composite linear models for incomplete multinomial data a composite linear model clm is a matrix model for incomplete multinomial data a clm provides a unified approach for maximum likelihood inference which is applicable to a wide variety of problems involving incomplete multinomial data by formulating a model as a clm one can simplify computation of maximum likelihood estimates and asymptotic standard errors as an example we use clm to test marginal homogeneity for ordered categories subject to both ignorable and non ignorable missing data mechanisms

towers and injective cohomology algebras since carlsson s paper 5 and the work of lannes and zarati 9 topo1ogists have learned to recognize many spaces y so h y is injective in u for example if g is a finite group with p sylow subgroup z p kn the h bg is injective further infinite examples include eb z p k and any of the many wedge summands of these spaces a complete classification of u injectives is given in 18 and from this it can be seen that the only such injectives which are not a unbounded are the brown gitler modules these have been adequately discussed elsewhere see 6 among many an immediate consequence of theorem a will be the following result suppose y satisfies the same hypotheses as in theorem a

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 bayesian hierarchical model for categorical data with nonignorable nonresponse les modeles log lineaires ont montre leur utilite pour lisser des tableaux de contingence lorsque les variables categorielles sont sujettes a des non reponses qu on ne peut ignorer un modele log lineaire peut etre ajuste a un tableau de donnees augmente qui inclut une variable indicatrice indiquant si les individus sont repondants ou non les estimateurs calcules par maximum de vraisemblance a partir de tableaux de donnees augmentes sont connus pour leur instabilite en raison de solutions frontieres park et brown 1994 et park 1998 ont developpe des modeles bayesiens empiriques qui tendent a lisser les estimateurs a distance de la frontiere avec ces approches les estimateurs pour les non repondants ont ete calcules en maximisant une distribution a posteriori par l algorithme em pour etendre leur precedent travail nous avons developpe un modele hierarchique bayesien qui incorpore un modele log lineaire pour specifier la distribution a priori de plus du fait de l incertitude dans le processus de selection des variables associe a un seul modele log lineaire un nombre fini des modeles en nombre fini ont ete consideres simultanement en utilisant la procedure stochastique de selection de variables ssvs de mcculloch 1997 l integration de la procedure ssvs dans un echantillonneur de monte carlo par chaines de markov mcmc est direct et conduit a des estimations des frequences dans les cellules pour les non repondants qui sont les moyennes resultantes de plusieurs modeles log lineaires les methodes sont demontrees avec un exemple de donnees de niveaux de creatinine chez des patients ayant survecus a des transplantations renales une etude de simulation a ete conduite afin de rechercher les proprietes du modele

a bivariate pseudolikelihood for incomplete longitudinal binary data with nonignorable nonmonotone missingness for analyzing longitudinal binary data with nonignorable and non monotone missing responses a full likelihood method is complicated algebraically and often requires intensive computation especially when there are many follow up times as an alternative a pseudo likelihood approach has been proposed in the literature under minimal parametric assumptions this formulation only requires specification of the marginal distributions of the responses and missing data mechanism and uses an independence working assumption however this estimator can be inefficient for estimating both time varying and time stationary effects under moderate to strong within subject associations among repeated responses in this article we propose an alternative estimator based on a bivariate pseudo likelihood and demonstrate in simulations that the proposed method can be much more efficient than the previous pseudo likelihood obtained under the assumption of independence we illustrate the method using longitudinal data on cd4 counts from two clinical trials of hiv infected patients

the analysis of repeated categorical measurements subject to nonignorable nonresponse abstract conditional likelihood methods have been proposed for analyzing the repeated measurement of categorical responses this article extends the methods to include partially classified subjects the extension allows for an ignorable or nonignorable nonresponse mechanism and uses standard statistical software for the computations two examples of incomplete longitudinal categorical data illustrate the method

multivariate logistic models for incomplete binary responses abstract in this article we describe a likelihood based regression model appropriate for analyzing incomplete multivariate binary responses we focus on marginal models that is models where the marginal mean or expectation of the binary response is related to a set of covariates the association between the binary responses is modeled in terms of conditional log odds ratios when the nonresponse mechanism is ignorable it is not necessary to specify a nonresponse model and valid inferences can be obtained provided that the likelihood for the responses has been correctly specified but when the nonresponse mechanism is nonignorable valid inferences can only be obtained by incorporating a model for nonresponse an unresolved issue with nonignorable models concerns the identifiability of the parameters so far no general and practically useful necessary and sufficient conditions for identifiability are available here we suggest some simple procedures for examining the identifiability status of nonign

