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

goodness of fit test for monotone proportional subdistribution hazards assumptions based on weighted residuals recently goodness of fit tests have been proposed for checking the proportional subdistribution hazards assumptions in the fine and gray regression model zhou fine and laird proposed weighted schoenfeld type residuals tests derived under an assumed model with specific form of time varying regression coefficients li sheike and zhang proposed an omnibus test based on cumulative sums of schoenfeld type residuals in this article we extend the class of weighted residuals tests by allowing random weights of schoenfeld type residuals at ordered event times in particular it is demonstrated that weighted residuals tests using monotone weight functions of time are consistent against monotone proportional subdistribution hazards assumptions extensive monte carlo studies were conducted to evaluate the finite sample performance of recent goodness of fit tests results from simulation studies show that weighted residuals tests using monotone random weight functions commonly used in non proportional hazards regression settings tend to be more powerful for detecting monotone departures than other goodness of fit tests assuming no specific time varying effect or misspecified time varying effects two examples using real data are provided for illustrations copyright 2016 john wiley sons ltd

parsimonious analysis of time dependent effects in the cox model cox s proportional hazards model can be extended to accommodate time dependent effects of prognostic factors we briefly review these extensions along with their varying degrees of freedom spending more degrees of freedom with conventional procedures a priori defined interactions with simple functions of time restricted natural splines piecewise estimation for partitions of the time axis allows the fitting of almost any shape of time dependence but at an increased risk of over fit this results in increased width of confidence intervals of time dependent hazard ratios and in reduced power to confirm any time dependent effect or even any effect of a prognostic factor by means of comparative empirical studies the consequences of over fitting time dependent effects have been explored we conclude that fractional polynomials and similarly penalized likelihood approaches today are the methods of choice avoiding over fit by parsimonious use of degrees of freedom but also permitting flexible modelling if time dependence of a usually a priori unknown shape is present in a data set the paradigm of a parsimonious analysis of time dependent effects is exemplified by means of a gastric cancer study

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

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

maximum likelihood estimation in the proportional odds model abstract we consider maximum likelihood estimation of the parameters in the proportional odds model with right censored data the estimator of the regression coefficient is shown to be asymptotically normal with efficient variance the maximum likelihood estimator of the unknown monotonic transformation of the survival time converges uniformly at a parametric rate to the true transformation estimates for the standard errors of the estimated regression coefficients are obtained by differentiation of the profile likelihood and are shown to be consistent a likelihood ratio test for the regression coefficient is also considered

a mathematical model of the chemostat with a general class of functions describing nutrient uptake a model of the chemostat involving n microorganisms competing for a single essential growth limiting substrate is considered instead of assuming the familiar michaelis menten kinetics for nutrient uptake a general class of functions is used which includes all monotone increasing uptake functions but which also allows uptake functions that describe inhibition by the substrate at high concentrations the qualitative behaviour of this generalized model is determined analytically it is shown that the behaviour depends intimately upon certain parameters provided that all the parameters are distinct which is a biologically reaonable assumption at most one competitor survives the substrate and the surviving competitor if one exists approach limiting values thus there is competitive exclusion however unlike the standard model in certain cases the outcome is initial condition dependent

global asymptotic behavior of the chemostat general response functions and different removal rates in this paper we consider a competition model between n species in a chemostat that incorporates both monotone and nonmonotone general response functions and distinct removal rates we show that only the species with the lowest break even concentration survives provided that the variation of distinct removal rates relative to the flow rate of the chemostat can be controlled by either the difference between the two lowest break even concentrations or by a parameter based on the structure of response functions lasalle s extension theorem of the lyapunov stability theory and fluctuation lemma are the main tools

analysis of a delayed monod type chemostat model with impulsive input the polluted nutrient in this paper a monod type chemostat model with delayed response in growth and impulsive input the polluted nutrient is considered using the discrete dynamical system determined by the stroboscopic map we obtain a microorganism extinction periodic solution further it is globally attractive the permanent condition of the investigated system is also obtained by the theory of impulsive delay differential equation our results reveal that the delayed response in growth plays an important role on the outcome of the chemostat

global asymptotic behavior of a chemostat model with discrete delays this paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat the model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations in the case where only two species are engaged in competition it is shown that competitive exclusion holds for any monotone growth response functions sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n species case in regard to the delay effects on the qualitative outcome of competition it is demonstrated that when the delays are relatively small the predictions of the model are identical with the predictions given by corresponding models without time delays however including large delays in the model may alter the predicted outcome of competition the techniques used also work when different removal rates are permitted an

