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

recent progress on the poincare conjecture and the classification of 3 manifolds if m is a closed 3 manifold with trivial fundamental group then is m diffeomorphic to s the poincare conjecture is that the answer to this question is yes developing tools to attack this problem formed the basis for much of the work in 3 dimensional topology over the last century including for example the proof of dehn s lemma and the loop theorem and the study of surgery on knots and links in the 1980 s thurston developed another approach to 3 manifolds see 24 and 4 he considered 3 manifolds with riemannian metrics of constant negative curvature 1 these manifolds which are locally isometric to hyperbolic 3 space are called hyperbolic manifolds there are fairly obvious obstructions showing that not every 3 manifold can admit such a metric thurston formulated a general conjecture that roughly says that the obvious obstructions are the only ones should they vanish for a particular 3 manifold then that manifold admits such a metric his proof of various important special cases of this conjecture led him to formulate a more general conjecture about the existence of locally homogeneous metrics hyperbolic or otherwise for all manifolds this is called thurston s geometrization conjecture for 3 manifolds the statement of this conjecture is somewhat complicated so it is deferred until section 2 an important point is that thurston s geometrization conjecture includes the poincare conjecture as a very special case in addition thurston s conjecture has two advantages over the poincare conjecture it applies to all closed oriented 3 manifolds it posits a close relationship between topology and geometry in dimension three

on the distribution of the peak to average power ratio in ofdm signals the distribution of the peak to average power ratio papr in strictly band limited orthogonal frequency division multiplexing ofdm signals is studied assuming that the base band ofdm signal is characterized as a band limited complex gaussian process we first attempt to derive the exact distribution of the papr in the band limited ofdm signals since this distribution cannot be expressed in a closed form we further develop a simple closed form approximation based on the level crossing rate analysis comparisons of the proposed distributions with those obtained by computer simulations show good agreement and convergence with an increase in the number of subcarriers

g estimation and artificial censoring problems challenges and applications in principle g estimation is an attractive approach for dealing with confounding by variables affected by treatment it has rarely been applied for estimation of the effects of treatment on failure time outcomes part of this is due to artificial censoring an analytic device which considers some subjects who actually were observed to fail as if they were censored artificial censoring leads to a lack of smoothness in the estimating function which can pose problems in variance estimation and in optimization it also can lead to failure to have solutions to the usual estimating functions which then raises questions about the appropriate criteria for optimization to improve performance of the optimization procedures we consider approaches for reducing the amount of artificial censoring propose the substitution of smooth for indicator functions and propose the use of estimating functions scaled to a measure of the information in the data we evaluate performance of these approaches using simulation we also consider appropriate optimization criteria in the presence of information loss due to artificial censoring we motivate and illustrate our approaches using observational data on the effect of erythropoietin on mortality among subjects on hemodialysis

lp dual affine surface area abstract according to the notion of l p affine surface area by lutwak in this paper we introduce the concept of l p dual affine surface area further we establish the affine isoperimetric inequality and the blaschke santalo inequality for l p dual affine surface area besides the dual brunn minkowski inequality for l p dual affine surface area is presented

estimation of controlled direct effects on a dichotomous outcome using logistic structural direct effect models we consider the problem of assessing whether an exposure affects a dichotomous outcome other than by modifying a given mediator the standard approach logistic regression adjusting for both exposure and the mediator is known to be biased in the presence of confounders for the mediator outcome relationship because additional regression adjustment for such confounders is only justified when they are not affected by the exposure inverse probability weighting has been advocated but is not ideally tailored to mediators that are continuous or have strong measured predictors we overcome this limitation by developing inference for a novel class of causal models that are closely related to robins logistic structural direct effect models but do not inherit their difficulties of estimation we study identification and efficient estimation under the assumption that all confounders for the exposure outcome and mediator outcome relationships have been measured and find adequate performance in simulation studies we discuss extensions to case control studies and relevant implications for the generic problem of adjustment for time varying confounding

semiparametric modeling of repeated measurements under outcome dependent follow up in regression analysis of repeated measurements that are taken at subject specific times the availability of the outcome data may be related to the past outcome and to other measured variables that are not in the intended regression model in this paper we propose a natural extension of the semiparametric regression procedure of lin and ying j am stat assoc 2001 96 103 126 by building a class of inverse intensity rate ratio weighted estimators that accommodate such outcome dependent follow up the estimators have a closed form are radicaln consistent asymptotically normal and do not require estimation of any infinite dimensional parameters we give several simulations to demonstrate the estimator s performance and show a sensitivity study under follow up with various degrees of dependence on outcome related variables we illustrate our approach using data from a randomized health services research study with noncompliance to scheduled visits

collapsing manifolds obtained by kummer type constructions we construct f structures on a bott manifold and on some other manifolds obtained by kummer type constructions we also prove that if m e x where e is a fiber bundle with structure group g and a fiber admitting a g invariant metric of non negative sectional curvature and x admits an f structure with one trivial covering then one can construct on m a sequence of metrics with sectional curvature uniformly bounded from below and volume tending to zero i e vol k m 0 as a corollary we prove that all the elements in the spin cobordism ring can be represented by manifolds m with vol k m 0

