bayesian sensitivity analysis for unmeasured confounding in observational studies we consider bayesian sensitivity analysis for unmeasured confounding in observational studies where the association between a binary exposure binary response measured confounders and a single binary unmeasured confounder can be formulated using logistic regression models a model for unmeasured confounding is presented along with a family of prior distributions that model beliefs about a possible unknown unmeasured confounder simulation from the posterior distribution is accomplished using markov chain monte carlo because the model for unmeasured confounding is not identifiable standard large sample theory for bayesian analysis is not applicable consequently the impact of different choices of prior distributions on the coverage probability of credible intervals is unknown using simulations we investigate the coverage probability when averaged with respect to various distributions over the parameter space the results indicate that credible intervals will have approximately nominal coverage probability on average when the prior distribution used for sensitivity analysis approximates the sampling distribution of model parameters in a hypothetical sequence of observational studies we motivate the method in a study of the effectiveness of beta blocker therapy for treatment of heart failure

a global sensitivity test for evaluating statistical hypotheses with nonidentifiable models we consider the problem of evaluating a statistical hypothesis when some model characteristics are nonidentifiable from observed data such a scenario is common in meta analysis for assessing publication bias and in longitudinal studies for evaluating a covariate effect when dropouts are likely to be nonignorable one possible approach to this problem is to fix a minimal set of sensitivity parameters conditional upon which hypothesized parameters are identifiable here we extend this idea and show how to evaluate the hypothesis of interest using an infimum statistic over the whole support of the sensitivity parameter we characterize the limiting distribution of the statistic as a process in the sensitivity parameter which involves a careful theoretical analysis of its behavior under model misspecification in practice we suggest a nonparametric bootstrap procedure to implement this infimum test as well as to construct confidence bands for simultaneous pointwise tests across all values of the sensitivity parameter adjusting for multiple testing the methodology s practical utility is illustrated in an analysis of a longitudinal psychiatric study

spectral gap for the zero range process with constant rate we solve an open problem concerning the relaxation time inverse spectral gap of the zero range process in z d lz d with constant rate proving a tight upper bound of o 1 2 l 2 where p is the density of particles

sensitivity analysis for certain permutation inferences in matched observational studies summary in observational studies treatments are not randomly assigned to experimental units so that randomization tests and their associated interval estimates are not generally applicable in an effort to compensate for the lack of randomization treated and control units are often matched on the basis of observed covariates however the possibility remains of bias due to residual imbalances in unobserved covariates a general though simple method is proposed for displaying the sensitivity of permutation inferences to a range of assumptions about unobserved covariates in matched observational studies the sensitivity analysis is applicable to wilcoxon s signed rank test to the mcnemar cox test for paired binary responses and to some matching problems with a variable number of controls

a stochastic burgers equation from a class of microscopic interactions we consider a class of nearest neighbor weakly asymmetric mass conservative particle systems evolving on mathbb z which includes zero range and types of exclusion processes starting from a perturbation of a stationary state when the weak asymmetry is of order o n gamma for 1 2 gamma leq1 we show that the scaling limit of the fluctuation field as seen across process characteristics is a generalized ornstein uhlenbeck process however at the critical weak asymmetry when gamma 1 2 we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic burgers equation derived from taking the gradient of the kpz equation the proofs make use of a sharp boltzmann gibbs estimate which improves on earlier bounds

randomization inference and sensitivity analysis for composite null hypotheses with binary outcomes in matched observational studies abstractwe present methods for conducting hypothesis testing and sensitivity analyses for composite null hypotheses in matched observational studies when outcomes are binary causal estimands discussed include the causal risk difference causal risk ratio and the effect ratio we show that inference under the assumption of no unmeasured confounding can be performed by solving an integer linear program while inference allowing for unmeasured confounding of a given strength requires solving an integer quadratic program through simulation studies and data examples we demonstrate that our formulation allows these problems to be solved in an expedient manner even for large datasets and for large strata we further exhibit that through our formulation one can assess the impact of various assumptions about the potential outcomes on the performed inference r scripts are provided that implement our methods supplementary materials for this article are available online

