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

perturbation selection and influence measures in local influence analysis cook s j roy statist soc ser b 48 1986 133 169 local influence approach based on normal curvature is an important diagnostic tool for assessing local influence of minor perturbations to a statistical model however no rigorous approach has been developed to address two fundamental issues the selection of an appropriate perturbation and the development of influence measures for objective functions at a point with a nonzero first derivative the aim of this paper is to develop a differential geometrical framework of a perturbation model called the perturbation manifold and utilize associated metric tensor and affine curvatures to resolve these issues we will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model moreover we will introduce new influence measures that are applicable to objective functions at any point examples including linear regression models and linear mixed models are examined to demonstrate the effectiveness of using new influence measures for the identification of influential observations

bayesian variable selection in linear regression abstract this article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable it is based on a bayesian approach intended to be as objective as possible a probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model the method is not fully bayesian however because the ultimate choice of prior distribution from this family is affected by the data it is assumed that the predictors represent distinct observables the corresponding regression coefficients are assigned independent prior distributions for each regression coefficient subject to deletion from the model the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere that is a spike and slab distribution the random error component is assigned a normal distribution with mean 0 and standard deviation

semiparametric methods for response selective and missing data problems in regression summary suppose that data are generated according to the model f ylx 0 g x where y is a response and x are covariates we derive and compare semiparametric likelihood and pseudolikelihood methods for estimating 0 for situations in which units generated are not fully observed and in which it is impossible or undesirable to model the covariate distribution the probability that a unit is fully observed may depend on y and there may be a subset of covariates which is observed only for a subsample of individuals our key assumptions are that the probability that a unit has missing data depends only on which of a finite number of strata that y x belongs to and that the stratum membership is observed for every unit applications include case control studies in epidemiology field reliability studies and broad classes of missing data and measurement error problems our results make fully efficient estimation of 0 feasible and they generalize and provide insight into a variety of methods that have been proposed for specific problems

vibration isolation from irregularity in a nearly periodic structure theory and measurements this article describes the theory and a simple experiment carried out to demonstrate the phenomenon of anderson localization in an acoustical context this is an effect whereby the propagation of vibration in a structure which is not entirely regular is impeded by the irregularities giving rise on the average to an exponential decay of vibration level away from the driving point even in the absence of any dissipation the structure used in the experiment was a stretched string with masses attached to it this string was studied with regular spacings of the masses and after the masses had been moved in a controlled way to provide a small degree of irregularity in both cases the transmission of energy from end to end of the string was measured as a function of frequency and also the mode shapes in the second and fourth passbands were measured so as to demonstrate the underlying physics of the localization phenomenon in which the individual modes making up each passband change from being extended throughout the structure in the regular case to being localized in specific areas of the structure in the presence of irregularity all measurements yielded satisfactory agreement with the theoretical predictions

semiparametric regression for repeated outcomes with nonignorable nonresponse abstract we consider inference about the parameter indexing the conditional mean of a vector of correlated outcomes given a vector of explanatory variables when some of the outcomes are missing in a subsample of the study and the probability of response depends on both observed and unobserved data values that is nonresponse is nonignorable we propose a class of augmented inverse probability of response weighted estimators that are consistent and asymptotically normal can for estimating when the response probabilities can be parametrically modeled and a can estimator exists the proposed estimators do not require full specification of a parametric likelihood and their computation does not require numerical integration our estimators can be viewed as an extension of generalized estimating equation estimators that allows for nonignorable nonresponse we show that our class essentially consists of all can estimators of we also show that the asymptotic variance of the optimal estimator in our

semiparametric efficiency in multivariate regression models with missing data abstract we consider the efficiency bound for the estimation of the parameters of semiparametric models defined solely by restrictions on the means of a vector of correlated outcomes y when the data on y are missing at random we show that the semiparametric variance bound is the asymptotic variance of the optimal estimator in a class of inverse probability of censoring weighted estimators and that this bound is unchanged if the data are missing completely at random for this case we study the asymptotic performance of the generalized estimating equations gee estimators of mean parameters and show that the optimal gee estimator is inefficient except for special cases the optimal weighted estimator depends on unknown population quantities but for monotone missing data we propose an adaptive estimator whose asymptotic variance can achieve the bound

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

pattern mixture models for multivariate incomplete data consider a random sample on variables x1 xv with some values of xv missing selection models specify the distribution of x1 xv over respondents and nonrespondents to xv and the conditional distribution that xv is missing given x1 xv in contrast pattern mixture models specify the conditional distribution of x 1 xv given that xv is observed or missing respectively and the marginal distribution of the binary indicator for whether or not xv is missing for multivariate data with a general pattern of missing values the literature has tended to adopt the selection modeling approach see for example little and rubin here pattern mixture models are proposed for this more general problem pattern mixture models are chronically underidentified in particular for the case of univariate nonresponse mentioned above there are no data on the distribution of xv given x1 xv 1 in the stratum with xv missing thus the models require restrictions or prior information to identify the paramet