estimation of controlled direct effects in time varying treatments using structural nested mean models application to a primary prevention trial for coronary events with pravastatin for the estimation of controlled direct effects i e direct effects controlling intermediates that are set at a fixed level for all members of the population without bias two fundamental assumptions must hold the absence of unmeasured confounding factors for treatment and outcome and for intermediate variables and outcome even if these assumptions hold one would nonetheless fail to estimate direct effects using standard methods for example stratification or regression modeling when the treatment influences confounding factors for such situations the sequential g estimation method for structural nested mean models has been developed for estimating controlled direct effects in point treatment situations in this study we demonstrate that this method can be applied to longitudinal data with time varying treatments and repeatedly measured intermediate variables we sequentially estimate the parameters in two structural nested mean models one for a repeatedly measured intermediate and the other one for direct effects of a time varying treatment the method was applied to data from a large primary prevention trial for coronary events in which pravastatin was used to lower the cholesterol levels in patients with moderate hypercholesterolemia

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

an approach to nonparametric regression for life history data using local linear fitting most hazard regression models in survival analysis specify a given functional form to describe the influence of the covariates on the hazard rate for instance cox s model assumes that the covariates act multiplicatively on the hazard rate and aalen s additive risk model stipulates that the covariates have a linear additive effect on the hazard rate in this paper we study a fully nonparametric model which makes no assumption on the association between the hazard rate and the covariates we propose a class of estimators for the conditional hazard function the conditional cumulative hazard function and the conditional survival function and study their large sample properties when the size of a data set is relatively large this fully nonparametric approach may provide more accurate information than that acquired from more restrictive models it may also be used to test whether a particular submodel gives a good fit to a given data set because our results are obtained under the multivariate counting process setting of aalen they apply to a number of models arising in survival analysis including various censoring and random truncation models our estimators are related to the conditional nelson aalen estimators proposed by beran for the random censorship model

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below ii we prove as our main theorem the finiteness of topological type of a complete open riemannian manifold m with a base point p m whose radial curvature at p is bounded from below by that of a non compact model surface of revolution m which admits a finite total curvature and has no pair of cut points in a sector here a sector is by definition a domain cut off by two meridians emanating from the base point p m notice that our model m does not always satisfy the diameter growth condition introduced by abresch and gromoll in order to prove the main theorem we need a new type of the toponogov comparison theorem as an application of the main theorem we present a partial answer to milnor s open conjecture on the fundamental group of complete open manifolds

predicting longitudinal trajectories of health probabilities with random effects multinomial logit regression researchers often encounter longitudinal health data characterized with three or more ordinal or nominal categories random effects multinomial logit models are generally applied to account for potential lack of independence inherent in such clustered data when parameter estimates are used to describe longitudinal processes however random effects both between and within individuals need to be retransformed for correctly predicting outcome probabilities this study attempts to go beyond existing work by developing a retransformation method that derives longitudinal growth trajectories of unbiased health probabilities variances of the predicted probabilities are estimated by using the delta method additionally the covariates regression coefficients on the multinomial logit function not substantively meaningful are transformed to the conditional effects on the predicted probabilities the empirical illustration uses the longitudinal data from the asset and health dynamics among the oldest old ahead our analysis compares three sets of the predicted probabilities of three health states at six time points obtained from respectively the retransformation method the blup and the fixed effects approach the results demonstrate that neglect of retransforming random errors in the random effects multinomial logit model results in severely biased longitudinal trajectories of health probabilities as well as overestimated effects of covariates on the probabilities

adjusting for time varying confounders in survival analysis using structural nested cumulative survival time models accounting for time varying confounding when assessing the causal effects of time varying exposures on survival time is challenging standard survival methods that incorporate time varying confounders as covariates generally yield biased effect estimates estimators using weighting by inverse probability of exposure can be unstable when confounders are highly predictive of exposure or the exposure is continuous structural nested accelerated failure time models aftms require artificial recensoring which can cause estimation difficulties here we introduce the structural nested cumulative survival time model sncstm this model assumes that intervening to set exposure at time t to zero has an additive effect on the subsequent conditional hazard given exposure and confounder histories when all subsequent exposures have already been set to zero we show how to fit it using standard software for generalized linear models and describe two more efficient double robust closed form estimators all three estimators avoid the artificial recensoring of aftms and the instability of estimators that use weighting by the inverse probability of exposure we examine the performance of our estimators using a simulation study and illustrate their use on data from the uk cystic fibrosis registry the sncstm is compared with a recently proposed structural nested cumulative failure time model and several advantages of the former are identified

