stable weights that balance covariates for estimation with incomplete outcome data weighting methods that adjust for observed covariates such as inverse probability weighting are widely used for causal inference and estimation with incomplete outcome data part of the appeal of such methods is that one set of weights can be used to estimate a range of treatment effects based on different outcomes or a variety of population means for several variables however this appeal can be diminished in practice by the instability of the estimated weights and by the difficulty of adequately adjusting for observed covariates in some settings to address these limitations this article presents a new weighting method that finds the weights of minimum variance that adjust or balance the empirical distribution of the observed covariates up to levels prespecified by the researcher this method allows the researcher to balance very precisely the means of the observed covariates and other features of their marginal and joint distributions such as variances and correlations and also for example the

leveraging prognostic baseline variables to gain precision in randomized trials we focus on estimating the average treatment effect in a randomized trial if baseline variables are correlated with the outcome then appropriately adjusting for these variables can improve precision an example is the analysis of covariance ancova estimator which applies when the outcome is continuous the quantity of interest is the difference in mean outcomes comparing treatment versus control and a linear model with only main effects is used ancova is guaranteed to be at least as precise as the standard unadjusted estimator asymptotically under no parametric model assumptions and also is locally semiparametric efficient recently several estimators have been developed that extend these desirable properties to more general settings that allow any real valued outcome e g binary or count contrasts other than the difference in mean outcomes such as the relative risk and estimators based on a large class of generalized linear models including logistic regression to the best of our knowledge we give the first simulation study in the context of randomized trials that compares these estimators furthermore our simulations are not based on parametric models instead our simulations are based on resampling data from completed randomized trials in stroke and hiv in order to assess estimator performance in realistic scenarios we provide practical guidance on when these estimators are likely to provide substantial precision gains and describe a quick assessment method that allows clinical investigators to determine whether these estimators could be useful in their specific trial contexts

he s variational iteration method for treating nonlinear singular boundary value problems this paper applies he s variational iteration method for solving nonlinear singular boundary value problems the solution process is illustrated and various physically relevant results are obtained comparison of the obtained results with exact solutions shows that the method used is an effective and highly promising method for treating various classes of both linear and nonlinear singular boundary value problems

model averaged double robust estimation researchers estimating causal effects are increasingly challenged with decisions on how to best control for a potentially high dimensional set of confounders typically a single propensity score model is chosen and used to adjust for confounding while the uncertainty surrounding which covariates to include into the propensity score model is often ignored and failure to include even one important confounder will results in bias we propose a practical and generalizable approach that overcomes the limitations described above through the use of model averaging we develop and evaluate this approach in the context of double robust estimation more specifically we introduce the model averaged double robust ma dr estimators which account for model uncertainty in both the propensity score and outcome model through the use of model averaging the ma dr estimators are defined as weighted averages of double robust estimators where each double robust estimator corresponds to a specific choice of the outcome model and the propensity score model the ma dr estimators extend the desirable double robustness property by achieving consistency under the much weaker assumption that either the true propensity score model or the true outcome model be within a specified possibly large class of models using simulation studies we also assessed small sample properties and found that ma dr estimators can reduce mean squared error substantially particularly when the set of potential confounders is large relative to the sample size we apply the methodology to estimate the average causal effect of temozolomide plus radiotherapy versus radiotherapy alone on one year survival in a cohort of 1887 medicare enrollees who were diagnosed with glioblastoma between june 2005 and december 2009

uniform central limit theorems for kernel density estimators let mathbb p n ast k h n x n 1 h n d sum i 1 n k left x x i h n right be the classical kernel density estimator based on a kernel k and n independent random vectors x i each distributed according to an absolutely continuous law mathbb p on mathbb r d it is shown that the processes f longmapsto sqrt n int fd mathbb p n ast k h n mathbb p f in mathcal f converge in law in the banach space ell infty mathcal f for many interesting classes mathcal f of functions or sets some mathbb p donsker some just mathbb p pregaussian the conditions allow for the classical bandwidths h n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error thus showing that subject to some natural conditions kernel density estimators are plug in estimators in the sense of bickel and ritov ann statist 31 1033 1053 2003 some new results on the uniform central limit theorem for smoothed empirical processes needed in the proofs are also included

he s variational iteration method for solving linear and non linear systems of ordinary differential equations abstract in this article he s variational iteration method vim is employed to solve a system of differential equations of first order since every ordinary differential equations of higher order can be converted into a system of differential of the first order this method can be used for solving most differential equations some examples are presented to show the ability of the method for linear and non linear systems of differential equations the results reveal that the method is very effective and simple