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

predictive accuracy and explained variation in cox regression summary we suggest a new measure of the proportion of the variation of possibly censored survival times explained by a given proportional hazards model the proposed measure termed v shares several favorable properties with an earlier v1 but also improves the handling of censoring the statistic contrasts distance measures between individual 1 0 survival processes and fitted survival curves with and without covariate information these distance measures dx and d respectively are themselves informative as summaries of absolute rather than relative predictive accuracy we recommend graphical comparisons of survival curves for prognostic index groups to improve the understanding of obtained values for v dx and d their use and interpretation is exemplified for a yorkshire lung cancer study on survival from this and an overview for several well known clinical data sets we show that the likely amount of relative or absolute predictive accuracy is often low even if there are highly significant and relatively strong prognostic factors

global dynamics of a mathematical model of competition in the chemostat general response functions and differential death rates a model of exploitative competition of n species in a chemostat for a single essential nonreproducing growth limiting resource is considered s b hsu siam j appl math 34 1978 pp 760 763 applies lasalle s extension theorem of lyapunov stability theory to study the asymptotic behavior of solutions in the special case that the response functions are modeled by michaelis menten dynamics g j butler and g s k wolkowicz siam j appl math 45 1985 pp 138 151 on the other hand allow more general response functions including monotone and nonmonotone functions but their analysis requires the assumption that the death rates of all the species are negligible in comparison with the washout rate and hence can be ignored by means of lyapunov stability theory the global dynamics of the model for a large class of response functions are studied including both monotone and nonmonotone functions though it is not as general as the class studied by butler and wolkowicz and the results in

global asymptotic behavior of a chemostat model with delayed response in growth this paper studies the global asymptotic behavior of solutions of a competition model between n species in the chemostat the model incorporates time delays that allow the competing organisms to be stored the nutrient so that the free nutrient concentration does not reflect the nutrient available for growth and allow growth response functions to be nonmonotone by introducing three auxiliary functions and using a lemma it is shown that at most one competitor survives and the substrate and the surviving competitor if two exist approach limiting values

estimation of the average hazard ratio summary simple estimators of the average hazard ratio are obtained for the two sample problem with censored failure time data these estimators are compared with estimators arising out of the partial likelihood within the proportional hazards class efficiency results are found to be generally quite favourable provided the hazard ratio is not too large some comments are made on extensions to the k sample problem

relaxed lyapunov criteria for robust global stabilisation of non linear systems the notion of the restricted robust control lyapunov function rclf is introduced and is exploited for the design of robust feedback stabilisers for non linear systems particularly it is shown for systems with input constraints that restricted rclfs can be easily obtained while rclfs are not available moreover it is shown that the use of restricted rclfs usually results in different feedback designs from the ones obtained by the use of the standard rclf methodology using the restricted rclfs feedback design methodology a simple controller that guarantees robust global stabilisation of a perturbed chemostat model is provided

a practical divergence measure for survival distributions that can be estimated from kaplan meier curves this paper introduces a new simple divergence measure between two survival distributions for two groups of patients the divergence measure between their associated survival distributions is based on the integral of the absolute difference in probabilities that a patient from one group dies at time t and a patient from the other group survives beyond time t and vice versa in the case of non crossing hazard functions the divergence measure is closely linked to the harrell concordance index c the mann whitney test statistic and the area under a receiver operating characteristic curve the measure can be used in a dynamic way where the divergence between two survival distributions from time zero up to time t is calculated enabling real time monitoring of treatment differences the divergence can be found for theoretical survival distributions or can be estimated non parametrically from survival data using kaplan meier estimates of the survivor functions the estimator of the divergence is shown to be generally unbiased and approximately normally distributed for the case of proportional hazards the constituent parts of the divergence measure can be used to assess the proportional hazards assumption the use of the divergence measure is illustrated on the survival of pancreatic cancer patients copyright 2016 john wiley sons ltd

isomorphisms of ordered structures of abelian c subalgebras of c algebras abstract the aim of this note is to study the interplay between the jordan structure of c algebra and the structure of its abelian c subalgebras let abel a be a system of unital c subalgebras of a unital c algebra a ordered by set theoretic inclusion we show that any order isomorphism abel a abel b can be uniquely written in the form c c s a i c s a where is a partially linear jordan isomorphism between self adjoint parts of unital c algebras a and b as a corollary we obtain that for certain class of c algebras including von neumann algebras ordered structure of abelian subalgebras completely determines the jordan structure the results extend hitherto known results for abelian c algebras and may be relevant to foundations of quantum theory