prior elicitation variable selection and bayesian computation for logistic regression models summary bayesian selection of variables is often difficult to carry out because of the challenge in specifying prior distributions for the regression parameters for all possible models specifying a prior distribution on the model space and computations we address these three issues for the logistic regression model for the first we propose an informative prior distribution for variable selection several theoretical and computational properties of the prior are derived and illustrated with several examples for the second we propose a method for specifying an informative prior on the model space and for the third we propose novel methods for computing the marginal distribution of the data the new computational algorithms only require gibbs samples from the full model to facilitate the computation of the prior and posterior model probabilities for all possible models several properties of the algorithms are also derived the prior specification for the first challenge focuses on the observables in that the elicitation is based on a prior prediction yo for the response vector and a quantity ao quantifying the uncertainty in yo then yo and ao are used to specify a prior for the regression coefficients semi automatically examples using real data are given to demonstrate the methodology

estimation for zero inflated beta binomial regression model with missing response data discrete data in the form of proportions with overdispersion and zero inflation can arise in toxicology and other similar fields in regression analysis of such data another problem that also may arise in practice is that some responses may be missing in this paper we develop estimation procedure for the parameters of a zero inflated overdispersed binomial model in the presence of missing responses under three different missing data mechanisms a weighted expectation maximization algorithm is used for the maximum likelihood estimation of the parameters involved extensive simulations are conducted to study the properties of the estimates in terms of average of estimates relative bias variance mean squared error and coverage probability of estimates simulations show much superior properties of the estimates obtained using the weighted expectation maximization algorithm some illustrative examples and a discussion are given

log linear analysis of censored survival data with partially observed covariates abstract log linear models provide a flexible means of extending life table techniques for the analysis of censored survival data with categorical covariates as discussed by holford 1980 and laird and olivier 1981 we extend this methodology to incorporate cases in which one or more of the categorical covariates are sometimes missing maximum likelihood estimates of the parameters are calculated using data from all cases this can result in large gains in efficiency over standard methods that require the exclusion of cases with incomplete data with this approach we assume that the hazard function conditional on the covariates is a stepwise function over disjoint intervals of time the model has two parts a log linear model describing the hazard parameters and a multinomial model describing the probabilities in the contingency table defined by the covariates the main interest is in the model for the hazard parameters we show how to calculate maximum likelihood estimates of parameters of the mo

a finite element method for the nonlinear tricomi problem we consider a finite element procedure for numerical solution of the nonlinear problem l u yu xx u yy r x y u f x y u in a simply connected regiong in thex y plane the boundary ofg consists of 0 1 and 2 and we impose the boundary condition u gamma 0 cup gamma 1 0 0 is assumed to be a piecewises smooth curve lying in the half planey 0 with endpointsa 1 0 andb 0 0 1 and 2 are characteristics of the operatorl issued froma andb which intersect at the pointc 1 2 y c an error analysis of the method is also given

a nonparametric method for dealing with mismeasured covariate data abstract mismeasurement of covariate data is a frequent problem in statistical data analysis however when true and mismeasured data are obtained for a subsample of the observations it is possible to estimate the parameters relating the outcome to the covariate of interest maximum likelihood methods that rely on parametric models for the mismeasurement have not met with much success realistic models for the mismeasurement process are difficult to construct the form of the likelihood is often intractable and more important such methods are not robust to model misspecification we propose an easily implemented method that is nonparametric with respect to the mismeasurement process and that is applicable when mismeasurement is due to the problem of missing data errors in variables or use of imperfect surrogate covariates specifically denote the outcome variable by y the covariate data subject to mismeasurement by x and the remaining covariates including perhaps surrogates or mismeasured values

multiple imputation of missing categorical and continuous values via bayesian mixture models with local dependence abstractwe present a nonparametric bayesian joint model for multivariate continuous and categorical variables with the intention of developing a flexible engine for multiple imputation of missing values the model fuses dirichlet process mixtures of multinomial distributions for categorical variables with dirichlet process mixtures of multivariate normal distributions for continuous variables we incorporate dependence between the continuous and categorical variables by 1 modeling the means of the normal distributions as component specific functions of the categorical variables and 2 forming distinct mixture components for the categorical and continuous data with probabilities that are linked via a hierarchical model this structure allows the model to capture complex dependencies between the categorical and continuous data with minimal tuning by the analyst we apply the model to impute missing values due to item nonresponse in an evaluation of the redesign of the survey of income and program partic

estimating propensity scores with missing covariate data using general location mixture models in many observational studies analysts estimate causal effects using propensity scores e g by matching sub classifying or inverse probability weighting based on the scores estimation of propensity scores is complicated when some values of the covariates are missing analysts can use multiple imputation to create completed data sets from which propensity scores can be estimated we propose a general location mixture model for imputations that assumes that the control units are a latent mixture of i units whose covariates are drawn from the same distributions as the treated units covariates and ii units whose covariates are drawn from different distributions this formulation reduces the influence of control units outside the treated units region of the covariate space on the estimation of parameters in the imputation model which can result in more plausible imputations in turn this can result in more reliable estimates of propensity scores and better balance in the true covariate distributions when matching or sub classifying we illustrate the benefits of the latent class modeling approach with simulations and with an observational study of the effect of breast feeding on children s cognitive abilities

proportional hazards regression with missing covariates abstract nonparametric maximum likelihood npml is used to estimate regression parameters in a proportional hazards regression model with missing covariates the npml estimator is shown to be consistent and asymptotically normally distributed under some conditions em type algorithms are applied to solve the maximization problem variance estimates of the regression parameters are obtained by a profile likelihood approach that uses em aided numerical differentiation simulation results indicate that the npml estimates of the regression parameters are more efficient than the approximate partial likelihood estimates and estimates from complete case analysis when missing covariates are missing completely at random and that the proposed method corrects for bias when the missing covariates are missing at random

