed test is more powerful than the log rank test simulations support these conclusions an example is given as an illustration

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

accelerated hazards regression model and its adequacy for censored survival data summary the accelerated hazards regression model is introduced to study the relationship between survival times and covariates through a scale change between hazard functions the model is also compared with several other popular classes of regression models for censored survival data in statistical literature test statistics are proposed and studied to assess the model s adequacy actual data from a randomized clinical trial of biodegradable carmustine polymer for treatment of brain cancer are analyzed to demonstrate the potential application of the regression model and the proposed test statistics

dualities in integrable systems and n 2 susy theories we discuss dualities in the integrable dynamics behind the exact solution to the n 2 susy yang mills theory it is shown that t duality in the string theory is related to the separation of variables procedure in the dynamical system we argue that there are analogues of s duality as well as three dimensional mirror symmetry in the many body systems of hitchin type governing the low energy effective actions

estimating the relative risk with censored data abstract we investigate a class of nonparametric estimators of relative risk in the two sample case of the proportional hazards model for censored data the asymptotic distribution of these estimators is derived using influence functions the optimal estimator in this class has the same influence functions and the same asymptotic distribution as the maximum partial likelihood estimator of cox 1972 the behavior of the influence functions is discussed briefly and the last section presents two examples from the literature

two sample goodness of fit tests for additive risk models with censored observations summary the additive risk model assumes that the hazard function associated with a set of covariates is the sum of the baseline hazard function and the regression function of covariates we propose two different test procedures for checking the adequacy of two sample additive risk models for randomly censored observations one is based on the martingale residuals and the other on the difference between weighted estimators of the excess risk the test statistics are shown to be asymptotically normal under appropriate regularity conditions and consistent under any model misspecifications finally two real examples are provided along with results of a simulation study

a generalized moments specification test of the proportional hazards model abstract this article describes a generalized moments specification test of the semiparametric proportional hazards model of cox in contrast to other specification tests for this model the generalized moments test is applicable in the presence of continuous explanatory variables and does not require assigning the data to predetermined cells the results of a monte carlo investigation suggest that the generalized moments test has good finite sample properties and that it is more powerful than other available tests for models with continuous explanatory variables when the alternative is an accelerated failure time model the test is based on generalized residuals for the proportional hazards model when the model is correct the generalized residuals have asymptotically the possibly censored unit exponential distribution the generalized moments are means of functions of the generalized residuals the generalized moments have known values when the generalized residuals have the unit exponential distribu

some diagnostic methods for cox regression models through hazard smoothing in this paper some graphical methods are proposed for evaluating the fit of cox regression models in survival analysis the simplest procedure proposed is to estimate the baseline hazard separately within subgroups by applying kernel based smoothing to standard cumulative hazard estimates the different estimates should be the same to within sampling variability when the model is correct a modification of this procedure gives a direct way of calculating smooth estimates of hazard ratios over time a further modification gives estimates of ratios as a function of a continuous covariate and time the methods are applied to data taken from a clinical trial of adjuvant therapy for breast cancer

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

a specification test for univariate and multivariate proportional hazards models this paper applies white s 1982 econometrica 50 1 25 information matrix im test for correct model specification to proportional hazards models of univariate and multivariate censored survival data several alternative estimators of the test statistic are presented and their size performance examined white also suggested an estimator of the parameter covariance matrix that was robust to certain forms of model misspecification this has been subsequently proposed by others e g royall 1986 international statistical review 54 221 226 and applied by huster brookmeyer and self 1989 biometrics 45 145 156 as part of an independence working model iwm approach to multivariate censored survival data we illustrate how the im test can be used for both univariate data and as part of the iwm approach to multivariate data

bayesian model selection for partially observed diffusion models summary we present an approach to bayesian model selection for finitely observed diffusion processes we use data augmentation by treating the paths between observed points as missing data for a fixed model formulation the strong dependence between the missing paths and the volatility of the diffusion can be broken down by adopting the method of roberts stramer 2001 we describe how this method may be extended to the case of model selection via reversible jump markov chain monte carlo in addition we extend the formulation of a diffusion model to capture a potential non markov state dependence in the drift issues of appropriate choices of priors and efficient transdimensional proposal distributions for the reversible jump algorithm are also addressed the approach is illustrated using simulated data and an example from finance

bayes factors for discrete observations from diffusion processes summary we present an approach to model selection for a time series of data on a fine time scale the underlying process generating the data is modelled as a continuous time stochastic process the underlying continuous processes are assumed to be diffusions with time varying drift and diffusion coefficient several approaches to modelling the diffusion coefficient are described to perform model selection we propose an approximation to the bayes factor that uses only the discrete data we illustrate our approach for several well known processes including brownian motion with drift the ornstein uhlenbeck process a mean reversion process with drift exponential brownian motion and a logistic growth model finally we apply our technique to data from the standard poor s 500 stock index by comparing a random walk to a mean reversion model

