doubly robust estimates for binary longitudinal data analysis with missing response and missing covariates longitudinal studies often feature incomplete response and covariate data likelihood based methods such as the expectation maximization algorithm give consistent estimators for model parameters when data are missing at random mar provided that the response model and the missing covariate model are correctly specified however we do not need to specify the missing data mechanism an alternative method is the weighted estimating equation which gives consistent estimators if the missing data and response models are correctly specified however we do not need to specify the distribution of the covariates that have missing values in this article we develop a doubly robust estimation method for longitudinal data with missing response and missing covariate when data are mar this method is appealing in that it can provide consistent estimators if either the missing data model or the missing covariate model is correctly specified simulation studies demonstrate that this method performs well in a variety of situations

missing data methods for generalized linear models a comparative review missing data is a major issue in many applied problems especially in the biomedical sciences we review four common approaches for inference in generalized linear models glms with missing covariate data maximum likelihood ml multiple imputation mi fully bayesian fb and weighted estimating equations wees there is considerable interest in how these four methodologies are related the properties of each approach the advantages and disadvantages of each methodology and computational implementation we examine data that are missing at random and nonignorable missing for ml we focus on techniques using the em algorithm and in particular discuss the em by the method of weights and related procedures as discussed by ibrahim for mi we examine the techniques developed by rubin for fb we review approaches considered by ibrahim et al for wee we focus on the techniques developed by robins et al we use a real dataset and a detailed simulation study to compare the four methods

assessing accuracy of mammography in the presence of verification bias and intrareader correlation la performance des tests diagnostiques medicaux est souvent evaluee en comparant le resultat du test au vrai etat de la maladie du patient l analyse des courbes de caracteristiques de fonctionnement roc peut etre utilisee pour resumer la precision du test cependant une telle analyse peut rencontrer plusieurs difficultes dans la pratique reelle une difficulte est la verification du biais c est a dire que l evaluation de l etat de la maladie peut etre partiellement disponible et la probabilite de determination de la maladie peut dependre aussi bien du resultat du test que des caracteristiques du sujet une deuxieme difficulte est que les tests interpretes par le meme lecteur peuvent ne pas etre independants a l aide des equations d estimation nous generalisons des methodes proposees pour prendre en compte ces problemes nous comparons la performance de differents estimateurs de la precision en utilisant les estimateurs robustes de la variance sandwich pour permettre une inference asymptotique valide nous montrons que dans le contexte d une cohorte d observation ou de bonnes informations sur les covariables sont disponibles une approche par des equations d estimations ponderees wee peut etre preferable pour sa robustesse a un modele mal specifie nous appliquons la methodologie a la mammographie comme realisee dans la communaute des radiologues

regression analysis with missing covariate data using estimating equations in regression analysis missing covariate data has been among the most common problems frequently practitioners adopt the so called complete case analysis i e performing the analysis on only a complete dataset after excluding records with missing covariates performing a complete case analysis is convenient with existing statistical packages but it may be inefficient since the observed outcomes and covariates on those records with missing covariates are not used it can even give misleading statistical inference if missing is not completely at random this paper introduces a joint estimating equation jee for regression analysis in the presence of missing observations on one covariate which may be thought of as a method in a general framework for the missing covariate data problem proposed by robins rotnitzky and zhao 1994 journal of the american statistical association 89 846 866 a generalization of jee to more than one such covariate is discussed the jee is generally applicable to estimating regression coefficients from a regression model including linear and logistic regression provided that the missing covariate data is either missing completely at random or missing at random in addition to mild regularity conditions estimates of regression coefficients from the jee are consistent and have an asymptotic normal distribution simulation results show that the asymptotic distribution of estimated coefficients performs well in finite samples also shown through the simulation study is that the validity of jee estimates depends on the correct specification of the probability function that characterizes the missing mechanism suggesting a need for further research on how to robustify the estimation from making this nuisance assumption finally the jee is illustrated with an application from a case control study of diet and thyroid cancer