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

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

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

a semiparametric change point regression model for longitudinal observations many longitudinal studies involve relating an outcome process to a set of possibly time varying covariates giving rise to the usual regression models for longitudinal data when the purpose of the study is to investigate the covariate effects when experimental environment undergoes abrupt changes or to locate the periods with different levels of covariate effects a simple and easy to interpret approach is to introduce change points in regression coefficients in this connection we propose a semiparametric change point regression model in which the error process stochastic component is nonparametric and the baseline mean function functional part is completely unspecified the observation times are allowed to be subject specific and the number locations and magnitudes of change points are unknown and need to be estimated we further develop an estimation procedure that combines the recent advance in semiparametric analysis based on counting process argument and multiple change points inference an

competition in the chemostat global asymptotic behavior of a model with delayed response in growth the asymptotic behavior of solutions of a model for competition between two microorganisms in the chemostat is studied the model incorporates time delays that allow for the cellular components of each competing species to be structured to include unassimilated or stored nutrient sufficient conditions on the parameters of the model that guarantee i competitive exclusion and ii persistence of at least one of the competing organisms are determined the results extend earlier results on the corresponding model for a single species and also extend these to include competition

a stochastic chemostat model with an inhibitor and noise independent of population sizes abstract in this paper a stochastic chemostat model with an inhibitor is considered here the inhibitor is input from an external source and two organisms in chemostat compete for a nutrient firstly we show that the system has a unique global positive solution secondly by constructing some suitable lyapunov functions we investigate that the average in time of the second moment of the solutions of the stochastic model is bounded for a relatively small noise that is the asymptotic behaviors of the stochastic system around the equilibrium points of the deterministic system are studied however the sufficient large noise can make the microorganisms become extinct with probability one although the solutions to the original deterministic model may be persistent finally the obtained analytical results are illustrated by computer simulations

competition in the chemostat a distributed delay model and its global asymptotic behavior in this paper we propose a two species competition model in a chemostat that uses a distributed delay to model the lag in the process of nutrient conversion and study the global asymptotic behavio

asymptotic behavior of a stochastic delayed chemostat model with nonmonotone uptake function abstract in this paper a stochastic delay differential equations chemostat model with nonmonotone uptake function is considered and the nutrient conversion process involves time delay first we verify that there is a unique global positive solution of the stochastic system second we find that the solutions of stochastic system will oscillate around the equilibria of the corresponding deterministic model moreover results show that time delay has critical effects on the extinction and persistence of the microorganism that is to say under small noise when the time delay is small microorganism is persistent when the time delay is large microorganism will be extinct in addition we can find by the computer simulation that large noise may lead to microorganism become extinct although microorganism is persistent in the deterministic systems when the time delay is small finally computer simulations are carried out to illustrate the obtained results and the existence of bistability is observed

a delayed chemostat model with general nonmonotone response functions and differential removal rates a chemostat model with general nonmonotone response functions is considered the nutrient conversion process involves time delay we show that under certain conditions when n species compete in the chemostat for a single resource that is allowed to be inhibitory at high concentrations the competitive exclusion principle holds in the case of insignificant death rates the result concerning the attractivity of the single species survival equilibrium already appears in the literature several times see h m el owaidy m ismail asymptotic behavior of the chemostat model with delayed response in growth chaos solitons fractals 13 2002 787 795 h m el owaidy a a moniem asymptotic behavior of a chemostat model with delayed response growth appl math comput 147 2004 147 161 s yuan m han z ma competition in the chemostat convergence of a model with delayed response in growth chaos solitons fractals 17 2003 659 667 however the proofs are all incorrect in this paper we provide a correct proof that also applies in the case of differential death rates in addition we provide a local stability analysis that includes sufficient conditions for the bistability of the single species survival equilibrium and the washout equilibrium thus showing the outcome can be initial condition dependent moreover we show that when the species specific death rates are included damped oscillations may occur even when there is no delay thus the species specific death rates might also account for the damped oscillations in transient behavior observed in experiments