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

stability results for the heat equation backward in time abstract for the heat equation backward in time u t u x x x r t 0 t u t l p r subject to the constraint u 0 l p r e with t 0 l p r 0 e 1 p being given we prove that if u 1 and u 2 are two solutions of the problem then there is a constant c 0 such that u 1 t u 2 t l p r c t t e 1 t t t 0 t in case p 2 we establish stability estimates of holder type for all derivatives with respect to x and t of the solutions we suggest a useful strategy of choosing mollification parameters which provides a continuity at t 0 when an additional condition on the smoothness of u x 0 is given furthermore we propose a stable marching difference scheme for this ill posed problem and test several related numerical methods for it

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

fourier regularization for a backward heat equation abstract in this paper a simple and convenient new regularization method for solving backward heat equation fourier regularization method is given meanwhile some quite sharp error estimates between the approximate solution and exact solution are provided a numerical example also shows that the method works effectively

covariate adjustment in randomized trials with binary outcomes targeted maximum likelihood estimation covariate adjustment using linear models for continuous outcomes in randomized trials has been shown to increase efficiency and power over the unadjusted method in estimating the marginal effect of treatment however for binary outcomes investigators generally rely on the unadjusted estimate as the literature indicates that covariate adjusted estimates based on the logistic regression models are less efficient the crucial step that has been missing when adjusting for covariates is that one must integrate average the adjusted estimate over those covariates in order to obtain the marginal effect we apply the method of targeted maximum likelihood estimation tmle to obtain estimators for the marginal effect using covariate adjustment for binary outcomes we show that the covariate adjustment in randomized trials using the logistic regression models can be mapped by averaging over the covariate s to obtain a fully robust and efficient estimator of the marginal effect which equals a targeted maximum likelihood estimator this tmle is obtained by simply adding a clever covariate to a fixed initial regression we present simulation studies that demonstrate that this tmle increases efficiency and power over the unadjusted method particularly for smaller sample sizes even when the regression model is mis specified

a piecewise variational iteration method for treating a nonlinear oscillator of a mass attached to a stretched elastic wire abstract in this paper we introduce a piecewise variational iteration method for treating a nonlinear oscillator of a mass attached to a stretched elastic wire for the nonlinear oscillator the present method can produce a good approximation to the exact solution in a very large region while the standard variational iteration method vim gives a good approximation only in a small region the numerical results obtained show that the present method does not share the drawback of the standard vim and can provide very accurate analytical approximate solutions for both small and large values of the oscillation amplitude and parameter

a splitting method for a backward parabolic equation with time dependent coefficients in this paper we propose a stable numerical method for an ill posed backward parabolic equation with time dependent coefficients in a parallelepiped the problem is reformulated as an ill posed least squares problem which is solved by the conjugate gradient method with an a posteriori stopping rule the least squares problem is discretized by a splitting method which reduces the large dimensions of the discretized problem we calculate the gradient of the objective functional of the discretized least squares problem by the aid of an adjoint discretized problem which enhances its accuracy the algorithm is tested on several examples that proves its efficiency

kernel based covariate functional balancing for observational studies covariate balance is often advocated for objective causal inference since it mimics randomization in observational data unlike methods that balance specific moments of covariates our proposal attains uniform approximate balance for covariate functions in a reproducing kernel hilbert space the corresponding infinite dimensional optimization problem is shown to have a finite dimensional representation in terms of an eigenvalue optimization problem large sample results are studied and numerical examples show that the proposed method achieves better balance with smaller sampling variability than existing methods

bounded efficient and multiply robust estimation of average treatment effects using instrumental variables instrumental variables ivs are widely used for estimating causal effects in the presence of unmeasured confounding under the standard iv model however the average treatment effect ate is only partially identifiable to address this we propose novel assumptions that allow for identification of the ate our identification assumptions are clearly separated from model assumptions needed for estimation so that researchers are not required to commit to a specific observed data model in establishing identification we then construct multiple estimators that are consistent under three different observed data models and multiply robust estimators that are consistent in the union of these observed data models we pay special attention to the case of binary outcomes for which we obtain bounded estimators of the ate that are guaranteed to lie between 1 and 1 our approaches are illustrated with simulations and a data analysis evaluating the causal effect of education on earnings