monte carlo em for missing covariates in parametric regression models we propose a method for estimating parameters for general parametric regression models with an arbitrary number of missing covariates we allow any pattern of missing data and assume that the missing data mechanism is ignorable throughout when the missing covariates are categorical a useful technique for obtaining parameter estimates is the em algorithm by the method of weights proposed in ibrahim 1990 journal of the american statistical association85 765 769 we extend this method t o continuous or mixed categorical and continuous covariates and for arbitrary parametric regression models by adapting a monte carlo version of the em algorithm as discussed by wei and tanner 1990 journal of the american statistical association85 699 704 in addition we discuss the gibbs sampler for sampling from the conditional distribution of the missing covariates given the observed data and show that the appropriate complete conditionals are log concave the log concavity property of the conditional distributions will facilitate a straightforward implementation of the gibbs sampler via the adaptive rejection algorithm of gilks and wild 1992 applied statistics41 337 348 we assume the model for the response given the covariates is an arbitrary parametric regression model such as a generalized linear model a parametric survival model or a nonlinear model we model the marginal distribution of the covariates as a product of one dimensional conditional distributions this allows us a great deal of flexibility in modeling the distribution of the covariates and reduces the number of nuisance parameters that are introduced in the e step we present examples involving both simulated and real data

a monte carlo implementation of the em algorithm and the poor man s data augmentation algorithms abstract the first part of this article presents the monte carlo implementation of the e step of the em algorithm given the current guess to the maximizer of the posterior distribution latent data patterns are generated from the conditional predictive distribution the expected value of the augmented log posterior is then updated as a mixture of augmented log posteriors mixed over the generated latent data patterns multiple imputations in the m step of the algorithm this mixture is maximized to obtain the update to the maximizer of the observed posterior the gradient and hessian of the observed log posterior are also expressed as mixtures mixed over the multiple imputations the relation between the monte carlo em mcem algorithm and the data augmentation algorithm is noted two modifications to the mcem algorithm the poor man s data augmentation algorithms which allow for the calculation of the entire posterior are then presented these approximations serve as diagnostics for the validity o

two decentralized algorithms for strong interaction fairness for systems with unbounded speed variability abstract we present two randomized algorithms one for message passing and the other for shared memory that with probability 1 schedule multiparty interactions in a strongly fair manner both algorithms improve upon a previous result by joung and smolka proposed in a shared memory model along with a straightforward conversion to the message passing paradigm in the following aspects first processes speeds as well as communication delays need not be bounded by any predetermined constant secondly our algorithms are completely decentralized and the shared memory solution makes use of only single writer variables finally both algorithms are symmetric in the sense that all processes execute the same code and no unique identifier is used to distinguish processes

toward a curse of dimensionality appropriate coda asymptotic theory for semi parametric models we argue that due to the curse of dimensionality there are major difficulties with any pure or smoothed likelihood based method of inference in designed studies with randomly missing data when missingness depends on a high dimensional vector of variables we study in detail a semi parametric superpopulation version of continuously stratified random sampling we show that all estimators of the population mean that are uniformly consistent or that achieve an algebraic rate of convergence no matter how slow require the use of the selection randomization probabilities we argue that in contrast to likelihood methods which ignore these probabilities inverse selection probability weighted estimators continue to perform well achieving uniform n 1 2 rates of convergence we propose a curse of dimensionality appropriate coda asymptotic theory for inference in non and semi parametric models in an attempt to formalize our arguments we discuss whether our results constitute a fatal blow to the likelihood principle and study the attitude toward these that a committed subjective bayesian would adopt finally we apply our coda theory to analyse the effect of the curse of dimensionality in several interesting semi parametric models including a model for a two armed randomized trial with randomization probabilities depending on a vector of continuous pretreatment covariates x we provide substantive settings under which a subjective bayesian would ignore the randomization probabilities in analysing the trial data we then show that any statistician who ignores the randomization probabilities is unable to construct nominal 95 per cent confidence intervals for the true treatment effect that have both i an expected length which goes to zero with increasing sample size and ii a guaranteed expected actual coverage rate of at least 95 per cent over the ensemble of trials analysed by the statistician during his or her lifetime however we derive a new interval estimator depending on the randomization probabilities that satisfies i and ii