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 pseudo bayesian shrinkage approach to regression with missing covariates we consider the linear regression of outcome y on regressors w and z with some values of w missing when our main interest is the effect of z on y controlling for w three common approaches to regression with missing covariates are i complete case analysis cc which discards the incomplete cases and ii ignorable likelihood methods which base inference on the likelihood based on the observed data assuming the missing data are missing at random rubin 1976b and iii nonignorable modeling which posits a joint distribution of the variables and missing data indicators another simple practical approach that has not received much theoretical attention is to drop the regressor variables containing missing values from the regression modeling dv for drop variables dv does not lead to bias when either i the regression coefficient of w is zero or ii w and z are uncorrelated we propose a pseudo bayesian approach for regression with missing covariates that compromises between the cc and dv estimates exploiting information in the incomplete cases when the data support dv assumptions we illustrate favorable properties of the method by simulation and apply the proposed method to a liver cancer study extension of the method to more than one missing covariate is also discussed

estimating equations with incomplete categorical covariates in the cox model incomplete covariate data is a common occurrence in many studies in which the outcome is survival time when a full likelihood is specified a useful technique for obtaining parameter estimates is the em algorithm we propose a set of estimating equations to estimate the parameters of cox s proportional hazards model when some covariate values are missing these estimating equations can be solved by an algorithm similar to the em algorithm because of the computational burden of finding a solution to these estimating equations we propose obtaining parameter estimates via monte carlo methods asymptotic variances of the parameter estimates are also derived we present a clinical trials example with three covariates two of which have some missing values

structure estimation for discrete graphical models generalized covariance matrices and their inverses we investigate the relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix we show that for certain graph structures the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph our work extends results that have previously been established only in the context of multivariate gaussian graphical models thereby addressing an open question about the significance of the inverse covariance matrix of a non gaussian distribution the proof exploits a combination of ideas from the geometry of exponential families junction tree theory and convex analysis these population level results have various consequences for graph selection methods both known and novel including a novel method for structure estimation for missing or corrupted observations we provide nonasymptotic guarantees for such methods and illustrate the sharpness of these predictions via simulations

a galerkin method for elliptic hyperbolic type equations a certain galerkin type method is proposed for the numerical solution of a class of second order linear mixed type partial differential equations convergence of such an approximation is proved in the whole region of definition error estimates are derived in a natural norm assuming smooth solutions numerical experiments are conducted using bicubic splines the obtained results show that solutions are approximated equally well in both elliptic and hyperbolic regions

a pole zero representation for the admittance matrix of elastic beam and plate systems the behavior of coupled elastic systems is often analyzed in terms of the admittances or impedances of the component systems in many applications the admittances are approximated by a small number of simple oscillators i e a low order truncation of the normal mode expansion such an approach is reasonable in the neighborhood of system resonances but is much less convenient and accurate in the neighborhood of antiresonances pole zero product expansions of the sort employed in circuit analysis offer a potential means of improving the accuracy of low order approximations the validity of such representations of the admittance matrix is explored here and it is noted that in general the off diagonal components cannot be written in this fashion nevertheless it is shown below that the entire admittance matrix of beam and plate strip systems can always be represented in the pole zero product form further the representation is shown to be valid even when the beam or plate is coupled to an arbitrary conservative elastic structure

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

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

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

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

the em algorithm an old folk song sung to a fast new tune celebrating the 20th anniversary of the presentation of the paper by dempster laird and rubin which popularized the em algorithm we investigate after a brief historical account strategies that aim to make the em algorithm converge faster while maintaining its simplicity and stability e g automatic monotone convergence in likelihood first we introduce the idea of a working parameter to facilitate the search for efficient data augmentation schemes and thus fast em implementations second summarizing various recent extensions of the em algorithm we formulate a general alternating expectation conditional maximization algorithm aecm that couples flexible data augmentation schemes with model reduction schemes to achieve efficient computations we illustrate these methods using multivariate t models with known or unknown degrees of freedom and poisson models for image reconstruction we show through both empirical and theoretical evidence the potential for a dramatic reduction in computational time with little increase in human effort we also discuss the intrinsic connection between em type algorithms and the gibbs sampler and the possibility of using the techniques presented here to speed up the latter the main conclusion of the paper is that with the help of statistical considerations it is possible to construct algorithms that are simple stable and fast

parameter expansion to accelerate em the px em algorithm summary the em algorithm and its extensions are popular tools for modal estimation but ar e often criticised for their slow convergence we propose a new method that can often make em much faster the intuitive idea is to use a covariance adjustment to correct the analysis of the m step capitalising on extra information captured in the imputed complete data the way we accomplish this is by parameter expansion we expand the complete data model while preserving the observed data model and use the expanded complete data model to generate em this parameter expanded ei m px em algorithm shares the simplicity and stability of ordinary em but has a faster rate of convergence since its m step performs a more efficient analysis the px em algorithm is illustrated for the multivariate t distribution a random effects model factor analysis probit regression and a poisson imaging model

estimating a structured covariance matrix from multilab measurements in high throughput biology we consider the problem of quantifying the degree of coordination between transcription and translation in yeast several studies have reported a surprising lack of coordination over the years in organisms as different as yeast and humans using diverse technologies however a close look at this literature suggests that the lack of reported correlation may not reflect the biology of regulation these reports do not control for between study biases and structure in the measurement errors ignore key aspects of how the data connect to the estimand and systematically underestimate the correlation as a consequence here we design a careful meta analysis of 27 yeast datasets supported by a multilevel model full uncertainty quantification a suite of sensitivity analyses and novel theory to produce a more accurate estimate of the correlation between mrna and protein levels a proxy for coordination from a statistical perspective this problem motivates new theory on the impact of noise model misspecif