numerical solution for an inverse heat source problem by an iterative method abstract in this paper we consider a typical inverse heat source problem that is we determine two separable source terms in a heat equation from the initial and boundary data along with two additional measurements by a simple transformation the original problem is changed into a nonlinear problem and then we use an iterative method to solve it after giving an algorithm we prove some holder convergence rates for both the reconstructed heat source terms and the temperature distribution subject to certain bounds of the data numerical results show that our method is accurate and efficient

empirical likelihood based inference in missing response problems and its application in observational studies summary the problem of missing response data is ubiquitous in medical and social science studies in the case of responses that are missing at random depending on some covariate information analyses focused only on the complete data may lead to biased results various debias methods have been extensively studied in the literature particularly the weighting method that was motivated by horvitz and thompson s estimators to improve efficiency robins rotnitzky and zhao proposed augmented estimating equations based on corrected complete case analyses a nice feature of the augmented method is its double robustness i e the estimator that is derived from the augmented method is asymptotically unbiased if either the underlying missing data mechanism or the underlying regression function is correctly specified furthermore the augmented estimator can achieve full efficiency if both the missing data mechanism and the regression function are correctly specified in general however it is very difficult to specify the regression function correctly especially when the dimension of covariates is high this is the so called curse of dimensionality problem the augmented estimator has much lower efficiency if the working regression model is not close to the true regression model in this paper the empirical likelihood method is employed to seek a constrained empirical likelihood estimation of mean response with the assumption that responses are missing at random the empirical likelihood based estimators enjoy the double robustness property moreover it is possible that the empirical likelihood based inference can produce asymptotically unbiased and efficient estimators even if the true regression function is not completely known simulation results indicate that the empirical likelihood based estimators are very robust to a misspecification of the propensity score and dominate other competitors in the sense of having smaller mean square errors methods that are developed in this paper have a nice application in observational causal inferences the propensity score is used to adjust for differences in pretreatment variables in the estimation of average treatment effects

solving the inverse problem of identifying an unknown source term in a parabolic equation the inverse problem of determining an unknown source term in a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition is considered the series pattern solution of the problem is obtained by using the weighted homotopy analysis method wham a description of the method for solving the problem and identifying the unknown source term is derived finally several examples are presented to illustrate this method

bounded efficient and doubly robust estimation with inverse weighting consider estimating the mean of an outcome in the presence of missing data or estimating population average treatment effects in causal inference a doubly robust estimator remains consistent if an outcome regression model or a propensity score model is correctly specified we build on a previous nonparametric likelihood approach and propose new doubly robust estimators which have desirable properties in efficiency if the propensity score model is correctly specified and in boundedness even if the inverse probability weights are highly variable we compare the new and existing estimators in a simulation study and find that the robustified likelihood estimators yield overall the smallest mean squared errors

enhanced precision in the analysis of randomized trials with ordinal outcomes we present a general method for estimating the effect of a treatment on an ordinal outcome in randomized trials the method is robust in that it does not rely on the proportional odds assumption our estimator leverages information in prognostic baseline variables and has all of the following properties i it is consistent ii it is locally efficient iii it is guaranteed to have equal or better asymptotic precision than both the inverse probability weighted and the unadjusted estimators to the best of our knowledge this is the first estimator of the causal relation between a treatment and an ordinal outcome to satisfy these properties we demonstrate the estimator in simulations based on resampling from a completed randomized clinical trial of a new treatment for stroke we show potential gains of up to 39 in relative efficiency compared to the unadjusted estimator the proposed estimator could be a useful tool for analyzing randomized trials with ordinal outcomes since existing methods either rely on model assumptions that are untenable in many practical applications or lack the efficiency properties of the proposed estimator we provide r code implementing the estimator

source term identification in 1 d ihcp we propose stable numerical solutions for the simultaneous identification of temperature temperature gradient and general source terms in the one dimensional inverse heat conduction problem ihcp the numerical solution consists of a regularization procedure based on the mollification method and a marching scheme for the solution of the stabilized problem the stability error analysis and implementation of the algorithm are presented together with a set of numerical results

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

a non overlap domain decomposition method for the forward backward heat equation a non overlap domain decomposition method is presented for the numerical solution of the forward backward heat equation the convergence of the given method is established the numerical experiments show that the given domain decomposition method is very effective

variational iteration method for solving burger s and coupled burger s equations by means of variational iteration method the solutions of burger s equation and coupled burger s equations are exactly obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

numerical solutions of nonlinear evolution equations using variational iteration method the variational iteration method is used to solve three kinds of nonlinear partial differential equations coupled nonlinear reaction diffusion equations hirota satsuma coupled kdv system and drinefel d sokolov wilson equations numerical solutions obtained by the variational iteration method are compared with the exact solutions revealing that the obtained solutions are of high accuracy he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomial in adomian method the method is straightforward and concise and it can also be applied to other nonlinear evolution equations in mathematical physics