nonparametric and semiparametric models for missing covariates in parametric regression robustness of covariate modeling for the missing covariate problem in parametric regression is studied under the missing at random assumption for a simple missing covariate pattern nonparametric covariate model is proposed and is shown to yield a consistent and semiparametrically efficient estimator for the regression parameter total robustness is achieved in this situation for more general missingcovariate patterns a novel semiparametric modeling approach is proposed for the covariates in this approach the covariate distribution is first decomposed into the product of a series of conditional distributions according to the overall missing data patterns and the conditional distributions are then represented in the general odds ratio form the general odds ratios are modeled parametrically and the other components of the covariate distribution are modeled nonparametrically maximum semiparametric likelihood is used to find the parameter estimates the proposed method yields a consistent estimator f

fixed points of generalized contractive multi valued mappings in a recent paper n mizoguchi and w takahashi gave a positive answer to the conjecture of s reich concerning the existence of fixed points of multi valued mappings that satisfy a certain contractive condition in this paper we provide an alternative and somewhat more straightforward proof for the theorem of mizoguchi and takahashi also the problems associated with fixed points of weakly contractive multi valued mappings are studied finally we make a few comments that improve other results from their paper j math anal appl 141 1989 177 188

multi valued nonlinear contraction mappings abstract two concepts of nonlinear contractions for multi valued mappings in complete metric spaces are introduced and three fixed point theorems are proved presented theorems are generalizations of very recent fixed point theorems due to klim and wardowski d klim d wardowski fixed point theorems for set valued contractions in complete metric spaces j math anal appl 334 2007 132 139 and feng and liu y feng s liu fixed point theorems for multi valued contractive mappings and multi valued caristi type mappings j math anal appl 317 2006 103 112 as well as a fixed point theorem of mizoguchi and takahashi n mizoguchi w takahash fixed point theorems for multi valued mappings on complete metric spaces j math anal appl 141 1989 177 188 and several others an example is given to show that presented results are genuine generalizations our results are also some contribution to an open problem raised by reich s reich fixed points of contractive functions boll unione mat ital 5 1972 26 42 s reich some fixed point problems atti acad naz lincei 57 1974 194 198 s reich some problems and results in fixed point theory contemp math 21 1983 179 187

three fixed point theorems for generalized contractions with constants in complete metric spaces abstract we prove three fixed point theorems for generalized contractions with constants in complete metric spaces which are generalizations of very recent fixed point theorems due to suzuki we also raise one problem concerning the constants

a weighted estimating equation for linear regression with missing covariate data linear regression is one of the most popular statistical techniques in linear regression analysis missing covariate data occur often a recent approach to analyse such data is a weighted estimating equation with weighted estimating equations the contribution to the estimating equation from a complete observation is weighted by the inverse probability of being observed in this paper we propose a weighted estimating equation in which we wrongly assume that the missing covariates are multivariate normal but still produces consistent estimates as long as the probability of being observed is correctly modelled in simulations these weighted estimating equations appear to be highly efficient when compared to the most efficient weighted estimating equation as proposed by robins et al and lipsitz et al however these weighted estimating equations in which we wrongly assume that the missing covariates are multivariate normal are much less computationally intensive than the weighted estimating equations given by lipsitz et al we compare the weighted estimating equations proposed in this paper to the efficient weighted estimating equations via an example and a simulation study we only consider missing data which are missing at random non ignorably missing data are not addressed in this paper