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

a missing composite covariate in survival analysis a case study of the chinese longitudinal health and longevity survey we estimate a cox proportional hazards model where one of the covariates measures the level of a subject s cognitive functioning by grading the total score obtained by the subject on the items of a questionnaire a case study is presented where the sample includes partial respondents who did not answer some questionnaire items the total score takes hence the form of an interval censored variable and as a result the level of cognitive functioning is missing on some subjects we handle the partial respondents by taking a likelihood based approach where survival time is jointly modelled with the censored total score and the size of the censoring interval estimates are obtained by an e m type algorithm that reduces to the iterative maximization of three complete log likelihood functions derived from two augmented data sets with case weights alternated with weights updating this methodology is exploited to assess the mini mental state examination index as a prognostic factor of survival in a sample of chinese older adults

local sensitivity approximations for selectivity bias summary observational data analysis is often based on tacit assumptions of ignorability or randomness the paper develops a general approach to local sensitivity analysis for selectivity bias which aims to study the sensitivity of inference to small departures from such assumptions if m is a model assuming ignorability we surround m by a small neighbourhood at defined in the sense of kullback leibler divergence and then compare the inference for models in iv with that for m interpretable bounds for such differences are developed applications to missing data and to observational comparisons are discussed local approximations to sensitivity analysis are model robust and can be applied to a wide range of statistical problems

acoustic radiation from fluid loaded infinite circular cylinders with doubly periodic ring supports analytic expressions are derived for the farfield acoustic radiation from a point driven fluid loaded circular cylindrical shell reinforced with doubly periodic ring supports the ring supports interact with the cylindrical shell only through normal forces the model includes the effects of fluid loading and interactions between supports predictions are compared to measured farfield acoustic radiation data taken on a finite circular cylindrical shell with doubly periodic ring supports characteristics of the acoustic radiation are examined by example and some physical interpretation of significant features are offered

two alternative expressions for the spherical wave expansion of the time domain scalar free space green s function and an application scattering by a soft sphere the importance of expanding green s functions particularly free space green s functions in terms of orthogonal wave functions is practically self evident when frequency domain scattering problems are of interest with the relatively recent and widespread interest in time domain scattering problems similar expansions of green s functions are expected to be useful in the time domain in this paper two alternative expressions expanded in terms of orthogonal spherical wave functions for the free space time domain scalar green s functions are presented although the two expressions are equivalent one of them is seen to be more convenient for the calculation of the scattered field for a known equivalent source density whereas the second expression is more suitable for setting up an integral equation for the equivalent source density such an integral equation may be setup for example by the application of a time domain equivalent of the t matrix extended boundary condition method

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

regression with missing x s a review abstract the literature of regression analysis with missing values of the independent variables is reviewed six classes of procedures are distinguished complete case analysis available case methods least squares on imputed data maximum likelihood bayesian methods and multiple imputation methods are compared and illustrated when missing data are confined to one independent variable and extensions to more general patterns are indicated attention is paid to the performance of methods when the missing data are not missing completely at random least squares methods that fill in missing x s using only data on the x s are contrasted with likelihood based methods that use data on the x s and y the latter approach is preferred and provides methods for elaboration of the basic normal linear regression model it is suggested that more widely distributed software is needed that advances beyond complete case analysis available case analysis and naive imputation methods bayesian simulation methods and mu

double semiparametric method for missing covariates in cox regression models the problem of nuisance covariate model specification is considered in cox regression where the maximum semiparametric likelihood method is used to handle the missing covariates a component of the covariates is modeled nonparametrically to achieve robustness against covariate model misspecification and to reduce the number of possibly intractable integrations involved in the parametric modeling of the covariates the statistical properties of the proposed method are examined it is found that in some important situations the maximum semiparametric likelihood can be applied without making any additional parametric model assumptions on covariates the proposed method can yield a more efficient estimator than the nonparametric imputation methods and does not require specification of the missingness mechanism when compared with the inverse probability weighting method a real data example is analyzed to demonstrate use of the proposed method

local influence for incomplete data models this paper proposes a method to assess the local influence in a minor perturbation of a statistical model with incomplete data the idea is to utilize cook s approach to the conditional expectation of the complete data log likelihood function in the em algorithm it is shown that the method proposed produces analytic results that are very similar to those obtained from a classical local influence approach based on the observed data likelihood function and has the potential to assess a variety of complicated models that cannot be handled by existing methods an application to the generalized linear mixed model is investigated some illustrative artificial and real examples are presented