nonparametric analyses for two level single stress accelerated life tests in accelerated life testing information is sought about a process at normal stress levels by studying it at higher stress levels let pr t x be the probability of failure by time t of an item subjected to level x of a stress in this article i consider the nonparametric proportional hazards model pr t x 1 e g x h t and the nonparametric accelerated failure time model pr f x f g x t where g and h are nonnegative nondecreasing functions of x and t respectively f is an arbitrary distribution function and g has sigmoid s shaped curvature i develop confidence bounds for low stress long time probabilities and quantiles i also discuss a goodness of fit test of the proportional hazards model the results which are primarily for data at two levels of stress accommodate simple right censoring

graphical methods for assessing violations of the proportional hazards assumption in cox regression a major assumption of the cox proportional hazards model is that the effect of a given covariate does not change over time if this assumption is violated the simple cox model is invalid and more sophisticated analyses are required this paper describes eight graphical methods for detecting violations of the proportional hazards assumption and demonstrates each on three published datasets with a single binary covariate i discuss the relative merits of these methods smoothed plots of the scaled schoenfeld residuals are recommended for assessing ph violations because they provide precise usable information about the time dependence of the covariate effects

action angle maps and scattering theory for some finite dimensional integrable systems i the pure soliton case we construct an action angle transformation for the calogero moser systems with repulsive potentials and for relativistic generalizations thereof this map is shown to be closely related to the wave transformations for a large classl of hamiltonians and is shown to have remarkable duality properties all dynamics inl lead to the same scattering transformation which is obtained explicitly and exhibits a soliton structure an auxiliary result concerns the spectral asymptotics of matrices of the formm exp td ast it pertains to diagonal matricesd whose diagonal elements have pairwise different real parts and to matricesm for which certain principal minors are non zero

score tests for homogeneity of regression effect in the proportional hazards model a simple model containing the proportional hazards regression model as a special case is presented the purpose of the model is to provide a framework in which specific alternatives to the proportional hazards assumption may be tested rank invariant score tests for linear quadratic or exponential trends can for instance all be undertaken within this framework in the case of the two sample problem the required calculations are shown to take a particularly simple form special consideration is given to the two sample case in which there is an inversion of the regression effect i e where the hazard functions cross at some given point both of the motivating examples are concerned with this problem computational aspects are relatively straightforward and some discussion on this is provided

pearson type goodness of fit tests the univariate case abstract chi squared tests are proposed for testing goodness of fit when the data may or may not be subject to random censoring both the simple and compositive null hypotheses are considered for the simple null hypothesis the resulting test statistic in the uncensored case differs from pearson s statistic only in that the expected frequency for cell aj is taken to be where is the empirical distribution function instead of n a j d f o this results in k df instead of the usual k 1 the theoretical justification lies in the behavior of the corresponding modified version of the usual empirical process it converges weakly to a brownian motion process instead of a tied down brownian motion under censoring the tests require no restrictive assumptions on the censoring distribution and the statistic has the same simple form for the composite null hypothesis the tests development is based on the weak convergence of the modified empirical process when parameters are estimated the raw data maximum li

erosion of regression effect in a survival study lack of persistence or erosion of the regression effect is an alternative to proportional hazards of particular interest in many medical applications such a departure from proportional hazards is often the most likely direction in which the model may be inadequate questions such as is the effect of treatment only transitory or to what extent does an initially measured prognostic variable maintain its impact frequently arise in the context of a simple changepoint model we propose a test of the null hypothesis of proportional hazards against the specific alternative of erosion of the regression effect the particular changepoint model used can be viewed as a first approximation to a more complex reality an approximation that enables us to avoid specifically modeling the functional form that any erosion might take practical guidelines for carrying out the test are provided the approach is illustrated in the context of a study on risk factors for breast cancer survival

on prequential model assessment in life history analysis summary we extend the prequential approach introduced by a p dawid to continuous time marked point processes a prequential forecasting system is expressed by means of the stochastic intensities of the point process the exact independence and 1 exponentiality of the derived compensators are used to assess the predictive performance of the model and the inferential procedure

smooth goodness of fit tests for the baseline hazard in cox s proportional hazards model abstract for the cox proportional hazards model the goodness of fit problem of testing whether the baseline hazard rate function equals some specified hazard rate function 0 in the presence of incomplete data is considered extending neyman s idea the goodness of fit tests are score tests obtained by embedding in a larger family of hazard rate functions developed through smooth and possibly random transformations of 0 but whereas their smooth alternatives were formulated through the density function my formulation is through the hazard rate function which is more natural in failure time models where model specification is now done through hazard rates furthermore my smooth alternatives are allowed to be data driven or dynamic through this machinery classes of omnibus as well as directional goodness of fit tests are obtained moreover this machinery provides a theoretically sound method for combining varied directional goodness of fit tests into omnibus tests some of the resul