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

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

causal models for patterns of nonresponse abstract the problem of missing data for categorical variables is examined from the perspective of modeling the mechanisms of nonresponse log linear causal models as formulated by goodman are studied for the relationship of the survey variables to response under some conditions several such models are estimable from the observed data for nested patterns of nonresponse a specific causal model represents exactly the assumption of ignorable response most causal models however imply nonignorable response mechanisms and yield alternative estimates for the distribution of the survey variables

bias in a binary risk behaviour model subject to inconsistent reports and dropout in a south african high school cohort study we describe a methodology for analysing self reported risk behaviour transitional patterns in a binary outcome variable subject to misclassification and a large loss to follow up the motivation stems from the analysis of self reported transitional patterns in responses to the question have you ever smoked a whole cigarette in a cohort of south african school children the partially complete records analysis pcra introduced estimates the transitional probability as the ratio of the joint probability of the response at two time points based on the complete records for this time sequence over the marginal probabilities of the response based on the complete records at the first time point and assumes a non informative missing pattern a comparison was made using un weighted complete records and inverse probability weighted logistic regression the estimates of the probabilities of reporting ever having smoked a cigarette obtained from the three methods were similar for a particular transition the pcra method lacked precision compared with the inverse probability weighted logistic regression a simulation study indicated an association between bias and reporting error in all three methods the pcra method can be considered as a method for the estimation of transition probabilities in a cohort study where there is consistency in the self reported risk behaviour pattern and the sample size is large at baseline the inverse probability weighting approach is more precise and is suitable for this setting in order to determine risk factors for the incidence of self reported substance used in a cohort with a high dropout rate

estimating and using propensity scores with partially missing data abstract investigators in observational studies have no control over treatment assignment as a result large differences can exist between the treatment and control groups on observed covariates which can lead to badly biased estimates of treatment effects propensity score methods are an increasingly popular method for balancing the distribution of the covariates in the two groups to reduce this bias for example using matching or subclassification sometimes in combination with model based adjustment to estimate propensity scores which are the conditional probabilities of being treated given a vector of observed covariates we must model the distribution of the treatment indicator given these observed covariates much work has been done in the case where covariates are fully observed we address the problem of calculating propensity scores when covariates can have missing values in such cases which commonly arise in practice the pattern of missing covariates can be prognostically important and

interval estimation for log linear models with one variable subject to non ignorable non response log linear models for multiway contingency tables where one variable is subject to non ignorable non response will often yield boundary solutions with the probability of non respondents being classified in some cells of the table estimated as 0 the paper considers the effect of this non standard behaviour on two methods of interval estimation based on the distribution of the maximum likelihood estimator the first method relies on the estimator being approximately normally distributed with variance equal to the inverse of the information matrix it is shown that the information matrix is singular for boundary solutions but intervals can be calculated after a simple transformation for the second method based on the bootstrap asymptotic results suggest that the coverage properties may be poor for boundary solutions both methods are compared with profile likelihood intervals in a simulation study based on data from the british general election panel study the results of this study indicate that all three methods perform poorly for a parameter of the non response model whereas they all perform well for a parameter of the margin model irrespective of whether or not there is a boundary solution

itemwise conditionally independent nonresponse modelling for incomplete multivariate data summary we introduce a nonresponse mechanism for multivariate missing data in which each study variable and its nonresponse indicator are conditionally independent given the remaining variables and their nonresponse indicators this is a nonignorable missingness mechanism in that nonresponse for any item can depend on values of other items that are themselves missing we show that under this itemwise conditionally independent nonresponse assumption one can define and identify nonparametric saturated classes of joint multivariate models for the study variables and their missingness indicators we also show how to perform sensitivity analysis with respect to violations of the conditional independence assumptions encoded by this missingness mechanism we illustrate the proposed modelling approach with data analyses

models for categorical data with nonignorable nonresponse abstract when categorical outcomes are subject to nonignorable nonresponse log linear models may be used to adjust for the nonresponse the models are fitted to the data in an augmented frequency table in which one index corresponds to whether or not the subject is a respondent the likelihood function is maximized over pseudo observed cell frequencies with respect to this log linear model using an em algorithm each e step of the em algorithm determines the pseudo observed cell frequencies and the m step yields the maximum likelihood estimators mle s of these pseudo observed cell frequencies this approach may produce boundary estimates for the expected cell frequencies of the nonrespondents in these cases the estimators of the log linear model parameters are not uniquely determined and may be unstable following the approach of clogg et al we propose a bayesian method that uses smoothing constants to adjust the pseudo observed cell frequencies so that the solution is not on the boundary the role

monte carlo em algorithm in logistic linear models involving non ignorable missing data abstract many data sets obtained from surveys or medical trials often include missing observations since ignoring the missing information usually cause bias and inefficiency an algorithm for estimating parameters is proposed based on the likelihood function of which the missing information is taken account a binomial response and normal exploratory model for the missing data are assumed we fit the model using the monte carlo em expectation and maximization algorithm the e step is derived by metropolis hastings algorithm to generate a sample for missing data and the m step is done by newton raphson to maximize the likelihood function asymptotic variances and the standard errors of the mle maximum likelihood estimates of parameters are derived using the observed fisher information

models for nonresponse in sample surveys abstract the literature on the analysis of incomplete data using models is reviewed in the context of nonresponse in sample surveys the modeling approach provides a large body of methods for handling unit and item nonresponse some of which cannot be derived from the randomization theory of inference for surveys key concepts from the literature on incomplete data such as factorizations of the likelihood for special data patterns the em algorithm for general data patterns and ignorability of the response mechanism are discussed within the survey context model based procedures are related to common methods for handling nonresponse in surveys such as weighting or imputation of means within subclasses of the population