estimating exposure effects by modelling the expectation of exposure conditional on confounders in order to estimate the causal effects of one or more exposures or treatments on an outcome of interest one has to account for the effect of confounding factors which both covary with the exposures or treatments and are independent predictors of the outcome in this paper we present regression methods which in contrast to standard methods adjust for the confounding effect of multiple continuous or discrete covariates by modelling the conditional expectation of the exposures or treatments given the confounders in the special case of a univariate dichotomous exposure or treatment this conditional expectation is identical to what rosenbaum and rubin have called the propensity score they have also proposed methods to estimate causal effects by modelling the propensity score our methods generalize those of rosenbaum and rubin in several ways first our approach straightforwardly allows for multivariate exposures or treatments each of which may be continuous ordinal or discrete second even in the case of a single dichotomous exposure our approach does not require subclassification or matching on the propensity score so that the potential for residual confounding i e bias due to incomplete matching is avoided third our approach allows a rather general formalization of the idea that it is better to use the estimated propensity score than the true propensity score even when the true score is known the additional power of our approach derives from the fact that we assume the causal effects of the exposures or treatments can be described by the parametric component of a semiparametric regression model to illustrate our methods we reanalyze the effect of current cigarette smoking on the level of forced expiratory volume in one second in a cohort of 2 713 adult white males we compare the results with those obtained using standard methods

some graphical displays and marginal regression analyses for recurrent failure times and time dependent covariates abstract recurrent event time data are common in medical research examples include infections in aids patients and seizures in epilepsy patients in this context as well as in the more usual context of a single failure time variable time dependent covariates are frequently of interest we suggest some rate functions that might be displayed when analyzing recurrent failure time data or when the effect of a categorical time dependent covariate is of interest estimators of these functions are provided along with two sample test statistics a new approach to regression modeling of these data is suggested and contrasted with existing methods our methods do not require that an explicit model be formulated for the probabilistic association between failure times within an individual this is in line with the currently popular generalized estimating equation approach to longitudinal data if the nature of such associations is known or is of particular interest then alternative methods may be appropriate

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

semiparametric theory for causal mediation analysis efficiency bounds multiple robustness and sensitivity analysis whilst estimation of the marginal total causal effect of a point exposure on an outcome is arguably the most common objective of experimental and observational studies in the health and social sciences in recent years investigators have also become increasingly interested in mediation analysis specifically upon evaluating the total effect of the exposure investigators routinely wish to make inferences about the direct or indirect pathways of the effect of the exposure not through or through a mediator variable that occurs subsequently to the exposure and prior to the outcome although powerful semiparametric methodologies have been developed to analyze observational studies that produce double robust and highly efficient estimates of the marginal total causal effect similar methods for mediation analysis are currently lacking thus this paper develops a general semiparametric framework for obtaining inferences about so called marginal natural direct and indirect causal effects while appropriately accounting for a large number of pre exposure confounding factors for the exposure and the mediator variables our analytic framework is particularly appealing because it gives new insights on issues of efficiency and robustness in the context of mediation analysis in particular we propose new multiply robust locally efficient estimators of the marginal natural indirect and direct causal effects and develop a novel double robust sensitivity analysis framework for the assumption of ignorability of the mediator variable

a boundary of the set of the riemannian manifolds with bounded curvatures and diameters on etudie la structure de la fermeture de l ensemble constitue par des classes d isometrie de varietes de riemann a courbures et diametres bornes

a radius sphere theorem the purpose of this paper is to present an optimal sphere theorem for metric spaces analogous to the celebrated rauch berger klingenberg sphere theorem and the diameter sphere theorem in riemannian geometry there has lately been considerable interest in studying spaces which are more singular than riemannian manifolds a natural reason for doing this is because gromov hausdorff limits of riemannian manifolds are almost never riemannian manifolds but usually only inner metric spaces with various nice properties the kind of spaces we wish to study here are the so called alexandrov spaces alexandrov spaces are finite dimensional inner metric spaces with a lower curvature bound in the distance comparison sense this definition might seem a little ambiguous since there are many ways in which one can define finite dimensionality and lower curvature bounds the foundational work by plaut in pi however shows that these different possibilities for definitions are equivalent the structure of alexandrov spaces was studied in bgp pi and p in particular if x is an alexandrov space and p e x then the space of directions ep at p is an alexandrov space of one less dimension and with curvature 1 furthermore a neighborhood of p in x is homeomorphic to the linear cone over one of the important implications of this is that the local structure of n dimensional alexandrov spaces is determined by the structure of n l dimensional alexandrov spaces with curvature 1 sphere theorems in this context seem to be particularly interesting for if one can give geometric characterizations of

lp mixed affine surface area abstract according to the three notions of mixed affine surface area l p affine surface area and l p mixed affine surface area proposed by lutwak in this article we give the concept of ith l p mixed affine surface area such that the first and second notions of lutwak are its special cases further some lutwak s results are extended associated with this concept besides applying this concept we establish an inequality for the volumes and dual quermassintegrals of a class of star bodies