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

a modified quasi boundary value method for ill posed problems in this paper we study a final value problem for first order abstract differential equation with positive self adjoint unbounded operator coefficient this problem is ill posed perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter we show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution we also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions finally we give explicit convergence rates

sharp nonparametric bounds and randomization inference for treatment effects on an ordinal outcome in clinical research investigators are interested in inferring the average causal effect of a treatment however the causal parameter that can be used to derive the average causal effect is not well defined for ordinal outcomes although some definitions have been proposed they are limited in that they are not identical to the well defined causal risk for a binary outcome which is the simplest ordinal outcome in this paper we propose the use of a causal parameter for an ordinal outcome defined as the proportion that a potential outcome under one treatment condition would not be smaller than that under the other condition for a binary outcome this proportion is identical to the causal risk unfortunately the proposed causal parameter cannot be identified even under randomization therefore we present a numerical method to calculate the sharp nonparametric bounds within a sample reflecting the impact of confounding when the assumption of independent potential outcomes is included the causal parameter can be identified when randomization is in play then we present exact tests and the associated confidence intervals for the relative treatment effect using the randomization based approach which are an extension of the existing methods for a binary outcome our methodologies are illustrated using data from an emetic prevention clinical trial

numerical solution of the inverse problem of determining an unknown source term in a two dimensional heat equation a numerical procedure for an inverse problem of determination of unknown source term in two dimensional parabolic equation is presented the approach of the proposed method is to approximate unknown function by a piecewise linear function whose coefficients are determined from the solution of minimisation problem based on the overspecified data some numerical examples are presented

application of the variational iteration method to inverse heat source problems this paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions the numerical solution is developed by using the variational iteration method this method is based on the use of lagrange multipliers for the identification of optimal values of parameters in a functional using this method a rapid convergent sequence can be obtained which tends to the exact solution of the problem furthermore the variational iteration method does not require the discretization of the problem thus the variational iteration method is suitable for finding the approximation of the solution without discretization of the problem two numerical examples are presented to illustrate the strength of the method

improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data considerable recent interest has focused on doubly robust estimators for a population mean response in the presence of incomplete data which involve models for both the propensity score and the regression of outcome on covariates the usual doubly robust estimator may yield severely biased inferences if neither of these models is correctly specified and can exhibit nonnegligible bias if the estimated propensity score is close to zero for some observations we propose alternative doubly robust estimators that achieve comparable or improved performance relative to existing methods even with some estimated propensity scores close to zero

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

on the existence of maximum likelihood estimates in logistic regression models summary the problems of existence uniqueness and location of maximum likelihood estimates in log linear models have received special attention in the literature haberman 1974 chapter 2 wedderburn 1976 silvapulle 1981 for multinomial logistic regression models we prove existence theorems by considering the possible patterns of data points which fall into three mutually exclusive and exhaustive categories complete separation quasicomplete separation and overlap our results suggest general rules for identifying infinite parameter estimates in log linear models for frequency tables

estimation of conditional and marginal odds ratios using the prognostic score introduced by hansen in 2008 the prognostic score pgs has been presented as the prognostic analogue of the propensity score pps pps based methods are intended to estimate marginal effects most previous studies evaluated the performance of existing pgs based methods adjustment stratification and matching using the pgs in situations in which the theoretical conditional and marginal effects are equal i e collapsible situations to support the use of pgs framework as an alternative to the pps framework applied researchers must have reliable information about the type of treatment effect estimated by each method we propose four new pgs based methods each developed to estimate a specific type of treatment effect we evaluated the ability of existing and new pgs based methods to estimate the conditional treatment effect cte the marginal average treatment effect on the whole population ate and the marginal average treatment effect on the treated population att when the odds ratio a non collapsible estimator is the measure of interest the performance of pgs based methods was assessed by monte carlo simulations and compared with pps based methods and multivariate regression analysis existing pgs based methods did not allow for estimating the ate and showed unacceptable performance when the proportion of exposed subjects was large when estimating marginal effects pps based methods were too conservative whereas the new pgs based methods performed better with low prevalence of exposure and had coverages closer to the nominal value when estimating cte the new pgs based methods performed as well as traditional multivariate regression copyright 2016 john wiley sons ltd