sieve maximum likelihood estimation for regression models with covariates missing at random missing covariates are common in regression problems we propose a new semiparametric method based on a fully nonparametric distribution for the missing covariates that are assumed to be missing at random the method of sieve maximum likelihood estimation is used to obtain the estimators of the regression coefficients these estimators are shown to be consistent and asymptotically normal with their asymptotic covariance matrix that achieves the semiparametric efficiency bound a bootstrap approach is used to estimate the asymptotic covariance matrix some practical modeling approaches for high dimensional covariates are proposed extensive simulation studies are conducted to examine the finite sample properties of the estimates and a real data set from a liver cancer clinical trial is analyzed using the proposed method

fixed point theorems for multivalued mappings on complete metric spaces on obtient une version multivaluee du theoreme de point fixe de caristi qui est equivalent au principe e variationnel du a ekeland on utilise ce resultat pour obtenir un theoreme qui deduit le theoreme de nadler et un theoreme de point fixe commun pour une application a 1 valeur et une application multivaluee

a fixed points of multi valued contractions soit x d un espace metrique complet 7b a un sous ensemble ferme non vide de 7b m t x et f t x 2 x une multifonction a valeur fermee on suppose que f est 7b a stable pour chaque 7b a et qu il existe 0 1 tel que d h f t x f y kd x y pour tout t t x y x on suppose qu il existe 0 7b a tel que sup t t d 0 t f t 0 t sous de telles hypotheses f admet un 7b a point fixe tel que d 0

semiparametric pseudoscore for regression with multidimensional but incompletely observed regressor we study the regression f y x z where y is the response z rd is a vector of fully observed regressors and x is the regressor with incomplete observation to handle missing data maximum likelihood estimation via expectation maximisation em is the most efficient but is sensitive to the specification of the distribution of x under a missing at random assumption we propose an em type estimation via a semiparametric pseudoscore like in em we derive the conditional expectation of the score function given y and z or the mean score over the incompletely observed units under a postulated distribution of x instead of directly using the mean score in estimating equation we use it as a working index to construct the semiparametric pseudoscore via nonparametric regression introduction of semiparametric pseudoscore into the em framework reduces sensitivity to the specified distribution of x it also avoids the curse of dimensionality when z is multidimensional the resulting regression estimator is more than doubly robust it is consistent if either the pattern of missingness in x is correctly specified or the working index is appropriately but not necessarily correctly specified it attains optimal efficiency when both conditions are satisfied numerical performance is explored by monte carlo simulations and a study on treating hepatitis c patients with hiv coinfection published 2017 this article is a u s government work and is in the public domain in the usa

semiparametric estimation of the covariate specific roc curve in presence of ignorable verification bias covariate specific roc curves are often used to evaluate the classification accuracy of a medical diagnostic test or a biomarker when the accuracy of the test is associated with certain covariates in many large scale screening tests the gold standard is subject to missingness due to high cost or harmfulness to the patient in this paper we propose a semiparametric estimation of the covariate specific roc curves with a partial missing gold standard a location scale model is constructed for the test result to model the covariates effect but the residual distributions are left unspecified thus the baseline and link functions of the roc curve both have flexible shapes with the gold standard missing at random mar assumption we consider weighted estimating equations for the location scale parameters and weighted kernel estimating equations for the residual distributions three roc curve estimators are proposed and compared namely imputation based inverse probability weighted and doubly robust estimators we derive the asymptotic normality of the estimated roc curve as well as the analytical form the standard error estimator the proposed method is motivated and applied to the data in an alzheimer s disease research

maximum likelihood estimation for mixed continuous and categorical data with missing values summary maximum likelihood procedures for analysing mixed continuous and categorical data with missing values are presented the general location model of olkin tate 1961 and extensions introduced by krzanowski 1980 1982 form the basis for our methods maximum likelihood estimation with incomplete data is achieved by an application of the em algorithm dempster laird rubin 1977 special cases of the algorithm include orchard woodbury s 1972 algorithm for incomplete normal samples fuchs s 1982 algorithms for log linear modelling of partially classified contingency tables and day s 1969 algorithm for multivariate normal mixtures applications include a imputation of missing values b logistic regression and discriminant analysis with missing predictors and unclassified observations c linear regression with missing continuous and categorical predictors and d parametric cluster analysis with incomplete data methods are illustrated using data from the st louis risk research project some key word8 cluster analysis discriminant analysis em algorithm incomplete data linear regression logistic regression log linear model mixture model