dynamic cox modelling based on fractional polynomials time variations in gastric cancer prognosis the most popular model used for survival analysis is the proportional hazards regression model proposed by cox this is mainly due to its exceptional simplicity nevertheless the fundamental assumption of the cox model is the proportionality of the hazards for many applications however this assumption is doubtful proposals to extend the cox model for non proportional hazards to allow for dynamic effect structures usually either depend on prespecifications or require non standard estimation techniques and are thus not favoured in application moreover tests to verify the dynamic effect structures are not straightforward or lack omnibus power in this paper we propose a flexible method for modelling dynamic effects in survival data within the cox framework the method is based on fractional polynomials as introduced by royston and altman this allows for a transformation of the dynamic predictor which leads back to the conventional cox model and hence fitting is straightforward using standard estimation techniques in addition it offers the possibility of easily verifying the existence of time variation we describe a model selection algorithm which selects time varying effects only when evidence is given in the data we illustrate the properties of the approach in a simulation study and compare it with other methods in a survival study of gastric cancer patients we apply the approach to analyse dynamic structures in the effects of prognostic factors

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

joint estimation of time dependent and non linear effects of continuous covariates on survival in order to yield more flexible models the cox regression model lambda t x lambda 0 t exp betax has been generalized using different non parametric model estimation techniques one generalization is the relaxation of log linearity in x lambda t x lambda 0 t exp r x another is the relaxation of the proportional hazards assumption lambda t x lambda 0 t exp beta t x these generalizations are typically considered independently of each other we propose the product model lambda t x lambda 0 t exp beta t r x which allows for joint estimation of both effects and investigate its properties the functions describing the time dependent beta t and non linear r x effects are modelled simultaneously using regression splines and estimated by maximum partial likelihood likelihood ratio tests are proposed to compare alternative models simulations indicate that both the recovery of the shapes of the two functions and the size of the tests are reasonably accurate provided they are based on the correct model by contrast type i error rates may be highly inflated and the estimates considerably biased if the model is misspecified applications in cancer epidemiology illustrate how the product model may yield new insights about the role of prognostic factors

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

causal estimands and confidence intervals associated with wilcoxon mann whitney tests in randomized experiments although the p value from a wilcoxon mann whitney test is used often with randomized experiments it is rarely accompanied with a causal effect estimate and its confidence interval the natural parameter for the wilcoxon mann whitney test is the mann whitney parameter which measures the probability that a randomly selected individual in the treatment arm will have a larger response than a randomly selected individual in the control arm plus an adjustment for ties we show that the mann whitney parameter may be framed as a causal parameter and show that it is not equal to a closely related and nonidentifiable causal effect the probability that a randomly selected individual will have a larger response under treatment than under control plus an adjustment for ties we review the paradox first expressed by hand that the parameter may imply that the treatment is worse or better than control while the mann whitney parameter shows the opposite unlike the mann whitney parameter is nonidentifiable from a randomized experiment we review some nonparametric assumptions that rule out hand s paradox through bounds on and use bootstrap methods to make inferences on those bounds we explore the relationship of the proportional odds parameter to hand s paradox showing that the paradox may occur for proportional odds parameters between 1 9 and 9 thus large effects are needed to ensure that if treatment appears better by the mann whitney parameter then treatment improves responses in most individuals we demonstrate these issues using a vaccine trial

the mackey gleason problem let a be a von neumann algebra with no direct summand of type i 2 and let p a be its lattice of projections let x be a banach space let m p a x be a bounded function such that m p q m p m q whenever p and q are orthogonal projections the main theorem states that m has a unique extension to a bounded linear operator from a to x in particular each bounded complex valued finitely additive quantum measure on p a has a unique extension to a bounded linear functional on a