estimating causal effects for multivalued treatments a comparison of approaches interventions with multivalued treatments are common in medical and health research such as when comparing the efficacy of competing drugs or interventions or comparing between various doses of a particular drug in recent years there has been a growing interest in the development of multivalued treatment effect estimators using observational data in this paper we compare the performance of commonly used regression based methods that estimate multivalued treatment effects based on the unconfoundedness assumption these estimation methods fall into three general categories i estimators based on a model for the outcome variable using conventional regression adjustment ii weighted estimators based on a model for the treatment assignment and iii doubly robust estimators that model both the treatment assignment and outcome variable within the same framework we assess the performance of these models using monte carlo simulation and demonstrate their application with empirical data our results show that i when models estimating both the treatment and outcome are correctly specified all adjustment methods provide similar unbiased estimates ii when the outcome model is misspecified regression adjustment performs poorly while all the weighting methods provide unbiased estimates iii when the treatment model is misspecified methods based solely on modeling the treatment perform poorly while regression adjustment and the doubly robust models provide unbiased estimates and iv when both the treatment and outcome models are misspecified all methods perform poorly given that researchers will rarely know which of the two models is misspecified our results support the use of doubly robust estimation

variational iteration method for solving two point boundary value problems variational iteration method is introduced to solve two point boundary value problems numerical results demonstrate that the method is promising and may overcome the difficulty arising in adomian decomposition method

semiparametric regression estimation in the presence of dependent censoring summary we propose a semiparametric estimation procedure for estimating the regression of an outcome y measured at the end of a fixed follow up period on baseline explanatory variables x measured prior to start of follow up in the presence of dependent censoring given x the proposed estimators are consistent when the data are missing at random but not missing completely at random rubin 1976 and do not require full specification of the complete data likelihood specifically we assume that the probability of censoring at time t is independent of the outcome y conditional on the recorded history up to t of a vector of time dependent covariates that are correlated with y our estimators can be used to adjust for dependent censoring and nonrandom noncompliance in randomised trials studying the effect of a treatment on the mean of a response variable of interest even with independent censoring our methods allow the investigator to increase efficiency by exploiting the correlation of the outcome with a vector of time dependent covariates

nonhomogeneous heat equation identification and regularization for the inhomogeneous term we study the nonhomogeneous heat equation under the form ut uxx t f x where the unknown is the pair of functions u f under various assumptions about the function and the final value in t 1 i e g x we propose different regularizations on this ill posed problem based on the fourier transform associated with a lebesgue measure for 0 the solution is unique

an a posteriori parameter choice rule for the truncation regularization method for solving backward parabolic problems the goal of this paper is to investigate a backward parabolic equation with time dependent coefficient by the truncation method tm the key point to our paper is that we give an a posteriori choice of regularization parameter for the tm with which the convergence estimates between the exact solution and the regularized approximation are obtained numerical implementation sheds light on the accuracy and efficiency of the proposed method

a theory of statistical models for monte carlo integration summary the task of estimating an integral by monte carlo methods is formulated as a statistical model using simulated observations as data the difficulty in this exercise is that we ordinarily have at our disposal all of the information required to compute integrals exactly by calculus or numerical integration but we choose to ignore some of the information for simplicity or computational feasibility our proposal is to use a semiparametric statistical model that makes explicit what information is ignored and what information is retained the parameter space in this model is a set of measures on the sample space which is ordinarily an infinite dimensional object none the less from simulated data the base line measure can be estimated by maximum likelihood and the required integrals computed by a simple formula previously derived by vardi and by lindsay in a closely related model for biased sampling the same formula was also suggested by geyer and by meng and wong using entirely different arguments by contrast with geyer s retrospective likelihood a correct estimate of simulation error is available directly from the fisher information the principal advantage of the semiparametric model is that variance reduction techniques are associated with submodels in which the maximum likelihood estimator in the submodel may have substantially smaller variance than the traditional estimator the method is applicable to markov chain and more general monte carlo sampling schemes with multiple samplers