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

weighted generalized estimating functions for longitudinal response and covariate data that are missing at random longitudinal studies often feature incomplete response and covariate data it is well known that biases can arise from naive analyses of available data but the precise impact of incomplete data depends on the frequency of missing data and the strength of the association between the response variables and covariates and the missing data indicators various factors may influence the availability of response and covariate data at scheduled assessment times and at any given assessment time the response may be missing covariate data may be missing or both response and covariate data may be missing here we show that it is important to take the association between the missing data indicators for these two processes into account through joint models inverse probability weighted generalized estimating equations offer an appealing approach for doing this here we develop these equations for a particular model generating intermittently missing at random data empirical studies demonstrate that the consistent e

score tests for association under response dependent sampling designs for expensive covariates this is a pre copyedited author produced pdf of an article accepted for publication in biometrika following peer review the version of record derkach a lawless j f and sun l 2015 score tests for association under response dependent sampling designs for expensive covariates biometrika 102 4 988 994 is available online at http biomet oxfordjournals org content 102 4 988 full pdf html doi 10 1093 biomet asv038

doubly robust type estimation for covariate adjustment in latent variable modeling due to the difficulty in achieving a random assignment a quasi experimental or observational study design is frequently used in the behavioral and social sciences if a nonrandom assignment depends on the covariates multiple group structural equation modeling that includes the regression function of the dependent variables on the covariates that determine the assignment can provide reasonable estimates under the condition of correct specification of the regression function however it is usually difficult to specify the correct regression function because the dimensions of the dependent variables and covariates are typically large therefore the propensity score adjustment methods have been proposed since they do not require the specification of the regression function and have been applied to several applied studies however these methods produce biased estimates if the assignment mechanism is incorrectly specified in order to make a more robust inference it would be more useful to develop an estimation method that integrates the regression approach with the propensity score methodology in this study we propose a doubly robust type estimation method for marginal multiple group structural equation modeling this method provides a consistent estimator if either the regression function or the assignment mechanism is correctly specified a simulation study indicates that the proposed estimation method is more robust than the existing methods

calculi for synchrony and asynchrony abstract a calculus for distributed computation is studied based upon four combinators a central idea is an abelian group of actions which models the interfaces between components of a distributed computing agent using a notion of bisimulation congruence relations are defined over computing agents and thence an algebraic theory is derived the calculus models both synchronous and asynchronous computation in particular it is shown that the author s calculus of communicating systems 1980 which is an asynchronous model is derivable from the calculus presented here

a weighted estimating equation for missing covariate data with properties similar to maximum likelihood abstract in regression analysis missing covariate data occurs often a recent approach to analyzing such data is weighted estimating equations with weighted estimating equations the contribution to the estimating equation from a complete observation is weighted by the inverse probability of being observed in this article we propose a weighted estimating equation that is almost identical to the maximum likelihood estimating equations as such we propose an em type algorithm to solve these weighted estimating equations although the weighted estimating equations are a special case of those proposed earlier by robins et al our em type algorithm to solve them is new similar to robins and ritov we give the result that to obtain a consistent estimate of the regression parameters either the missing data mechanism or the distribution of the missing data given the observed data must be correctly specified we compare the weighted estimating equations to maximum likelihood via two examples a simulation a

suzuki type fixed point theorems for generalized multi valued mappings on a set endowed with two b metrics abstract in this paper we obtained suzuki type fixed point results for a generalized multi valued mapping on a set equipped with two b metrics as a consequence existence and uniqueness of solution of functional equation arising in dynamical programming is also derived