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

competitive exclusion in delayed chemostat models with differential removal rates competitive exclusion is frequently observed in the chemostat when multiple species compete exploitatively for a single growth limiting nutrient however rigorous mathematical proof for competitive exclusion is very limited especially for delayed chemostat models with differential removal rates in this paper by employing the method of liapunov functionals and placing two assumptions on the response functions we prove that competitive exclusion holds under a generic condition for chemostat models with differential removal rates and discrete also finitely distributed delays for delayed chemostat models with a large variety of widely used response functions including holling types i ii and iii and even some nonmonotone response functions our results show that the competition outcome is completely determined by the species break even concentrations it is the species with the lowest break even concentration that survives in the chemostat and drives other species to extinction

distribution free inference on contrasts of arbitrary summary measures of survival we present an approach for inference on contrasts of clinically meaningful functionals of a survivor distribution e g restricted mean quantiles that can avoid strong parametric or semiparametric assumptions on the underlying structure of the data in this multistage approach we first use an adaptive predictive model to estimate conditional survival distributions based on covariates we then estimate nonparametrically one or more functionals of survival from the covariate specific survival curves and evaluated contrasts of those functionals we find that the use of an adaptive nonparametric tree based predictive model leads to minimal loss in precision when semiparametric assumptions hold and provides marked improvement in accuracy when those assumptions are invalid therefore this work as a whole supports the use of survival summaries appropriate to a given medical application whether that be for example the median or 75th percentile in some settings or perhaps a restricted mean in others the approach is also compared with the mayo r score for primary biliary cirrhosis prognosis copyright 2012 john wiley sons ltd

evaluation of long term survival use of diagnostics and robust estimators with cox s proportional hazards model we consider methodological problems in evaluating long term survival in clinical trials in particular we examine the use of several methods that extend the basic cox regression analysis in the presence of a long term observation the proportional hazard ph assumption may easily be violated and a few long term survivors may have a large effect on parameter estimates we consider both model selection and robust estimation in a data set of 474 ovarian cancer patients enrolled in a clinical trial and followed for between 7 and 12 years after randomization two diagnostic plots for assessing goodness of fit are introduced one shows the variation in time of parameter estimates and is an alternative to ph checking based on time dependent covariates the other takes advantage of the martingale residual process in time to represent the lack of fit with a metric of the type observed minus expected number of events robust estimation is carried out by maximizing a weighted partial likelihood which downweights the contribution to estimation of influential observations this type of complementary analysis of long term results of clinical studies is useful in assessing the soundness of the conclusions on treatment effect in the example analysed here the difference in survival between treatments was mostly confined to those individuals who survived at least two years beyond randomization

spline based tests in survival analysis this paper examines a method for testing hypotheses on covariate effects in a proportional hazards model and also on how effects change over time in regression analysis of survival data the technique used is very general and can be applied to testing many other aspects of parametric and semiparametric models the basic idea is to formulate a flexible parametric alternative using fixed knot splines together with a penalty function that penalizes noisy alternatives more than smooth ones to focus the power of the tests toward smooth alternatives the test statistics are the analogs of ordinary likelihood based statistics only computed from a penalized likelihood formed by subtracting the penalty function from the ordinary log likelihood large sample approximations to the distributions are found when the number of knots is held fixed as the sample size increases numerical results suggest these approximations may be adequate with moderate sized samples

misspecified poisson regression models for large scale registry data inference for large n and small p poisson regression is an important tool in register based epidemiology where it is used to study the association between exposure variables and event rates in this paper we will discuss the situation with large n and small p where n is the sample size and p is the number of available covariates specifically we are concerned with modeling options when there are time varying covariates that can have time varying effects one problem is that tests of the proportional hazards assumption of no interactions between exposure and other observed variables or of other modeling assumptions have large power due to the large sample size and will often indicate statistical significance even for numerically small deviations that are unimportant for the subject matter another problem is that information on important confounders may be unavailable in practice this situation may lead to simple working models that are then likely misspecified to support and improve conclusions drawn from such models we discuss methods for sensitivity analysis for estimation of average exposure effects using aggregated data and a semi parametric bootstrap method to obtain robust standard errors the methods are illustrated using data from the danish national registries investigating the diabetes incidence for individuals treated with antipsychotics compared with the general unexposed population

analysis of survival data by the proportional odds model a model is presented for the analysis of lifetime data in which the rates of mortality for separate groups of patients converge with time a non parametric estimate is given for the survivor function the theoretical basis for the model assumes that prognostic factors have a multiplicative effect on the odds against survival beyond any given time the model is fitted to data using maximum likelihood estimation and an example of its use in the analysis of a lung cancer trial is given