missing covariates in generalized linear models when the missing data mechanism is non ignorable we propose a method for estimating parameters in generalized linear models with missing covariates and a non ignorable missing data mechanism we use a multinomial model for the missing data indicators and propose a joint distribution for them which can be written as a sequence of one dimensional conditional distributions with each one dimensional conditional distribution consisting of a logistic regression we allow the covariates to be either categorical or continuous the joint covariate distribution is also modelled via a sequence of one dimensional conditional distributions and the response variable is assumed to be completely observed we derive the e and m steps of the em algorithm with non ignorable missing covariate data for categorical covariates we derive a closed form expression for the e and m steps of the em algorithm for obtaining the maximum likelihood estimates mles for continuous covariates we use a monte carlo version of the em algorithm to obtain the mles via the gibbs sampler computational techniques for gibbs sampling are proposed and implemented the parametric form of the assumed missing data mechanism itself is not testable from the data and thus the non ignorable modelling considered here can be viewed as a sensitivity analysis concerning a more complicated model therefore although a model may have passed the tests for a certain missing data mechanism this does not mean that we have captured even approximately the correct missing data mechanism hence model checking for the missing data mechanism and sensitivity analyses play an important role in this problem and are discussed in detail several simulations are given to demonstrate the methodology in addition a real data set from a melanoma cancer clinical trial is presented to illustrate the methods proposed

bayesian inference for multivariate meta regression with a partially observed within study sample covariance matrix multivariate meta regression models are commonly used in settings where the response variable is naturally multidimensional such settings are common in cardiovascular and diabetes studies where the goal is to study cholesterol levels once a certain medication is given in this setting the natural multivariate endpoint is low density lipoprotein cholesterol ldl c high density lipoprotein cholesterol hdl c and triglycerides tg ldl c hdl c tg in this article we examine study level aggregate multivariate meta data from 26 merck sponsored double blind randomized active or placebo controlled clinical trials on adult patients with primary hypercholesterolemia our goal is to develop a methodology for carrying out bayesian inference for multivariate meta regression models with study level data when the within study sample covariance matrix s for the multivariate response data is partially observed specifically the proposed methodology is based on postulating a multivariate random effects re

analysis of multivariate missing data with nonignorable nonresponse we consider multivariate regression analysis with missing data in the outcome variables when the nonresponse mechanism depends on the underlying values of the responses and hence is nonignorable related problems include response biased sampling where data are sampled with probability depending only on the univariate response our methods do not require specification of the form of the nonresponse mechanism we show that under certain regularity conditions all the regression parameters can be identified from a conditional likelihood based on the complete cases if the marginal distribution of the covariates is known if the marginal distribution of the covariates is estimated from the data then the regression parameters are identified from a pseudolikelihood resulting from substituting the estimated marginal distribution of the covariates in the above conditional likelihood simulation studies suggest that the pseudolikelihood method is approximately unbiased in order to identify the model parameters usually the dimension of the covariates and observed responses is required to be at least as large as the dimension of the missing responses the method can also be modified to handle partial information about the missing data mechanism we also consider the special case where the missing data have a monotone pattern where better use of the incomplete information can be made under certain assumptions

the ecme algorithm a simple extension of em and ecm with faster monotone convergence a generalisation of the ecm algorithm meng rubin 1993 which is itselfan extension of the em algorithm dempster laird rubin 1977 can be obtained by replacing some cm steps of ecm which maximise the constrained expected complete data loglikelihood function with steps that maximise the correspondingly constrained actual likelihood function this algorithm which we call ecme algorithm for expectation conditional maximisation either shares with both em and ecm their stable monotone convergence and basic simplicity of implementation relative to competing faster converging methods moreover ecme can have a substantially faster convergence rate than either em or ecm measured using either the number of iterations or actual computer time

bias reduction and a solution for separation of logistic regression with missing covariates logistic regression is an important statistical procedure used in many disciplines the standard software packages for data analysis are generally equipped with this procedure where the maximum likelihood estimates of the regression coefficients are obtained iteratively it is well known that the estimates from the analyses of small or medium sized samples are biased also in finding such estimates often a separation is encountered in which the likelihood converges but at least one of the parameter estimates diverges to infinity standard approaches of finding such estimates do not take care of these problems moreover the missingness in the covariates adds an extra layer of complexity to the whole process in this article we address these three practical issues bias separation and missing covariates by means of simple adjustments we have applied the proposed technique using real and simulated data the proposed method always finds a solution and the estimates are less biased a sas macro that implements the proposed method can be obtained from the authors

non ignorable missing covariates in generalized linear models we propose a likelihood method for estimating parameters in generalized linear models with missing covariates and a non ignorable missing data mechanism in this paper we focus on one missing covariate we use a logistic model for the probability that the covariate is missing and allow this probability to depend on the incomplete covariate we allow the covariates including the incomplete covariate to be either categorical or continuous we propose an em algorithm in this case for a missing categorical covariate we derive a closed form expression for the e and m steps of the em algorithm for obtaining the maximum likelihood estimates mles for a missing continuous covariate we use a monte carlo version of the em algorithm to obtain the mles via the gibbs sampler the methodology is illustrated using an example from a breast cancer clinical trial in which time to disease progression is the outcome and the incomplete covariate is a quality of life physical well being score taken after the start of therapy this score may be missing because the patients are sicker so this covariate could be non ignorably missing

maximum likelihood estimation and model selection in contingency tables with missing data abstract in many studies the values of one or more variables are missing for subsets of the original sample this article focuses on the problem of obtaining maximum likelihood estimates mle for the parameters of log linear models under this type of incomplete data the appropriate systems of equations are presented and the expectation maximization em algorithm dempster laird and rubin 1977 is suggested as one of the possible methods for solving them the algorithm has certain advantages but other alternatives may be computationally more effective tests of fit for log linear models in the presence of incomplete data are considered the data from the protective services project for older persons blenkner bloom and nielsen 1971 blenkner bloom and weber 1974 are used to illustrate the procedures discussed in the article