doubly robust inference for targeted minimum loss based estimation in randomized trials with missing outcome data missing outcome data is a crucial threat to the validity of treatment effect estimates from randomized trials the outcome distributions of participants with missing and observed data are often different which increases bias causal inference methods may aid in reducing the bias and improving efficiency by incorporating baseline variables into the analysis in particular doubly robust estimators incorporate 2 nuisance parameters the outcome regression and the missingness mechanism ie the probability of missingness conditional on treatment assignment and baseline variables to adjust for differences in the observed and unobserved groups that can be explained by observed covariates to consistently estimate the treatment effect one of these nuisance parameters must be consistently estimated traditionally nuisance parameters are estimated using parametric models which often precludes consistency particularly in moderate to high dimensions recent research on missing data has focused on data adaptive estimation to help achieve consistency but the large sample properties of such methods are poorly understood in this article we discuss a doubly robust estimator that is consistent and asymptotically normal under data adaptive estimation of the nuisance parameters we provide a formula for an asymptotically exact confidence interval under minimal assumptions we show that our proposed estimator has smaller finite sample bias compared to standard doubly robust estimators we present a simulation study demonstrating the enhanced performance of our estimators in terms of bias efficiency and coverage of the confidence intervals we present the results of an illustrative example a randomized double blind phase 2 3 trial of antiretroviral therapy in hiv infected persons

analysis of semiparametric regression models for repeated outcomes in the presence of missing data abstract we propose a class of inverse probability of censoring weighted estimators for the parameters of models for the dependence of the mean of a vector of correlated response variables on a vector of explanatory variables in the presence of missing response data the proposed estimators do not require full specification of the likelihood they can be viewed as an extension of generalized estimating equations estimators that allow for the data to be missing at random but not missing completely at random these estimators can be used to correct for dependent censoring and nonrandom noncompliance in randomized clinical trials studying the effect of a treatment on the evolution over time of the mean of a response variable the likelihood based parametric g computation algorithm estimator may also be used to attempt to correct for dependent censoring and nonrandom noncompliance but because of possible model misspecification the parametric g computation algorithm estimator in contrast with the proposed w

doubly robust estimation of generalized partial linear models for longitudinal data with dropouts we develop a doubly robust estimation of generalized partial linear models for longitudinal data with dropouts our method extends the highly efficient aggregate unbiased estimating function approach proposed in qu et al 2010 to a doubly robust one in the sense that under missing at random mar our estimator is consistent when either the linear conditional mean condition is satisfied or a model for the dropout process is correctly specified we begin with a generalized linear model for the marginal mean and then move forward to a generalized partial linear model allowing for nonparametric covariate effect by using the regression spline smoothing approximation we establish the asymptotic theory for the proposed method and use simulation studies to compare its finite sample performance with that of qu s method the complete case generalized estimating equation gee and the inverse probability weighted gee the proposed method is finally illustrated using data from a longitudinal cohort study

variational iteration method some recent results and new interpretations this paper is an elementary introduction to the concepts of variational iteration method first the main concepts in variational iteration method such as general lagrange multiplier restricted variation correction functional are explained heuristically subsequently the solution procedure is systematically addressed in particular for nonlinear oscillators particular attention is paid throughout the paper to give an intuitive grasp for the method the main motivation is to put things together in a convenient form for later reference and systematic use

issues for covariance analysis of dichotomous and ordered categorical data from randomized clinical trials and non parametric strategies for addressing them analysis of covariance is an effective method for addressing two considerations for randomized clinical trials one is reduction of variance for estimates of treatment effects and thereby the production of narrower confidence intervals and more powerful statistical tests the other is the clarification of the magnitude of treatment effects through adjustment of corresponding estimates for any random imbalances between the treatment groups with respect to the covariables the statistical basis of covariance analysis can be either non parametric with reliance only on the randomization in the study design or parametric through a statistical model for a postulated sampling process for non parametric methods there are no formal assumptions for how a response variable is related to the covariables but strong correlation between response and covariables is necessary for variance reduction computations for these methods are straightforward through the application of weighted least squares to fit linear models to the differences between treatment groups for the means of the response variable and the covariables jointly with a specification that has null values for the differences that correspond to the covariables moreover such analysis is similarly applicable to dichotomous indicators ranks or integers for ordered categories and continuous measurements since non parametric covariance analysis can have many forms the ones which are planned for a clinical trial need careful specification in its protocol a limitation of non parametric analysis is that it does not directly address the magnitude of treatment effects within subgroups based on the covariables or the homogeneity of such effects for this purpose a statistical model is needed when the response criterion is dichotomous or has ordered categories such a model may have a non linear nature which determines how covariance adjustment modifies results for treatment effects insight concerning such modifications can be gained through their evaluation relative to non parametric counterparts such evaluation usually indicates that alternative ways to compare treatments for a response criterion with adjustment for a set of covariables mutually support the same conclusion about the strength of treatment effects this robustness is noteworthy since the alternative methods for covariance analysis have substantially different rationales and assumptions since findings can differ in important ways across alternative choices for covariables as opposed to methods for covariance adjustment the critical consideration for studies with covariance analyses planned as the primary method for comparing treatments is the specification of the covariables in the protocol or in an amendment or formal plan prior to any unmasking of the study