graphical comparison of cumulative hazards for two populations summary we consider graphs confidence procedures and tests that can be used with censored survival data to compare the hazard experience of a treatment group with that of a control group in particular we consider the relative change a t in a cumulative rate function which is used to define a generalized proportional hazard model this model is equivalent to the model obtained by introducing a gamma distributed frailty term in the proportional hazard model we introduce an estimate a t or a t based on the kaplan meier and nelson aalen estimates and obtain the asymptotic distribution of the process a t a t appropriately scaled this result is used to construct asymptotic pointwise confidence intervals for a t under the proportional hazard model assumption doob s transformation of gaussian processes is used to derive asymptotic simultaneous confidence bands for a t from a general class of test statistics defined as sup norms of weighted differences between a t and its estimate a t we also consider procedures based on estimates r u of the relative trend function r u between cumulative hazards the methods are illustrated on a data set from a study of the effect of radiation on mice and on a data set from a study of the effect of tumour size on the survival of cancer patients

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

comparing two failure time distributions in the presence of dependent censoring summary in a randomised clinical trial to compare two groups of patients suppose that the time to disease occurrence the major response variable may be subject to dependent censoring by death or selective patient withdrawal while the patients who have reached the disease endpoint are followed for their secondary endpoints or survival information to adjust for the dependent censoring in assessing the group difference in disease occurrence we assume that on a logarithmic scale the times to disease occurrence and dependent censoring for the two groups satisfy a bivariate location shift model with a completely unspecified underlying distribution rank based procedures are constructed for making inferences about the location shift parameter model checking techniques are also developed numerical studies show that the proposed methods are appropriate for practical use an illustration with data taken from a recent aids clinical trial is provided

proportional hazards tests and diagnostics based on weighted residuals summary nonproportional hazards can often be expressed by extending the cox model to include time varying coefficients e g for a single covariate the hazard function for subject i is modelled as exp fl t zi t a common example is a treatment effect that decreases with time we show that the function 3 t can be directly visualized by smoothing an appropriate residual plot also many tests of proportional hazards including those of cox 1972 gill schumacher 1987 harrell 1986 lin 1991 moreau o quigley mesbah 1985 nagelkerke oosting hart 1984 o quigley pessione 1989 schoenfeld 1980 and wei 1984 are related to time weighted score tests of the proportional hazards hypothesis and can be visualized as a weighted least squares line fitted to the residual plot

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

testing goodness of fit for proportional hazards model with censored observations abstract a numerical omnibus test of fit for the two sample proportional hazards model is proposed for randomly censored observations the test is derived from cox s partial likelihood and does not need a dummy time dependent covariate or any partition of the time axis consistency of the test is established examples are provided for illustration

testing the hypothesis of proportional hazards in two populations the assumption of proportional hazards in different populations leads to elegant analysis of survival data as initiated by cox 1972 however this model is not appropriate in all situations in recent literature there have been several instances of crossing hazards which may be modelled by an increasing hazard ratio we present a u statistic test for checking constancy of the hazard ratio against this alternative the test is suitable for censored data and is found to have reasonable power in certain parametric families next we propose a test for checking proportionality of the hazards of two competing risks in the presence of a third risk against the alternative of increasing hazard ratio analyses of a few real data sets and some results based on computer simulations are also given

a graphical method for assessing goodness of fit in cox s proportional hazards model abstract suggested here is a simple graphical method for studying the goodness of fit in cox s regression model for survival data the method is easy to use as it does not require the estimation of alternative models and only involves quantities similar to those already appearing in the partial likelihood expression that is needed in the parameter estimation the rationale behind the graphs is very intuitive they make a direct comparison between observed and expected failure frequencies as estimated from the model in a correctly specified model one anticipates an approximate balance between such frequencies otherwise there will typically be groups of individuals for which the expected frequencies are systematically too high or too low to match with the data and this shows in the graphs introduced here in the concrete applications of the method the individuals are stratified in a way that depends on what aspect of the model is being checked against data there is always one graph for each stratum s

analysis of transformation models with censored data summary in this paper we consider a class of semi parametric transformation models under which an unknown transformation of the survival time is linearly related to the covariates with various completely specified error distributions this class of regression models includes the proportional hazards and proportional odds models inference procedures derived from a class of generalised estimating equations are proposed to examine the covariate effects with censored observations numerical studies are conducted to investigate the properties of our proposals for practical sample sizes these transformation models coupled with the new simple inference procedures provide many useful alternatives to the cox regression model in survival analysis

on graphical tests for proportionality of hazards in two samples in this paper we present a class of graphical tests of the proportional hazards hypothesis for two sample censored survival data the proposed tests are improvements over some existing tests based on asymptotic confidence bands of certain functions of the estimated cumulative hazard functions the new methods are based on the comparison of unrestricted estimates of the said functions and their restricted versions under the hypothesis they combine the rigour of analytical tests with the descriptive value of plots monte carlo simulations suggest that the proposed asymptotic procedures have reasonable small sample properties the power is much higher than existing graphical tests and comparable with existing analytical tests the method is then illustrated through the analysis of a data set on bone marrow transplantation for leukemia patients