incomplete data in generalized linear models abstract this article examines incomplete data for the class of generalized linear models in which incompleteness is due to partially missing covariates on some observations under the assumption that the missing data are missing at random it is shown that the e step of the em algorithm for any generalized linear model can be expressed as a weighted complete data log likelihood when the unobserved covariates are assumed to come from a discrete distribution with finite range expressing the e step in this manner allows for a straightforward maximization in the m step thus leading to maximum likelihood estimates mle s for the parameters asymptotic variances of the mle s are also derived and results are illustrated with two examples

doubly robust estimation in missing data and causal inference models the goal of this article is to construct doubly robust dr estimators in ignorable missing data and causal inference models in a missing data model an estimator is dr if it remains consistent when either but not necessarily both a model for the missingness mechanism or a model for the distribution of the complete data is correctly specified because with observational data one can never be sure that either a missingness model or a complete data model is correct perhaps the best that can be hoped for is to find a dr estimator dr estimators in contrast to standard likelihood based or nonaugmented inverse probability weighted estimators give the analyst two chances instead of only one to make a valid inference in a causal inference model an estimator is dr if it remains consistent when either a model for the treatment assignment mechanism or a model for the distribution of the counterfactual data is correctly specified because with observational data one can never be sure that a model for the treatment assignment mechanism or a model for the counterfactual data is correct inference based on dr estimators should improve upon previous approaches indeed we present the results of simulation studies which demonstrate that the finite sample performance of dr estimators is as impressive as theory would predict the proposed method is applied to a cardiovascular clinical trial

a conditional model for incomplete covariates in parametric regression models summary incomplete covariate data arise in many data sets when the missing covariates are categorical a useful technique for obtaining parameter estimates is the em algorithm by the method of weights proposed in ibrahim 1990 this method requires the estimation of many nuisance parameters for the distribution of the covariates unfortunately in data sets when the percentage of missing data is high and the missing covariate patterns are highly non monotone the estimates of the nuisance parameters can lead to highly unstable estimates of the parameters of interest we propose a conditional model for the covariate distribution that has several modelling advantages for the e step and provides a reduction in the number of nuisance parameters thus providing more stable estimates in finite samples we present a clinical trials example with six covariates five of which have some missing values

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

semiparametric estimation exploiting covariate independence in two phase randomized trials recent results for case control sampling suggest when the covariate distribution is constrained by gene environment independence semiparametric estimation exploiting such independence yields a great deal of efficiency gain we consider the efficient estimation of the treatment biomarker interaction in two phase sampling nested within randomized clinical trials incorporating the independence between a randomized treatment and the baseline markers we develop a newton raphson algorithm based on the profile likelihood to compute the semiparametric maximum likelihood estimate spmle our algorithm accommodates both continuous phase one outcomes and continuous phase two biomarkers the profile information matrix is computed explicitly via numerical differentiation in certain situations where computing the spmle is slow we propose a maximum estimated likelihood estimator mele which is also capable of incorporating the covariate independence this estimated likelihood approach uses a one step empirical covariate distribution thus is straightforward to maximize it offers a closed form variance estimate with limited increase in variance relative to the fully efficient spmle our results suggest exploiting the covariate independence in two phase sampling increases the efficiency substantially particularly for estimating treatment biomarker interactions

empirical likelihood in missing data problems missing data is a ubiquitous problem in medical and social sciences it is well known that inferences based only on the complete data may not only lose efficiency but may also lead to biased results if the data is not missing completely at random mcar the inverse probability weighting method proposed by horvitz and thompson 1952 is a popular alternative when the data is not mcar the horvitz thompson method however is sensitive to the inverse weights and may suffer from loss of efficiency in this paper we propose a unified empirical likelihood approach to missing data problems and explore the use of empirical likelihood to effectively combine unbiased estimating equations when the number of estimating equations is greater than the number of unknown parameters one important feature of this approach is the separation of the complete data unbiased estimating equations from the incomplete data unbiased estimating equations the proposed method can achieve semiparametric efficiency if the probabilit