a modified method for a backward heat conduction problem we consider a backward heat conduction problem in a strip where data is given at the final time t t t 0 and a solution for 0 t t is sought the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data in order to numerically solve the problem we study a modification of the equation where a third order mixed derivative term is added error estimates for this problem are given which show that the modified problem is stable and its solution is an approximation of the backward heat conduction problem some numerical tests illustrate that the proposed method is feasible and effective c 2006 elsevier inc all rights reserved

nonparametric regression with missing outcomes using weighted kernel estimating equations we consider nonparametric regression of a scalar outcome on a covariate when the outcome is missing at random mar given the covariate and other observed auxiliary variables we propose a class of augmented inverse probability weighted aipw kernel estimating equations for nonparametric regression under mar we show that aipw kernel estimators are consistent when the probability that the outcome is observed that is the selection probability is either known by design or estimated under a correctly specified model in addition we show that a specific aipw kernel estimator in our class that employs the fitted values from a model for the conditional mean of the outcome given covariates and auxiliaries is double robust that is it remains consistent if this model is correctly specified even if the selection probabilities are modeled or specified incorrectly furthermore when both models happen to be right this double robust estimator attains the smallest possible asymptotic variance of all aipw kernel

on the role of the propensity score in efficient semiparametric estimation of average treatment effects in this paper the role of the propensity score in the efficient estimation of average treatment effects is examined under the assumption that the treatment is ignorable given some observed characteristics it is shown that the propensity score is ancillary for estimation of the average treatment effects the propensity score is not ancillary for estimation of average treatment effects on the treated it is suggested that the marginal value of the propensity score lies entirely in the dimension reduction efficient semiparametric estimators of average treatment effects and average treatment effects on the treated are shown to take the form of relevant sample averages of the data completed by the nonparametric imputation method it is shown that the projection on the propensity score is not necessary for efficient semiparametric estimation of average treatment effects on the treated even if the propensity score is known an application to the experimental data reveals that conditioning on the propensity score may even result in a loss of efficiency

regularization for ill posed cauchy problems associated with generators of analytic semigroups this paper is concerned with the ill posed cauchy problem associated with a densely defined linear operator a in a banach space our main result is that if a is the generator of an analytic semigroup then there exists a family of regularizing operators for such an ill posed cauchy problem by using the quasi reversibility method fractional powers and semigroups of linear operators the applications to ill posed partial differential equations are also given

application of the restrained optimal perturbation method to study the backward heat conduction problem in this paper a restrained optimal perturbation method is firstly proposed to solve the backward heat conduction problem the initial temperature distribution will be identified from the overspecified data a regularization term is introduced in the objective functional for overcoming the ill posedness of this problem spectral projected gradient algorithm is used to solve the optimal problem and we give the sensitivity analysis of the initial value the results of numerical experiments are also presented

a note on overadjustment in inverse probability weighted estimation standardized means commonly used in observational studies in epidemiology to adjust for potential confounders are equal to inverse probability weighted means with inverse weights equal to the empirical propensity scores more refined standardization corresponds with empirical propensity scores computed under more flexible models unnecessary standardization induces efficiency loss however according to the theory of inverse probability weighted estimation propensity scores estimated under more flexible models induce improvement in the precision of inverse probability weighted means this apparent contradiction is clarified by explicitly stating the assumptions under which the improvement in precision is attained

estimation with missing data beyond double robustness summary we propose an estimator that is more robust than doubly robust estimators based on weighting completecasesusingweightsotherthaninverseprobabilitywhenestimatingthepopulationmean of a response variable subject to ignorable missingness we allow multiple models for both the propensity score and the outcome regression our estimator is consistent if any of the multiple models is correctly specified such multiple robustness against model misspecification is a significant improvement over double robustness which allows only one propensity score model and one outcome regression model our estimator attains the semiparametric efficiency bound when one propensity score model and one outcome regression model are correctly specified without requiring knowledge of which models are correct