bayesian analysis for generalized linear models with nonignorably missing covariates we propose bayesian methods for estimating parameters in generalized linear models glms with nonignorably missing covariate data we show that when improper uniform priors are used for the regression coefficients phi of the multinomial selection model for the missing data mechanism the resulting joint posterior will always be improper if i all missing covariates are discrete and an intercept is included in the selection model for the missing data mechanism or ii at least one of the covariates is continuous and unbounded this impropriety will result regardless of whether proper or improper priors are specified for the regression parameters beta of the glm or the parameters alpha of the covariate distribution to overcome this problem we propose a novel class of proper priors for the regression coefficients phi in the selection model for the missing data mechanism these priors are robust and computationally attractive in the sense that inferences about beta are not sensitive to the choice of the hyperparameters of the prior for phi and they facilitate a gibbs sampling scheme that leads to accelerated convergence in addition we extend the model assessment criterion of chen dey and ibrahim 2004a biometrika 91 45 63 called the weighted l measure to glms and missing data problems as well as extend the deviance information criterion dic of spiegelhalter et al 2002 journal of the royal statistical society b 64 583 639 for assessing whether the missing data mechanism is ignorable or nonignorable a novel markov chain monte carlo sampling algorithm is also developed for carrying out posterior computation several simulations are given to investigate the performance of the proposed bayesian criteria as well as the sensitivity of the prior specification real datasets from a melanoma cancer clinical trial and a liver cancer study are presented to further illustrate the proposed methods

a class of pattern mixture models for normal incomplete data summary likelihood based methods are developed for analyzing a random sample on two continuous variables when values of one of the variables are missing normal maximum likelihood estimates when values are missing completely at random were derived by anderson 1957 they are also maximum likelihood providing the missing data mechanism is ignorable in rubin s 1976 sense that the mechanism depends only on observed data a new class of pattern mixture models little 1993 is described for the situation where missingness is assumed to depend on an arbitrary unspecified function of a linear combination of the two variables maximum likelihood for models in this class is straightforward and yields the estimates of anderson 1957 when missingness depends solely on the completely observed variable and the estimates of brown 1990 when missingness depends solely on the incompletely observed variable another choice of linear combination yields estimates from complete case analysis large sample and bayesian methods are described for this model the data do not supply information about the ratio of the coefficients of the linear combination that controls missingness if this ratio is not welldetermined based on prior knowledge a prior distribution can be specified and bayesian inference is then readily accomplished alternatively sensitivity of inferences can be displayed for a variety of choices of the ratio

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

a comparison of multiple imputation and fully augmented weighted estimators for cox regression with missing covariates several approaches exist for handling missing covariates in the cox proportional hazards model the multiple imputation mi is relatively easy to implement with various software available and results in consistent estimates if the imputation model is correct on the other hand the fully augmented weighted estimators fawes recover a substantial proportion of the efficiency and have the doubly robust property in this paper we compare the fawes and the mi through a comprehensive simulation study for the mi we consider the multiple imputation by chained equation and focus on two imputation methods bayesian linear regression imputation and predictive mean matching simulation results show that the imputation methods can be rather sensitive to model misspecification and may have large bias when the censoring time depends on the missing covariates in contrast the fawes allow the censoring time to depend on the missing covariates and are remarkably robust as long as getting either the conditional expectations or the selection probability correct due to the doubly robust property the comparison suggests that the fawes show the potential for being a competitive and attractive tool for tackling the analysis of survival data with missing covariates

selection models for repeated measurements with non random dropout an illustration of sensitivity the outcome based selection model of diggle and kenward for repeated measurements with non random dropout is applied to a very simple example concerning the occurrence of mastitis in dairy cows in which the occurrence of mastitis can be modelled as a dropout process it is shown through sensitivity analysis how the conclusions concerning the dropout mechanism depend crucially on untestable distributional assumptions this example is exceptional in that from a simple plot of the data two outlying observations can be identified that are the source of the apparent evidence for non random dropout and also provide an explanation of the behaviour of the sensitivity analysis it is concluded that a plausible model for the data does not require the assumption of non random dropout

parameter estimation from incomplete data in binomial regression when the missing data mechanism is nonignorable we propose a method for estimating parameters in binomial regression models when the response variable is missing and the missing data mechanism is nonignorable we assume throughout that the covariates are fully observed using a logit model for the missing data mechanism we show how parameter estimation can be accomplished using the em algorithm by the method of weights proposed in ibrahim 1990 journal of the american statistical association 85 765 769 an example from the six cities study ware et al 1984 american review of respiratory diseases 129 366 374 is presented to illustrate the method

variable selection via gibbs sampling abstract a crucial problem in building a multiple regression model is the selection of predictors to include the main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets this procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices in this framework the promising subsets of predictors can be identified as those with higher posterior probability the computational burden is then alleviated by using the gibbs sampler to indirectly sample from this multinomial posterior distribution on the set of possible subset choices those subsets with higher probability the promising ones can then be identified by their more frequent appearance in the gibbs sample