normal structure and orbital fixed point conditions abstract in some sense all of the results discussed in this paper are an outgrowth of the concept of normal structure introduced by brodskii and milman in 1948 and following related idea introduced by belluce and kirk in 1969 a mapping t of a metric space k into itself is said to have diminishing orbital diameters if given any x k it is the case that lim n d i a m o t n x d i a m o x whenever d i a m o x 0 where o x x t x t 2 x among other things belluce and kirk observed that the assumption of diminishing orbital diameters on a mapping t k k is sufficient to assure that t has a fixed point if t is nonexpansive and if k is a weakly compact convex subset of a banach space it was later shown by kirk that the convexity assumption on k can be dropped the purpose of this discussion is to illustrate how the above ideas are related to a number of similar ones that have been introduced more recently among these is the following concept introduced recently by amini harandi et al a mapping t k k of a subset k of a banach space is said to be nonexpansive wrt orbits if for all x y k t x t y r x o y where r x o y sup x u u o y it is shown that if k is a weakly compact subset of a banach space and if t k k is nonexpansive wrt orbits then the following condition always assures the existence of a fixed point for t for each x k with t x x inf m n lim sup n t m x t n x d i a m o x some open questions are also discussed

sensitivity analyses for unmeasured confounding assuming a marginal structural model for repeated measures robins introduced marginal structural models msms and inverse probability of treatment weighted iptw estimators for the causal effect of a time varying treatment on the mean of repeated measures we investigate the sensitivity of iptw estimators to unmeasured confounding we examine a new framework for sensitivity analyses based on a nonidentifiable model that quantifies unmeasured confounding in terms of a sensitivity parameter and a user specified function we present augmented iptw estimators of msm parameters and prove their consistency for the causal effect of an msm assuming a correct confounding bias function for unmeasured confounding we apply the methods to assess sensitivity of the analysis of hernan et al who used an msm to estimate the causal effect of zidovudine therapy on repeated cd4 counts among hiv infected men in the multicenter aids cohort study under the assumption of no unmeasured confounders a 95 per cent confidence interval for the treatment effect includes zero we show that under the assumption of a moderate amount of unmeasured confounding a 95 per cent confidence interval for the treatment effect no longer includes zero thus the analysis of hernan et al is somewhat sensitive to unmeasured confounding we hope that our research will encourage and facilitate analyses of sensitivity to unmeasured confounding in other applications

assessing the sensitivity of regression results to unmeasured confounders in observational studies this paper presents a general approach for assessing the sensitivity of the point and interval estimates of the primary exposure effect in an observational study to the residual confounding effects of unmeasured variable after adjusting for measured covariates the proposed method assumes that the true exposure effect can be represented in a regression model that includes the exposure indicator as well as the measured and unmeasured confounders one can use the corresponding reduced model that omits the unmeasured confounder to make statistical inferences about the true exposure effect by specifying the distributions of the unmeasured confounder in the exposed and unexposed groups along with the effects of the unmeasured confounder on the outcome variable under certain conditions there exists a simple algebraic relationship between the true exposure effect in the full model and the apparent exposure effect in the reduced model one can then estimate the true exposure effect by making a simple adjustment to the point and interval estimates of the apparent exposure effect obtained from standard software or published reports the proposed method handles both binary response and censored survival time data accommodates any study design and allows the unmeasured confounder to be discrete or normally distributed we describe applications on two major medical studies

