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

a simple test for goodness of fit of cox s proportional hazards model cox s proportional hazards model for survival analysis 1972 has gained widespread attention from both applied and theoretical statisticians and is central to most work on survival analysis to verify the assumption of proportional hazards underlying the model several methods have recently been put forward cox 1979 schoenfeld 1980 andersen 1982 kalbfleisch and prentice 1980 these methods consist either of graphical techniques designed so as to visualize departures from the proportional hazards ph assumption or of formal tests based on parameterizing the interaction between covariates and time in this paper a simple test statistic is proposed for detecting departures from the ph assumption the proposed test is computationally and conceptually straightforward expressions for the first two moments of the test statistic are presented

adding a variable in generalized linear models the likelihood ratio statistic can be used to determine the significance of an explanatory variable in a generalized linear model in order to obtain such a statistic however we need two sets of iterations for two maximum likelihoods moreover the statistic is not directed to detect influential or outlying observations that affect the importance of the variable considered therefore we develop relatively simple procedures to help the analyst select an appropriate model and detect the effects of such observations on adding a variable into any model two examples are given for illustration

tests of loglinear and linear relative risks for cox s model summary a score test statistic is derived to test for the loglinear relative risk function in the cox proportional hazards model for failure time data the loglinear and linear relative risk functions are embedded within a power family transformation of the linear predictor in two simple cases some empirical results are obtained for the score test statistic using expected information and for the likelihood ratio statistic both statistics have distributions well approximated by the chi squared distribution problems of identifiability and problems associated with the linear relative risk model are also discussed a real example is given

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

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

heuristics for the stochastic eulerian tour problem the stochastic eulerian tour problem setp seeks the eulerian tour of minimum expected length on an undirected eulerian graph when demand on the arcs that have to be serviced is probabilistic the setp is np hard and in this paper we develop three constructive heuristics for this problem the first two are greedy tour construction heuristics while the third is a sub tour concatenation heuristic our experimental results show that for grid networks the sub tour concatenation heuristic performs well when the probability of service of each edge is greater than 0 1 for euclidean networks as the number of edges increases the second heuristic performs the best among the three also the expected length of our overall best solution is lower than the expected length of a random tour by up to 10 on average for grid networks and up to 2 for euclidean networks

on derivatives of artin l series we present a universal theory of refined stark conjectures we give evidence in support of these explicit and very general conjectures including a full proof in several important cases and explain the connection to previous conjectures of bloch and kato of lichtenbaum and of serre and tate we also deduce a wide range of unconditional consequences of these results concerning the annihilation as galois modules of ideal class groups by explicit elements constructed from the values of higher order derivatives of non abelian artin l series

a priori solution of a travelling salesman problem in which a random subset of the customers are visited consider a problem of routing through a set of n points on any given instance of the problem only a subset consisting of k out of n points 0 k n has to be visited with the number k random with known probability distribution we wish to find a priori a tour through all n points on any given instance the k points present will then be visited in the same order as they appear in the a priori tour the problem of finding such a tour of minimum length in the expected value sense is defined as a probabilistic traveling salesman problem ptsp what distinguishes one ptsp from another is the probability distribution or more generally the probability law that specifies the number k and the identity of the points that need to be visited on any given instance of the problem after motivating the problem by applications we first derive closed form expressions for computing efficiently the expected length of any given tour under very general probabilistic assumptions we then provide in a unified way

the graphical evaluation of explanatory variables in proportional hazard regression models summary a graphical technique for assessing explanatory variables in cox s proportional hazards regression model is examined the method arises from consideration of the cumulative hazard transformation and the partial likelihood score function and consists of permuting the observed rank statistic to account for the effects of explanatory variables fitted to the model an example is provided

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

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

new insertion and postoptimization procedures for the traveling salesman problem this paper describes a new insertion procedure and a new postoptimization routine for the traveling salesman problem the combination of the two methods results in an efficient algorithm genius which outperforms known alternative heuristics in terms of solution quality and computing time

diagnostic plots in cox s regression model two diagnostic plots are presented for validating the fitting of a cox proportional hazards model the added variable plot is developed to assess the effect of adding a covariate to the model the constructed variable plot is applied to detect nonlinearity of a fitted covariate both plots are also useful for identifying influential observations on the issues of interest the methods are illustrated on examples of multiple myeloma and lung cancer data

further results on the probabilistic traveling salesman problem abstract in 1985 jaillet introduced the probabilistic traveling salesman problem ptsp a variant of the classical tsp in which only a subset of the nodes may be present in any given instance of the problem the goal is to find an a priori tour of minimal expected length with the strategy of visiting the present nodes in a particular instance in the same order as they appear in the a priori tour in this paper we reexamine the ptsp using a variety of theoretical and computational approaches we sharpen the best known bounds for the ptsp derive several asymptotic relations and compare from various veiwpoints the ptsp with the re optimization strategy i e finding an optimal tour in every problem instance when a euclidean metric is used and the nodes are uniformly distributed in the unit square a heuristic for the ptsp is shown to be very close to the re optimization strategy we examine some ptsp heuristics with provable worst case performance and address the question of finding constant guarantee heuristics implementations of various heuristics some based on sorting and some on local optimality permit us to discuss the qualitative and quantitative properties of computational problems with up to 5000 nodes

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

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

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