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

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

maximum likelihood methods for cure rate models with missing covariates we propose maximum likelihood methods for parameter estimation for a novel class of semiparametric survival models with a cure fraction in which the covariates are allowed to be missing we allow the covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one dimensional conditional distributions we propose a novel em algorithm for maximum likelihood estimation and derive standard errors by using louis s formula louis 1982 journal of the royal statistical society series b 44 226 233 computational techniques using the monte carlo em algorithm are discussed and implemented a real data set involving a melanoma cancer clinical trial is examined in detail to demonstrate the methodology

a mean score method for missing and auxiliary covariate data in regression models summary we consider regression analysis when incomplete or auxiliary covariate data are available for all study subjects and in addition for a subset called the validation sample true covariate data of interest have been ascertained the term auxiliary data refers to data not in the regression model but thought to be informative about the true missing covariate data of interest we discuss a method which is nonparametric with respect to the association between available and missing data allows missingness to depend on available response and covariate values and is applicable to both cohort and case control study designs the method previously proposed by flanders greenland 1991 and by zhao lipsitz 1992 is generalised and asymptotic theory is derived our expression for the asymptotic variance of the estimator provides intuition regarding performance of the method optimal sampling strategies for the validation set are also suggested by the asymptotic results

likelihood methods for incomplete longitudinal binary responses with incomplete categorical covariates summary we consider longitudinal studies in which the outcome observed over time is binary and the covariates of interest are categorical with no missing responses or covariates one specifies a multinomial model for the responses given the covariates and uses maximum likelihood to estimate the parameters unfortunately incomplete data in the responses and covariates are a common occurrence in longitudinal studies here we assume the missing data are missing at random rubin 1976 biometrika63 581 592 since all of the missing data responses and covariates are categorical a useful technique for obtaining maximum likelihood parameter estimates is the em algorithm by the method of weights proposed in ibrahim 1990 journal of the american statistical association85 765 769 in using the em algorithm with missing responses and covariates one specifies the joint distribution of the responses and covariates here we consider the parameters of the covariate distribution as a nuisance in data sets where the percentage of missing data is high 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 modeling advantages for the em algorithm and provides a reduction in the number of nuisance parameters thus providing more stable estimates in finite samples

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

likelihood based methods for missing covariates in the cox proportional hazards model problems associated with missing covariate data are well known but often ignored we present a method for estimating the parameters in the cox proportional hazards model when the missing data are missing at random mar and censoring is noninformative due to the computational burden of this method we introduce an approximation that allows us to use a weighted expectation maximization em algorithm to estimate the parameters more easily when the missing covariates are continuous rather than categorical we implement a monte carlo version of the em algorithm along with the gibbs sampler to obtain parameter estimates we also give the asymptotic distribution of these estimates the primary advantage of this method over complete case analysis is that it produces more efficient parameter estimates and corrects for bias in the mar setting to motivate the methodology we present an analysis of a phase iii melanoma clinical trial conducted by the eastern cooperative oncology group

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

regression analysis for categorical variables with outcome subject to nonignorable nonresponse abstract we develop a log linear model for categorical response subject to nonignorable nonresponse the paper differs from fay 1986 in its focus on estimation and hypothesis testing in a regression setting as opposed to imputation in a multivariate setting we present several new results concerning the existence of solutions on the boundary of the parameter space and the construction of confidence intervals for estimates we illustrate the method by estimating the proportion of voters preferring truman in a 1948 preelection poll mosteller hyman mccarthy marks and truman 1949 results may depend strongly on the model assumed for nonresponse goodness of fit tests are available for comparing alternative models

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

pattern mixture models for multivariate incomplete data with covariates pattern mixture models stratify incomplete data by the pattern of missing values and formulate distinct models within each stratum pattern mixture models are developed for analyzing a random sample on continuous variables y 1 y 2 when values of y 2 are nonrandomly missing methods for scalar y 1 and y 2 are here generalized to vector y 1 and y 2 with additional fixed covariates x parameters in these models are identified by alternative assumptions about the missing data mechanism models may be underidentified in which case additional assumptions are needed just identified or overidentified maximum likelihood and bayesian methods are developed for the latter two situations using the em and sem algorithms direct and interactive simulation methods the methods are illustrated on a data set involving alternative dosage regimens for the treatment of schizophrenia using haloperidol and on a regression example sensitivity to alternative assumptions about the missing data mechanism is assessed and the new methods are compared with complete case analysis and maximum likelihood for a probit selection model

fast em type implementations for mixed effects models the mixed effects model in its various forms is a common model in applied statistics a useful strategy for fitting this model implements em type algorithms by treating the random effects as missing data such implementations however can be painfully slow when the variances of the random effects are small relative to the residual variance in this paper we apply the working parameter approach to derive alternative em type implementations for fitting mixed effects models which we show empirically can be hundreds of times faster than the common em type implementations in our limited simulations they also compare well with the routines in s plus and stata in terms of both speed and reliability the central idea of the working parameter approach is to search for efficient data augmentation schemes for implementing the em algorithm by minimizing the augmented information over the working parameter and in the mixed effects setting this leads to a transfer of the mixed effects variances into the regression slope parameters we also describe a variation for computing the restricted maximum likelihood estimate and an adaptive algorithm that takes advantage of both the standard and the alternative em type implementations