pseudomartingale estimating equations for modulated renewal process models summary we adapt martingale estimating equations based on gap time information to a gen eral intensity model for a single realization of a modulated renewal process the consistency and asymptotic normality of the estimators is proved under ergodicity conditions previous work has considered either parametric likelihood analysis or semiparametric multiplicative models using partial likelihood the framework is generally applicable to semiparametric and paramet ric models including additive and multiplicative specifications and periodic models it facilitates a semiparametric extension of a popular parametric earthquake model simulations and empir ical analyses of taiwanese earthquake sequences illustrate the methodology s practical utility

partly functional temporal process regression with semiparametric profile estimating functions summary marginal mean models of temporal processes in event time data analysis are gaining more attention for their milder assumptions than the traditional intensity models recent work on fully functional temporal process regression tpr offers great flexibility by allowing all the regression coefficients to be nonparametrically time varying the existing esti mation procedure however prevents successive goodness of fit test for covariate coefficients in comparing a sequence of nested models this article proposes a partly functional tpr model in the line of marginal mean models some covari ate effects are time independent while others are completely unspecified in time this class of models is very rich includ ing the fully functional model and the semiparametric model as special cases to estimate the parameters we propose semiparametric profile estimating equations which are solved via an iterative algorithm starting at a consistent estimate from a fully functional model in the existing work no smoothing is needed in contrast to other varying coefficient meth ods the weak convergence of the resultant estimators are developed using the empirical process theory successive tests of time varying effects and backward model selection procedure can then be carried out the practical usefulness of the methodology is demonstrated through a simulation study and a real example of recurrent exacerbation among cystic fibrosis patients

augmented designs to assess principal strata direct effects summary many studies involving causal questions are often concerned with understanding the causal pathways by which a treatment affects an outcome thus the concept of direct versus indirect effects comes into play we tackle the problem of disentangling direct and indirect effects by investigating new augmented experimental designs where the treatment is randomized and the mediating variable is not forced but only randomly encouraged there are two key features of our framework we adopt a principal stratification approach and we mainly focus on principal strata effects avoiding involving a priori counterfactual outcomes using non parametric identification strategies we provide a set of assumptions which allow us to identify partially the causal estimands of interest the principal strata direct effects some examples are shown to illustrate our design and causal estimands of interest large sample bounds for the principal strata average direct effects are provided and a simple hypothetical example is used to show how our augmented design can be implemented and how the bounds can be calculated finally our augmented design is compared and contrasted with a standard randomized design

identification and efficient estimation of the natural direct effect among the untreated the natural direct effect nde or the effect of an exposure on an outcome if an intermediate variable was set to the level it would have been in the absence of the exposure is often of interest to investigators in general the statistical parameter associated with the nde is difficult to estimate in the non parametric model particularly when the intermediate variable is continuous or high dimensional in this paper we introduce a new causal parameter called the natural direct effect among the untreated discus identifiability assumptions propose a sensitivity analysis for some of the assumptions and show that this new parameter is equivalent to the nde in a randomized controlled trial we also present a targeted minimum loss estimator tmle a locally efficient double robust substitution estimator for the statistical parameter associated with this causal parameter the tmle can be applied to problems with continuous and high dimensional intermediate variables and can be used to estimate the nde in a randomized controlled trial with such data additionally we define and discuss the estimation of three related causal parameters the natural direct effect among the treated the indirect effect among the untreated and the indirect effect among the treated

semiparametric transformation models for causal inference in time to event studies with all or nothing compliance we consider causal inference in randomized survival studies with right censored outcomes and all or nothing compliance using semiparametric transformation models to estimate the distribution of survival times in treatment and control groups conditional on covariates and latent compliance type estimands depending on these distributions for example the complier average causal effect cace the complier effect on survival beyond time t and the complier quantile effect are then considered maximum likelihood is used to estimate the parameters of the transformation models using a specially designed expectation maximization em algorithm to overcome the computational difficulties created by the mixture structure of the problem and the infinite dimensional parameter in the transformation models the estimators are shown to be consistent asymptotically normal and semiparametrically efficient inferential procedures for the causal parameters are developed a simulation study is conducted to evaluate the finite sample performance of the estimated causal parameters we also apply our methodology to a randomized study conducted by the health insurance plan of greater new york to assess the reduction in breast cancer mortality due to screening

transforming cumulative hazard estimates summarytime to event outcomes are often evaluated on the hazard scale but interpreting hazards may be difficult recently in the causal inference literature concerns have been raised that hazards actually have a built in selection bias that prevents simple causal interpretations this is a problem even in randomized controlled trials where hazard ratios have become a standard measure of treatment effects modelling on the hazard scale is nevertheless convenient for example to adjust for covariates using hazards for intermediate calculations may therefore be desirable in this paper we present a generic method for transforming hazard estimates consistently to other scales at which these built in selection biases are avoided the method is based on differential equations and generalizes a well known relation between the nelson aalen and kaplan meier estimators using the martingale central limit theorem we show that covariances can be estimated consistently for a large class of estimators thus allowing for rapid calculation of confidence intervals hence given cumulative hazard estimates based on for example aalen s additive hazard model we can obtain many other parameters without much more effort we give several examples and the associated estimators coverage and convergence speed are explored via simulations and the results suggest that reliable estimates can be obtained in real life scenarios