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

splitting based conjugate gradient method for a multi dimensional linear inverse heat conduction problem in this paper we consider a multi dimensional inverse heat conduction problem with time dependent coefficients in a box which is well known to be severely ill posed by a variational method the gradient of the functional to be minimized is obtained by the aid of an adjoint problem and the conjugate gradient method with a stopping rule is then applied to this ill posed optimization problem to enhance the stability and the accuracy of the numerical solution to the problem we apply this scheme to the discretized inverse problem rather than to the continuous one the difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easy to solve one dimensional problems the numerical results provided by our method are very good and the techniques seem to be very promising

a non local boundary value problem method for parabolic equations backward in time abstract the ill posed parabolic equation backward in time u t a u 0 0 t t u t f subject to the constraint u 0 e with the positive self adjoint unbounded operator a and e 0 being given is regularized by the well posed non local boundary value problem u t a u 0 0 t t u 0 u t f 0 the error estimates of holder type of the regularized solutions are obtained these estimates improve the related results by mel nikova denche and bessila

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

variable selection for covariate adjusted semiparametric inference in randomized clinical trials extensive baseline covariate information is routinely collected on participants in randomized clinical trials and it is well recognized that a proper covariate adjusted analysis can improve the efficiency of inference on the treatment effect however such covariate adjustment has engendered considerable controversy as post hoc selection of covariates may involve subjectivity and may lead to biased inference whereas prior specification of the adjustment may exclude important variables from consideration accordingly how to select covariates objectively to gain maximal efficiency is of broad interest we propose and study the use of modern variable selection methods for this purpose in the context of a semiparametric framework under which variable selection in modeling the relationship between outcome and covariates is separated from estimation of the treatment effect circumventing the potential for selection bias associated with standard analysis of covariance methods we demonstrate that such objective variable selection techniques combined with this framework can identify key variables and lead to unbiased and efficient inference on the treatment effect a critical issue in finite samples is validity of estimators of uncertainty such as standard errors and confidence intervals for the treatment effect we propose an approach to estimation of sampling variation of estimated treatment effect and show its superior performance relative to that of existing methods copyright 2012 john wiley sons ltd

on estimating regression based causal effects using sufficient dimension reduction summary in many causal inference problems the parameter of interest is the regression causal effect defined as the conditional mean difference in the potential outcomes given covariates in this paper we discuss how sufficient dimension reduction can be used to aid causal inference and we propose a new estimator of the regression causal effect inspired by minimum average variance estimation the estimator requires a weaker common support condition than propensity score based approaches and can be used to estimate the average causal effect for which it is shown to be asymptotically super efficient its finite sample properties are illustrated by simulation

a backward parabolic equation with a time dependent coefficient regularization and error estimates we consider the problem of determining the temperature u x t for x t 0 p x 0 t in the parabolic equation with a time dependent coefficient this problem is severely ill posed i e the solution if it exists does not depend continuously on the given data in this paper we use a modified method for regularizing the problem and derive an optimal stability estimation a numerical experiment is presented for illustrating the estimate

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

fourth order integro differential equations using variational iteration method in this paper the variational iteration method proposed by ji huan he is applied to solve both linear and nonlinear boundary value problems for fourth order integro differential equations the numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the variational iteration method is of high accuracy more convenient and efficient for solving integro differential equations

improved double robust estimation in missing data and causal inference models recently proposed double robust estimators for a population mean from incomplete data and for a finite number of counterfactual means can have much higher efficiency than the usual double robust estimators under misspecification of the outcome model in this paper we derive a new class of double robust estimators for the parameters of regression models with incomplete cross sectional or longitudinal data and of marginal structural mean models for cross sectional data with similar efficiency properties unlike the recent proposals our estimators solve outcome regression estimating equations in a simulation study the new estimator shows improvements in variance relative to the standard double robust estimator that are in agreement with those suggested by asymptotic theory

doubly robust nonparametric inference on the average treatment effect doubly robust estimators are widely used to draw inference about the average effect of a treatment such estimators are consistent for the effect of interest if either one of two nuisance parameters is consistently estimated however if flexible data adaptive estimators of these nuisance parameters are used double robustness does not readily extend to inference we present a general theoretical study of the behaviour of doubly robust estimators of an average treatment effect when one of the nuisance parameters is inconsistently estimated we contrast different methods for constructing such estimators and investigate the extent to which they may be modified to also allow doubly robust inference we find that while targeted minimum loss based estimation can be used to solve this problem very naturally common alternative frameworks appear to be inappropriate for this purpose we provide a theoretical study and a numerical evaluation of the alternatives considered our simulations highlight the need for and usefulness of these approaches in practice while our theoretical developments have broad implications for the construction of estimators that permit doubly robust inference in other problems