variance estimation when using inverse probability of treatment weighting iptw with survival analysis propensity score methods are used to reduce the effects of observed confounding when using observational data to estimate the effects of treatments or exposures a popular method of using the propensity score is inverse probability of treatment weighting iptw when using this method a weight is calculated for each subject that is equal to the inverse of the probability of receiving the treatment that was actually received these weights are then incorporated into the analyses to minimize the effects of observed confounding previous research has found that these methods result in unbiased estimation when estimating the effect of treatment on survival outcomes however conventional methods of variance estimation were shown to result in biased estimates of standard error in this study we conducted an extensive set of monte carlo simulations to examine different methods of variance estimation when using a weighted cox proportional hazards model to estimate the effect of treatment we considered three variance estimation methods i a naive model based variance estimator ii a robust sandwich type variance estimator and iii a bootstrap variance estimator we considered estimation of both the average treatment effect and the average treatment effect in the treated we found that the use of a bootstrap estimator resulted in approximately correct estimates of standard errors and confidence intervals with the correct coverage rates the other estimators resulted in biased estimates of standard errors and confidence intervals with incorrect coverage rates our simulations were informed by a case study examining the effect of statin prescribing on mortality 2016 the authors statistics in medicine published by john wiley sons ltd

spectral gap estimates for interacting particle systems via a bochner type identity abstract we develop a general technique based on a bochner type identity to estimate spectral gaps of a class of markov operator we apply this technique to various interacting particle systems in particular we give a simple and short proof of the diffusive scaling of the spectral gap of the kawasaki model at high temperature similar results are derived for kawasaki type dynamics in the lattice without exclusion and in the continuum new estimates for glauber type dynamics are also obtained

balancing covariates via propensity score weighting abstractcovariate balance is crucial for unconfounded descriptive or causal comparisons however lack of balance is common in observational studies this article considers weighting strategies for balancing covariates we define a general class of weights the balancing weights that balance the weighted distributions of the covariates between treatment groups these weights incorporate the propensity score to weight each group to an analyst selected target population this class unifies existing weighting methods including commonly used weights such as inverse probability weights as special cases general large sample results on nonparametric estimation based on these weights are derived we further propose a new weighting scheme the overlap weights in which each unit s weight is proportional to the probability of that unit being assigned to the opposite group the overlap weights are bounded and minimize the asymptotic variance of the weighted average treatment effect among the class of balancing wei

stratification and weighting via the propensity score in estimation of causal treatment effects a comparative study estimation of treatment effects with causal interpretation from observational data is complicated because exposure to treatment may be confounded with subject characteristics the propensity score the probability of treatment exposure conditional on covariates is the basis for two approaches to adjusting for confounding methods based on stratification of observations by quantiles of estimated propensity scores and methods based on weighting observations by the inverse of estimated propensity scores we review popular versions of these approaches and related methods offering improved precision describe theoretical properties and highlight their implications for practice and present extensive comparisons of performance that provide guidance for practical use

adjusting for nonignorable drop out using semiparametric nonresponse models abstract consider a study whose design calls for the study subjects to be followed from enrollment time t 0 to time t t at which point a primary endpoint of interest y is to be measured the design of the study also calls for measurements on a vector v t of covariates to be made at one or more times t during the interval 0 t we are interested in making inferences about the marginal mean 0 of y when some subjects drop out of the study at random times q prior to the common fixed end of follow up time t the purpose of this article is to show how to make inferences about 0 when the continuous drop out time q is modeled semiparametrically and no restrictions are placed on the joint distribution of the outcome and other measured variables in particular we consider two models for the conditional hazard of drop out given v t y where v t denotes the history of the process v t through time t t 0 t in the first model we assume that q t v t y exp 0 y where 0 is a scalar paramet

entropy dissipation estimates in a zero range dynamics we study the exponential decay of relative entropy functionals for zero range processes on the complete graph for the standard model with rates increasing at infinity we prove entropy dissipation estimates uniformly over the number of particles and the number of vertices

spectral gap for zero range dynamics we give a lower bound on the spectral gap for symmetric zero range processes under some conditions on the rate function we show that the gap shrinks as n 2 independent of the density for the dynamics localized on a cube of size n d we follow the method outlined by lu and yau where a similar spectral gap is proved for kawasaki dynamics