a general dynamical statistical model with causal interpretation summary we develop a general dynamical model as a framework for causal interpretation we first state a criterion of local independence in terms of measurability of processes that are involved in the doob meyer decomposition of stochastic processes then we define direct and indirect influence we propose a definition of causal influence using the concepts of a physical system this framework makes it possible to link descriptive and explicative statistical models and encompasses quantitative processes and events one of the features of the paper is the clear distinction between the model for the system and the model for the observation we give a dynamical representation of a conventional joint model for human immunodeficiency virus load and cd4 cell counts we show its inadequacy to capture causal influences whereas in contrast known mechanisms of infection by the human immunodeficiency virus can be expressed directly through a system of differential equations

counterfactual links to the proportion of treatment effect explained by a surrogate marker in a randomized clinical trial a statistic that measures the proportion of treatment effect on the primary clinical outcome that is explained by the treatment effect on a surrogate outcome is a useful concept we investigate whether a statistic proposed to estimate this proportion can be given a causal interpretation as defined by models of counterfactual variables for the situation of binary surrogate and outcome variables two counterfactual models are considered both of which include the concept of the proportion of the treatment effect which acts through the surrogate in general the statistic does not equal either of the two proportions from the counterfactual models and can be substantially different conditions are given for which the statistic does equal the counterfactual model proportions a randomized clinical trial with potential surrogate endpoints is undertaken in a scientific context this context will naturally place constraints on the parameters of the counterfactual model we conducted a simulation experiment to investigate what impact these constraints had on the relationship between the proportion explained pe statistic and the counterfactual model proportions we found that observable constraints had very little impact on the agreement between the statistic and the counterfactual model proportions whereas unobservable constraints could lead to more agreement

nonparametric k sample tests with panel count data we study the nonparametric k sample test problem with panel count data the asymptotic normality of a smooth functional of the nonparametric maximum pseudolikelihood estimator wellner zhang 2000 is established under some mild conditions we construct a class of easy to implement nonparametric tests for comparing mean functions of k populations based on this asymptotic normality we conduct various simulations to validate and compare the tests the simulations show that the tests perform quite well and generally have good power to detect differences among the mean functions the method is illustrated with a real life example

armitage lecture 2010 understanding treatment effects the value of integrating longitudinal data and survival analysis there is a single minded focus on events in survival analysis and we often ignore longitudinal data that are collected together with the event data this is due to a lack of methodology but also a result of the artificial distinction between survival and longitudinal data analyses understanding the dynamics of such processes is important but has been hampered by a lack of appreciation of the difference between confirmatory and exploratory causal inferences the latter represents an attempt at elucidating mechanisms by applying mediation analysis to statistical data and will usually be of a more tentative character than a confirmatory analysis the concept of local independence and the associated graphs are useful this is related to granger causality an important method from econometrics that is generally undervalued by statisticians this causality concept is different from the counterfactual one since it lacks lacks the intervention aspect the notion that one can intervene at will in naturally occurring processes which seems to underly much of modern causal inference is problematic when studying mediation and mechanisms it is natural to assume a stochastic process point of view when analyzing dynamic relationships we present some examples to illustrate this it is not clear how survival analysis must be developed to handle the complex life history data that are increasingly being collected today we give some suggestions copyright 2012 john wiley sons ltd

estimation of direct effects for survival data by using the aalen additive hazards model summary we extend the definition of the controlled direct effect of a point exposure on a survival outcome other than through some given time fixed intermediate variable to the additive hazard scale we propose two stage estimators for this effect when the exposure is dichotomous and randomly assigned and when the association between the intermediate variable and the survival outcome is confounded only by measured factors which may themselves be affected by the exposure the first stage of the estimation procedure involves assessing the effect of the intermediate variable on the survival outcome via aalen s additive regression for the event time given exposure intermediate variable and confounders the second stage involves applying aalen s additive model given the exposure alone to a modified stochastic process i e a modification of the observed counting process based on the first stage estimates we give the large sample properties of the estimator proposed and investigate its small sample properties by monte carlo simulation a real data example is provided for illustration

towards a general asymptotic theory for cox model with staggered entry a general asymptotic theory is established for the two parameter cox score process with staggered entry data it extends in several directions the existing theory developed by sellke and siegmund slud and gu and lai an essential tool employed here is a modern empirical process theory as elucidated in a recent monograph by pollard

inverse odds ratio weighted estimation for causal mediation analysis an important scientific goal of studies in the health and social sciences is increasingly to determine to what extent the total effect of a point exposure is mediated by an intermediate variable on the causal pathway between the exposure and the outcome a causal framework has recently been proposed for mediation analysis which gives rise to new definitions formal identification results and novel estimators of direct and indirect effects in the present paper the author describes a new inverse odds ratio weighted iorw approach to estimate so called natural direct and indirect effects the approach which uses as a weight the inverse of an estimate of the odds ratio function relating the exposure and the mediator is universal in that it can be used to decompose total effects in a number of regression models commonly used in practice specifically the approach may be used for effect decomposition in generalized linear models with a nonlinear link function and in a number of other commonly used models such as the cox proportional hazards regression for a survival outcome the approach is simple and can be implemented in standard software provided a weight can be specified for each observation an additional advantage of the method is that it easily incorporates multiple mediators of a categorical discrete or continuous nature