multiple imputation for missing values through conditional semiparametric odds ratio models multiple imputation is a practically useful approach to handling incompletely observed data in statistical analysis parameter estimation and inference based on imputed full data have been made easy by rubin s rule for result combination however creating proper imputation that accommodates flexible models for statistical analysis in practice can be very challenging we propose an imputation framework that uses conditional semiparametric odds ratio models to impute the missing values the proposed imputation framework is more flexible and robust than the imputation approach based on the normal model it is a compatible framework in comparison to the approach based on fully conditionally specified models the proposed algorithms for multiple imputation through the markov chain monte carlo sampling approach can be straightforwardly carried out simulation studies demonstrate that the proposed approach performs better than existing commonly used imputation approaches the proposed approach is applied to imputing missing values in bone fracture data

a finite element method for a boundary value problem of mixed type we consider the numerical solution of the boundary value problem 1 lu k y u xx u yy c x y u f u gamma 0 cup gamma 1 0 where k y gtreqless 0 for y gtreqless 0 0 and 1 are parts of the boundary of a bounded simply connected regiong inr 2 g is bounded fory 0 by a piecewise smooth curve 0 which intersects the liney 0 at the pointsa 1 0 andb 0 0 fory 0 g is bounded by a piecewise smooth curve 1 througha which meets the characteristic of 1 issued fromb at pointc and by the curve 2 which consists of the portioncb of the characteristic throughb using a weak formulation based on different spaces of test and trial functions we construct a galerkin procedure for the above boundary value problem existence uniqueness and uniform stability of an approximate solution is proven and a priori error bounds are given

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

inference for non random samples observational data are often analysed as if they had resulted from a controlled study and yet the tacit assumption of randomness can be crucial for the validity of inference we take some simple statistical models and supplement them by adding a parameter which reflects the degree of non randomness in the sample for a randomized study is known to be 0 we examine the profile log likelihood for and the sensitivity of inference to small non zero values of particular models cover the analysis of survey data with item non response the paired comparison t test and two group comparisons using observational data with covariates some practical examples are discussed allowing for sampling bias increases the uncertainty of estimation and weakens the significance of treatment effects sometimes substantially so

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

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

sensitivity analysis for nonrandom dropout a local influence approach diggle and kenward 1994 applied statistics 43 49 93 proposed a selection model for continuous longitudinal data subject to nonrandom dropout it has provoked a large debate about the role for such models the original enthusiasm was followed by skepticism about the strong but untestable assumptions on which this type of model invariably rests since then the view has emerged that these models should ideally be made part of a sensitivity analysis this paper presents a formal and flexible approach to such a sensitivity assessment based on local influence cook 1986 journal of the royal statistical society series b 48 133 169 the influence of perturbing a missing at random dropout model in the direction of nonrandom dropout is explored the method is applied to data from a randomized experiment on the inhibition of testosterone production in rats

missing responses in generalised linear mixed models when the missing data mechanism is nonignorable summary we propose a method for estimating parameters in the generalised linear mixed model with nonignorable missing response data and with nonmonotone patterns of missing data in the response variable we develop a monte carlo em algorithm for estimating the parameters in the model via the gibbs sampler for the normal random effects model we derive a novel analytical form for the e and m steps which is facilitated by integrating out the random effects this form leads to a computationally feasible and extremely efficient monte carlo em algorithm for computing maximum likelihood estimates and standard errors in addition we propose a very general joint multinomial model for the missing data indicators which can be specified via a sequence of one dimensional conditional distributions this multinomial model allows for an arbitrary correlation structure between the missing data indicators and has the potential of reducing the number of nuisance parameters real datasets from the international breast cancer study group and an environmental study involving dyspnoea in cotton workers are presented to illustrate the proposed methods

semiparametric bayesian estimation for marginal parametric potential outcome modeling application to causal inference we propose a new semiparametric bayesian model for causal inference in which assignment to treatment depends on potential outcomes the model uses the probit stick breaking process mixture proposed by chung and dunson 2009 a variant of the dirichlet process mixture modeling in contrast to previous bayesian models the proposed model directly estimates the parameters of the marginal parametric model of potential outcomes while it relaxes the strong ignorability assumption and requires no parametric model assumption for the assignment model and conditional distribution of the covariate vector the proposed estimation method is more robust than maximum likelihood estimation in that it does not require knowledge of the full joint distribution of potential outcomes covariates and assignments in addition the method is more efficient than fully nonparametric bayes methods we apply this model to infer the differential effects of cognitive and noncognitive skills on the wages of production and nonprodu

explicit hard bounding functions for boundary value problems for elliptic partial differential equations abstract we obtain tight super and sub solutions or hard bounding functions for the probabilistic representation of the solution to the boundary value problem for second order elliptic partial differential equations in an explicit polynomial form unlike accurate approximate solutions by the existing discretization based methods our hard bounding polynomial functions act as pointwise 100 confidence intervals within which the solution is guaranteed to exist our methodology can deal with a wide range of boundary conditions even of fully mixed type with dirichlet neumann and robin components without increasing the complexity the quality of hard bounds can be improved by introducing piecewise polynomial functions and domain decomposition convergence of hard bounding functions is proved under technical conditions numerical results are presented throughout the paper to support our theoretical developments and to illustrate the effectiveness of our methodology

the existence of generalized solutions for a class of quasi linear equations of mixed type we consider equations in a bounded simply connected region g where k 1 0 for y 0 and g is bounded by the curves r f fz hereby r is a piecewise smooth curve lying in the half plane y o which intersects the line y o in the points a 1 0 and b 0 0 f is a piecewise smooth curve through a in y 0 which meets the characteristic of 1 1 issued from

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