inference from nonrandomly missing categorical data an example from a genetic study on turner s syndrome abstract the process leading to partial classification with categorical data is sometimes nonrandom a particular model accounting for incomplete data which allows the probability of uncertain classification to depend on category identity is utilized for an analysis of data obtained from a genetic study on turner s syndrome estimates of population proportions are obtained from maximum likelihood a method for handling nonrandomly missing data arrayed in contingency tables is discussed sensitivity analyses incorporating parameters related to the missing data mechanism are recommended for estimation and testing

model based inference for categorical survey data subject to non ignorable non response we consider non response models for a single categorical response with categorical covariates whose values are always observed we present bayesian methods for ignorable models and a particular non ignorable model and we argue that standard methods of model comparison are inappropriate for comparing ignorable and non ignorable models uncertainty about ignorability of non response is incorporated by introducing parameters describing the extent of non ignorability into a pattern mixture specification and integrating over the prior uncertainty associated with these parameters our approach is illustrated using polling data from the 1992 british general election panel survey we suggest sample size adjustments for surveys when non ignorable non response is expected

on the link between l1 pca and ica principal component analysis pca based on l1 norm maximization is an emerging technique that has drawn growing interest in the signal processing and machine learning research communities especially due to its robustness to outliers the present work proves that l1 norm pca can perform independent component analysis ica under the whitening assumption however when the source probability distributions fulfil certain conditions the l1 norm criterion needs to be minimized rather than maximized which can be accomplished by simple modifications on existing optimal algorithms for l1 pca if the sources have symmetric distributions we show in addition that l1 pca is linked to kurtosis optimization a number of numerical experiments illustrate the theoretical results and analyze the comparative performance of different algorithms for ica via l1 pca although our analysis is asymptotic in the sample size this equivalence opens interesting new perspectives for performing ica using optimal algorithms for l1 pca with guaranteed global convergence while inheriting the increased robustness to outliers of the l1 norm criterion

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

bivariate binary data analysis with nonignorably missing outcomes one of the objectives in the northern manhattan stroke study is to investigate the impact of stroke subtype on the functional status 2 years after the first ischemic stroke a challenge in this analysis is that the functional status at 2 years after stroke is not completely observed in this paper we propose a method to handle nonignorably missing binary functional status when the baseline value and the covariates are completely observed the proposed method consists of fitting four separate binary regression models for the baseline outcome the outcome 2 years after the stroke the product of the previous two and finally the missingness indicator we then conduct a sensitivity analysis by varying the assumptions about the third and the fourth binary regression models our method belongs to an imputation paradigm and can be an alternative to the weighting method of rotnitzky and robins 1997 statistics in medicine 16 81 102 a jackknife variance estimate is proposed for the variance of the resulting estimate the proposed analysis can be implemented using statistical software such as sas

a method of moments estimation procedure for categorical quality of life data with nonignorable missingness abstract quality of life outcomes collected during clinical trials often have considerable amounts of missing data which if not appropriately accounted for may lead to bias in inferences we introduce a method of moments mm estimating procedure for a model designed to handle nonignorable missingness arising in categorical data measured on independent populations the missingness mechanism is assumed to be the same across the populations we derive necessary and sufficient conditions for the identifiability of the model and fit the model to quality of life data collected as part of a breast cancer clinical trial we compare the mm estimator to the maximum likelihood estimator in a simulation study

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

imputation of missing values when the probability of response depends on the variable being imputed abstract a method is developed for imputing missing values when the probability of response depends upon the variable being imputed the missing data problem is viewed as one of parameter estimation in a regression model with stochastic censoring of the dependent variable the prediction approach to imputation is used to solve this estimation problem wages and salaries are imputed to non respondents in the current population survey and the results are compared to the nonrespondents irs wage and salary data the stochastic censoring approach gives improved results relative to a prediction approach that ignores the response mechanism

qualms about bootstrap confidence intervals abstract the percentile method and bias corrected percentile method of efron 1981 1982 are discussed when these methods are used to construct nonparametric confidence intervals for the variance of a normal distribution the coverage probabilities are substantially below the nominal level for small to moderate samples this is due to the inapplicability of assumptions underlying the methods these assumptions are difficult or impossible to check in the complicated situations for which the bootstrap is intended therefore bootstrap confidence intervals should be used with caution in complex problems

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

building an identifiable latent class model with covariate effects on underlying and measured variables in recent years latent class models have proven useful for analyzing relationships between measured multiple indicators and covariates of interest such models summarize shared features of the multiple indicators as an underlying categorical variable and the indicators substantive associations with predictors are built directly and indirectly in unique model parameters in this paper we provide a detailed study on the theory and application of building models that allow mediated relationships between primary predictors and latent class membership but that also allow direct effects of secondary covariates on the indicators themselves theory for model identification is developed we detail an expectation maximization algorithm for parameter estimation standard error calculation and convergent properties comparison of the proposed model with models underlying existing latent class modeling software is provided a detailed analysis of how visual impairments affect older persons functioning requiring distance vision is used for illustration