restrictions on collapsing with a lower sectional curvature bound we obtain new topological information about the local structure of collapsing under a lower sectional curvature bound as an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every alexandrov space can be obtained as a limit of a sequence of riemannian manifolds with sectional curvature bounded from below

semiparametric regression analysis of longitudinal data with informative observation times statistical analysis of longitudinal data has been discussed by many authors and a number of methods have been proposed most of the research have focused on situations where observation times are independent of or carry no information about the response variable and therefore rely on conditional inference procedures given the observation times this article considers a different situation where the independence assumption may not hold that is the observation times may carry information about the response variable for inference estimating equation approaches are proposed and both large sample and final sample properties of the proposed methods are established the methodology is applied to a bladder cancer study that motivated this investigation

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

collapsing sequences of solutions to the ricci flow on 3 manifolds with almost nonnegative curvature we study sequences of 3 dimensional solutions to the ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity such sequences may arise from the limits of dilations about singularities of type iib in particular we study the case when the sequence collapses which may occur when dilating about infinite time singularities in this case we classify the possible gromov hausdorff limits and construct 2 dimensional virtual limits the virtual limits are constructed using fukaya theory of the limits of local covers we then show that the virtual limit arising from appropriate dilations of a type iib singularity is always hamilton s cigar soliton solution

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

some simple robust methods for the analysis of recurrent events nelson discussed a method of estimating the cumulative mean function for identically distributed processes of recurrent events we show that a similar approach can be used with more general models including regression the key idea is to use point estimates based on poisson models and to develop robust variance estimates that are valid more generally the methods are illustrated on reliability and warranty data

sensitivity analysis for unobserved confounding of direct and indirect effects using uncertainty intervals to estimate direct and indirect effects of an exposure on an outcome from observed data strong assumptions about unconfoundedness are required since these assumptions cannot be tested using the o

volume collapsed three manifolds with a lower curvature bound in this paper we determine the topology of three dimensional complete orientable riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small

methods for dealing with time dependent confounding longitudinal studies where data are repeatedly collected on subjects over a period are common in medical research when estimating the effect of a time varying treatment or exposure on an outcome of interest measured at a later time standard methods fail to give consistent estimators in the presence of time varying confounders if those confounders are themselves affected by the treatment robins and colleagues have proposed several alternative methods that provided certain assumptions hold avoid the problems associated with standard approaches they include the g computation formula inverse probability weighted estimation of marginal structural models and g estimation of structural nested models in this tutorial we give a description of each of these methods exploring the links and differences between them and the reasons for choosing one over the others in different settings

estimation of controlled direct effects when regression models adjust for mediators on the causal path from exposure to outcome the regression coefficient of exposure is commonly viewed as a measure of the direct exposure effect this interpretation can be misleading even with a randomly assigned exposure this is because adjustment for post exposure measurements introduces bias whenever their association with the outcome is confounded by more than just the exposure by the same token adjustment for such confounders stays problematic when these are themselves affected by the exposure robins accommodated this by introducing linear structural nested direct effect models with direct effect parameters that can be estimated by using inverse probability weighting by a conditional distribution of the mediator the resulting estimators are consistent but inefficient and can be extremely unstable when the mediator is absolutely continuous we develop direct effect estimators which are not only more efficient but also consistent under a less demanding model for a conditional expectation of the outcome we find that the one estimator which avoids inverse probability weighting altogether performs best this estimator is intuitive computationally straightforward and as demonstrated by simulation competes extremely well with ordinary least squares estimators in settings where standard regression is valid

papr analysis as a ratio of two random variables application to multicarrier systems with low subcarriers number the distribution of the peak to average power ratio papr of multicarrier systems is studied in the context of signals featuring low subcarriers number considering the mean power of the signal as a random variable a general cumulative distribution function of the papr is suggested we show that the new formulation is valid not only for low subcarriers number but also for larger subcarriers number thus an asymptotic analysis using a large subcarriers number proves a perfect concordance between the suggested approach and the state of the art derivations only valid for large subcarriers number all the theoretical developments are supported through simulations and we discuss the case of oversampled signals as well

mediation analysis with time varying exposures and mediators in this paper we consider causal mediation analysis when exposures and mediators vary over time we give non parametric identification results discuss parametric implementation and also provide a weighting approach to direct and indirect effects based on combining the results of two marginal structural models we also discuss how our results give rise to a causal interpretation of the effect estimates produced from longitudinal structural equation models when there are time varying confounders affected by prior exposure and mediator natural direct and indirect effects are not identified however we define a randomized interventional analogue of natural direct and indirect effects that are identified in this setting the formula that identifies these effects we refer to as the mediational g formula when there is no mediation the mediational g formula reduces to robins regular g formula for longitudinal data when there are no time varying confounders affected by prior exposure and mediator values then the mediational g formula reduces to a longitudinal version of pearl s mediation formula however the mediational g formula itself can accommodate both mediation and time varying confounders and constitutes a general approach to mediation analysis with time varying exposures and mediators