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

spectral gap for stochastic energy exchange model with nonuniformly positive rate function we give a lower bound on the spectral gap for a class of stochastic energy exchange models in 2011 grigo et al introduced the model and showed that for a class of stochastic energy exchange models with a uniformly positive rate function the spectral gap of an n component system is bounded from below by a function of order n 2 in this paper we consider the case where the rate function is not uniformly positive for this case the spectral gap depends not only on n but also on the averaged energy e which is the conserved quantity under the dynamics under some assumption we obtain a lower bound of the spectral gap which is of order c e n 2 where c e is a positive constant depending on e as a corollary of the result a lower bound of the spectral gap for the mesoscopic energy exchange process of billiard lattice studied by gaspard and gilbert j stat mech theory exp 2008 2008 p11021 j stat mech theory exp 2009 2009 p08020 and the stick process studied by feng et al stochastic process appl 66 1997 147 182 are obtained

asymptotic pointwise contractions a simple and elementary without using the ultrapower technique proof for the existence of a fixed point of an asymptotic pointwise contraction is provided the existence of fixed points for asymptotic pointwise nonexpansive mappings in a uniformly convex banach space is also investigated

approximate fixed points of α nonexpansive mappings abstract in this paper we define the class of nonexpansive mappings which is properly larger than the class of nonexpansive mappings and prove that every nonexpansive mapping t c c has an approximate fixed point sequence where c is a nonempty bounded subset of a banach space x 0 and 0 this in particular gives an affirmative answer to the open question posed by ariza ruiz and et al concerning the existence of an approximate fixed point sequence for nonexpansive mappings ariza ruiz et al 2016 4

mixing of the symmetric exclusion processes in terms of the corresponding single particle random walk we prove an upper bound for the e mixing time of the symmetric exclusion process on any graph g with any feasible number of particles our estimate is proportional to trw g ln v e where v is the number of vertices in g and trw g is the 1 4 mixing time of the corresponding single particle random walk this bound implies new results for symmetric exclusion on expanders percolation clusters the giant component of the erdos renyi random graph and poisson point processes in rd our technical tools include a variant of morris s chameleon process

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

interval estimation of population means under unknown but bounded probabilities of sample selection applying concepts from partial identification to the domain of finite population sampling we propose a method for interval estimation of a population mean when the probabilities of sample selection lie within a posited interval the interval estimate is derived from sharp bounds on the hajek 1971 estimator of the population mean we demonstrate the method s utility for sensitivity analysis by applying it to a sample of needles collected as part of a syringe tracking and testing programme in new haven connecticut copyright 2013 oxford university press

on asymptotic pointwise contractions in metric spaces abstract we discuss the existence of fixed points of asymptotic pointwise mappings in metric spaces this is the nonlinear version of some known results proved in banach spaces we also discuss the case of multivalued mappings

dual and simultaneous sensitivity analysis for matched pairs summary when a study shows an association between a treatment and a response before concluding that there is a causal relationship it is useful to assess whether or not an unobserved variable u might explain the observed association sensitivity analysis clarifies the properties u must have in terms of its relationship to the response and its imbalance in the groups being compared a substantial literature has investigated the imbalance or association between u and group assignment this paper develops a dual sensitivity analysis for matched pairs focusing on the strength of the relationship between u and the response required to reduce an observed association to non significance a third simultaneous form of sensitivity analysis models both relationships between u and treatment assignment and u and response the simultaneous form allows one to compare results from the sensitivity analysis to subject matter knowledge about both relationships the methods are illustrated by several examples