structural nested cumulative failure time models to estimate the effects of interventions in the presence of time varying confounders affected by prior treatment standard statistical methods for failure time analysis may be biased methods that correctly adjust for this type of covariate include the parametric g formula inverse probability weighted estimation of marginal structural cox proportional hazards models and g estimation of structural nested accelerated failure time models in this article we propose a novel method to estimate the causal effect of a time dependent treatment on failure in the presence of informative right censoring and time dependent confounders that may be affected by past treatment g estimation of structural nested cumulative failure time models sncftms an sncftm considers the conditional effect of a final treatment at time m on the outcome at each later time k by modeling the ratio of two counterfactual cumulative risks at time k under treatment regimes that differ only at time m inverse probability weights are used to adjust for informative censoring we a

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

on doubly robust estimation of the hazard difference the estimation of conditional treatment effects in an observational study with a survival outcome typically involves fitting a hazards regression model adjusted for a high dimensional covariate standard estimation of the treatment effect is then not entirely satisfactory as the misspecification of the effect of this covariate may induce a large bias such misspecification is a particular concern when inferring the hazard difference because it is difficult to postulate additive hazards models that guarantee non negative hazards over the entire observed covariate range we therefore consider a novel class of semiparametric additive hazards models which leave the effects of covariates unspecified the efficient score under this model is derived we then propose two different estimation approaches for the hazard difference and hence also the relative chance of survival both of which yield estimators that are doubly robust the approaches are illustrated using simulation studies and data on right heart catheterization and mortality from the support study

a semiparametric recurrent events model with time varying coefficients we consider a recurrent events model with time varying coefficients motivated by two clinical applications a random effects gaussian frailty model is used to describe the intensity of recurrent events the model can accommodate both time varying and time constant coefficients the penalized spline method is used to estimate the time varying coefficients laplace approximation is used to evaluate the penalized likelihood without a closed form the smoothing parameters are estimated in a similar way to variance components we conduct simulations to evaluate the performance of the estimates for both time varying and time independent coefficients we apply this method to analyze two data sets a stroke study and a child wheeze study

bias reduced doubly robust estimation over the past decade doubly robust estimators have been proposed for a variety of target parameters in causal inference and missing data models these are asymptotically unbiased when at least one of two nuisance working models is correctly specified regardless of which while their asymptotic distribution is not affected by the choice of root n consistent estimators of the nuisance parameters indexing these working models when all working models are correctly specified this choice of estimators can have a dramatic impact under misspecification of at least one working model in this article we will therefore propose a simple and generic estimation principle for the nuisance parameters indexing each of the working models which is designed to improve the performance of the doubly robust estimator of interest relative to the default use of maximum likelihood estimators for the nuisance parameters the proposed approach locally minimizes the squared first order asymptotic bias of the doubly robust estim

the generalized integro exponential function the generalized integro exponential function is defined in terms of the exponential integral incomplete gamma function and its derivatives with respect to order a compendium of analytic results is given in one section rational minimax approximations sufficient to permit the computation of the first six first order functions are reported in another section

vanishing of cohomology groups and large eigenvalues of the laplacian on p forms for collapsing of closed riemannian manifolds the first positive eigenvalue of the laplacian on p forms may or may not tend to infinity in special cases we show that the existence of the first positive eigenvalue tending to infinity is related to vanishing of cohomology groups of generic fibers

collapsing and pinching under a lower curvature bound in this paper we are concerned with collapsing phenomena and pinching problems of riemannian manifolds whose sectional curvatures are uniformly bounded from below for a positive integer n and for d 0 let x be the set of compact riemannian n manifolds m with sectional curvatures km 1 and diameters diam m d by the weak compactness theorem of gromov glp x is relatively compact in the set of compact metric spaces with respect to the hausdorff distance furthermore gromov proved in g2 that the sum of betti numbers of any element in x is uniformly bounded in terms of the given constants these results suggest that x is an object of study on which we could develop some geometry and topology in fact when volumes are uniformly bounded away from zero the following results are known let v be the subset of x with volume vol m v grove and petersen gp1 proved that the set of homotopy types of elements in 4 v is finite related results are in y2 for pinching problems otsu shiohama and the author osy y3 obtained some differential sphere theorems in the class x v according to a recent paper gpw if n 3 4 then v contains at most finitely many diffeomorphism types let mi i 1 2 be a convergent sequence in a and x be the limit of them we say that mi collapses to x if the hausdorff dimension of x is less than n by the volume condition no collapsing occurs in 4 v from this point of view it is quite natural to ask what collapsing phenomena occur and what types of pinching theorems hold in the class x in this situation it seems difficult to determine the singularities of x some examples are given in section 1 in this paper we study collapsing phenomena in the case when x is a riemannian manifold and establish a pinching theorem for a riemannian manifold m we denote by inj m the injectivity radius of m when n is written in place of x and the metric of n is normalized our