estimation and confidence regions for parameter sets in econometric models this paper develops a framework for performing estimation and inference in econometric models with partial identification focusing particularly on models characterized by moment inequalities and equalities applications of this framework include the analysis of game theoretic models revealed preference restrictions regressions with missing and corrupted data auction models structural quantile regressions and asset pricing models specifically we provide estimators and confidence regions for the set of minimizers i of an econometric criterion function q in applications the criterion function embodies testable restrictions on economic models a parameter value that describes an economic model satisfies these restrictions if q attains its minimum at this value interest therefore focuses on the set of minimizers called the identified set we use the inversion of the sample analog q n of the population criterion q to construct estimators and confidence regions for the identified set and develop consistency rates of convergence and inference results for these estimators and regions to derive these results we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification

confidence intervals for partially identified parameters recently a growing body of research has studied inference in settings where parameters of interest are partially identified in many cases the parameter is real valued and the identification region is an interval whose lower and upper bounds may be estimated from sample data for this case confidence intervals cis have been proposed that cover the entire identification region with fixed probability here we introduce a conceptually different type of confidence interval rather than cover the entire identification region with fixed probability we propose cis that asymptotically cover the true value of the parameter with this probability however the exact coverage probabilities of the simplest version of our new cis do not converge to their nominal values uniformly across different values for the width of the identification region to avoid the problems associated with this we modify the proposed ci to ensure that its exact coverage probabilities do converge uniformly to their nominal values we motivate this modified ci through exact results for the gaussian case

diametrically complete sets and normal structure abstract we prove that in some classes of reflexive banach spaces every maximal diametral set must be diametrically complete thus showing that diametrically complete sets may have empty interior also in reflexive spaces as a consequence we prove that in those spaces normal structure is equivalent to the weaker property that for every bounded set the absolute chebyshev radius is strictly smaller than the diameter moreover we prove that in any normed space the two classes of diametrically complete sets and of sets of constant radius from the boundary coincide if the space has normal structure

pattern mixture and selection models for analysing longitudinal data with monotone missing patterns summary we examine three pattern mixture models for making inference about parameters of the distribution of an outcome of interest y that is to be measured at the end of a longitudinal study when this outcome is missing in some subjects we show that these pattern mixture models also have an interpretation as selection models because these models make unverifiable assumptions we recommend that inference about the distribution of y be repeated under a range of plausible assumptions we argue that of the three models considered only one admits a parameterization that facilitates the examination of departures from the assumption of sequential ignorability the three models are nonparametric in the sense that they do not impose restrictions on the class of observed data distributions owing to the curse of dimensionality the assumptions that are encoded in these models are sufficient for identification but not for inference we describe additional flexible and easily interpretable assumptions under which it is possible to construct estimators that are well behaved with moderate sample sizes these assumptions define semiparametric models for the distribution of the observed data we describe a class of estimators which up to asymptotic equivalence comprise all the consistent and asymptotically normal estimators of the parameters of interest under the postulated semiparametric models we illustrate our methods with the analysis of data from a randomized clinical trial of contracepting women

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

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

poincare modified logarithmic sobolev and isoperimetric inequalities for markov chains with non negative ricci curvature abstract we study functional inequalities for markov chains on discrete spaces with entropic ricci curvature bounded from below our main results are that when curvature is non negative but not necessarily positive the spectral gap the cheeger isoperimetric constant and the modified logarithmic sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space with respect to a suitable metric these estimates are discrete analogues of classical results of riemannian geometry obtained by li and yau buser and wang

cross screening in observational studies that test many hypotheses abstractwe discuss observational studies that test many causal hypotheses either hypotheses about many outcomes or many treatments to be credible an observational study that tests many causal hypotheses must demonstrate that its conclusions are neither artifacts of multiple testing nor of small biases from nonrandom treatment assignment in a sense that needs to be defined carefully hidden within a sensitivity analysis for nonrandom assignment is an enormous correction for multiple testing in the absence of bias it is extremely improbable that multiple testing alone would create an association insensitive to moderate biases we propose a new strategy called cross screening different from but motivated by recent work of bogomolov and heller on replicability cross screening splits the data in half at random uses the first half to plan a study carried out on the second half then uses the second half to plan a study carried out on the first half and reports the more favorable conclusions of the tw