a distributional approach for causal inference using propensity scores drawing inferences about the effects of treatments and actions is a common challenge in economics epidemiology and other fields we adopt rubin s potential outcomes framework for causal inference and propose two methods serving complementary purposes one can be used to estimate average causal effects assuming no confounding given measured covariates the other can be used to assess how the estimates might change under various departures from no confounding both methods are developed from a nonparametric likelihood perspective the propensity score plays a central role and is estimated through a parametric model under the assumption of no confounding the joint distribution of covariates and each potential outcome is estimated as a weighted empirical distribution expectations from the joint distribution are estimated as weighted averages or equivalently to first order regression estimates the likelihood estimator is at least as efficient and the regression estimator is at least as efficient and r

marginal regression models for multivariate failure time data abstract in this article we propose a general cox type regression model to formulate the marginal distributions of multivariate failure time data this model has a nested structure in that it allows different baseline hazard functions among distinct failure types and imposes a common baseline hazard function on the failure times of the same type we prove that the maximum quasi partial likelihood estimator for the vector of regression parameters under the independence working assumption is consistent and asymptotically normal with a covariance matrix for which a consistent estimator is provided furthermore we establish the uniform consistency and joint weak convergence of the aalen breslow type estimators for the cumulative baseline hazard functions and develop a resampling technique to approximate the joint distribution of these processes which enables one to make simultaneous inference about the survival functions over the time axis and across failure types finally we assess the small sample pro

nilpotent structures and invariant metrics on collapsed manifolds let mn be a complete riemannian manifold of bounded curvature say iki 0 we put mn w n c u fn c where 1n e consists of those points at which the injectivity radius of the exponential map is e the complementary set n c is called the e collapsed part of mn if x e 4 n r e then the metric ball bx r is quasi isometric with small distortion to the flat ball b0 r in the euclidean space rn after slightly

estimation of the time dependent accelerated failure time model in the presence of confounding factors summary cox oakes 1984 p 66 introduced the strong version of the accelerated failure time model with time dependent exposures we provide conditions under which this model could be used to estimate from observational data the causal effect of a time varying exposure or treatment on time to an event of interest in the presence of time dependent confounding variables we propose a class of semiparametric tests and estimators for the model parameters this class contains an estimator that is semipara metric efficient in the sense of begun et al 1983 the purpose of this paper is first to provide conditions under which it would be possible to estimate from observational data the causal effect of a time dependent treatment or exposure on time to an event of interest in the presence of time dependent confounding covariates and then to construct estimators for the treatment effect under these condi tions our approach will be based on the semiparametric estimation of the parameters of the strong version of the accelerated failure time model with time dependent exposures or treatments introduced by cox oakes 1984 p 66 the usual approach to the estimation of the effect of a time varying treatment on survival is to model the hazard of failure at t as a function of past treatment history using a time dependent proportional hazards model we show that the usual approach may be biased whether or not one further adjusts for past confounder history in the analysis when a there exists a time dependent risk factor for or predictor of the event of interest that also predicts subsequent treatment and b past treatment history predicts subsequent risk factor level the following four examples demonstrate conditions a and b will often be true in an observational study in which there is treatment by indication the drug zidovudine formerly azt used in the treatment of the acquired immunodeficiency syndrome is a direct red blood cell toxin that is contra indicated in anaemic subjects i e subjects with depressed red cell counts since the toxic effects of zidovudine can worsen the anaemia further anaemic patients are at increased risk of death thus in an observational study of the effect of zidovudine on survival of patients with the acquired immunodeficiency syndrome anaemia is both a risk factor for death

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

semiparametric regression for the mean and rate functions of recurrent events the counting process with the cox type intensity function has been commonly used to analyse recurrent event data this model essentially assumes that the underlying counting process is a time transformed poisson process and that the covariates have multiplicative effects on the mean and rate function of the counting process recently pepe and cai and lawless and co workers have proposed semiparametric procedures for making inferences about the mean and rate function of the counting process without the poisson type assumption in this paper we provide a rigorous justification of such robust procedures through modern empirical process theory furthermore we present an approach to constructing simultaneous confidence bands for the mean function and describe a class of graphical and numerical techniques for checking the adequacy of the fitted mean rate model the advantages of the robust procedures are demonstrated through simulation studies an illustration with multiple infection data taken from a clinical study on chronic granulomatous disease is also provided

multiple robustness in factorized likelihood models we consider inference under a nonparametric or semiparametric model with likelihood that factorizes as the product of two or more variation independent factors we are interested in a finite dimensional parameter that depends on only one of the likelihood factors and whose estimation requires the auxiliary estimation of one or several nuisance functions we investigate general structures conducive to the construction of so called multiply robust estimating functions whose computation requires postulating several dimension reducing models but which have mean zero at the true parameter value provided one of these models is correct