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

marginalized transition models for longitudinal binary data with ignorable and non ignorable drop out we extend the marginalized transition model of heagerty to accommodate non ignorable monotone drop out using a selection model weakly identified drop out parameters are held constant and their effects evaluated through sensitivity analysis for data missing at random mar efficiency of inverse probability of censoring weighted generalized estimating equations ipcw gee is as low as 40 per cent compared to a likelihood based marginalized transition model mtm with comparable modelling burden mtm and ipcw gee regression parameters both display misspecification bias for mar and non ignorable missing data and both reduce bias noticeably by improving model fit

approximate maximum entropy joint feature inference consistent with arbitrary lower order probability constraints application to statistical classification we propose a new learning method for discrete space statistical classifiers similar to chow and liu 1968 and cheeseman 1983 we cast classification inference within the more general framework of estimating the joint probability mass function p m f for the feature vector class label pair cheeseman s proposal to build the maximum entropy me joint p m f consistent with general lower order probability constraints is in principle powerful allowing general dependencies between features however enormous learning complexity has severely limited the use of this approach alternative models such as bayesian networks bns require explicit determination of conditional independencies these may be difficult to assess given limited data here we propose an approximate me method which like previous methods incorporates general constraints while retaining quite tractable learning the new method restricts joint p m f support during learning to a small subset of the full feature space classification gains are realized over dependence trees tree augmented naive bayes networks bns trained by the kutato algorithm and multilayer perceptrons extensions to more general inference problems are indicated we also propose a novel exact inference method when there are several missing features

group comparisons involving missing data in clinical trials a comparison of estimates and power size for some simple approaches when using intent to treat approaches to compare outcomes between groups in clinical trials analysts face a decision regarding how to account for missing observations most model based approaches can be summarized as a process whereby the analyst makes assumptions about the distribution of the missing data in an attempt to obtain unbiased estimates that are based on functions of the observed data although pointed out by rubin as often leading to biased estimates of variances an alternative approach that continues to appear in the applied literature is to use fixed value imputation of means for missing observations the purpose of this paper is to provide illustrations of how several fixed value mean imputation schemes can be formulated in terms of general linear models that characterize the means of distributions of missing observations in terms of the means of the distributions of observed data we show that several fixed value imputation strategies will result in estimated intervention effects that correspond to maximum likelihood estimates obtained under analogous assumptions if the missing data process has been correctly characterized hypothesis tests based on variances estimated using maximum likelihood techniques asymptotically have the correct size in contrast hypothesis tests performed using the uncorrected variance obtained by applying standard complete data formula to singly imputed data can provide either conservative or anticonservative results surprisingly under several non ignorable non response scenarios maximum likelihood based analyses can yield equivalent hypothesis tests to those obtained when analysing only the observed data

estimation and comparison of changes in the presence of information right censoring by modeling the censoring process abstract in estimating and comparing the rates of change of a continuous variable between two groups the unweighted averages of individual simple least squares estimates from each group are often used under a linear random effects model when all individuals have completed observations at identical time points these statistics are maximum likelihood estimates for the expected rates of change however with censored of missing data these estimates are no longer efficient when compared to generalized least squares estimates when in addition the right censoring process is dependent upon the individual rates of change i e informative right censoring the generalized least squares estimates will be biased likelihood ratio test for informativeness of the censoring process and maximum likelihood estimates for the expected rates of change and the parameters of the right censoring process are developed under a linear random effect models with a probit model for the right censoring process in realistic situations we illustrate that the bias in estimating group rate of change and the reduction of power in comparing group difference could be substantial when strong dependency of the right censoring process on individual rates of change is ignored author

rapidly decreasing functions in reduced algebras of groups let r be a group we associate to any length function l on r the space h r of rapidly decreasing functions on r with respect to l which coincides with the space of smooth functions on the k dimensional torus when r zk we say that r has property rd if there exists a length function l on r such that h r is contained in the reduced c algebra c r of r we study the stability of property rd with respect to some constructions of groups such as subgroups over groups of finite index semidirect and amalgamated products finally we show that the following groups have property rd 1 finitely generated groups of polynomial growth 2 discrete cocompact subgroups of the group of all isometries of any hyperbolic space

slope estimation in the presence of informative right censoring modeling the number of observations as a geometric random variable summary a method is proposed for the estimation of rate of change from incomplete longitudinal data where the number of observations made for each subject is assumed to vary depending on the level of the response variable the proposed method involves a random slope model in which the number of observations is modeled as a geometric distribution with its mean dependent on the individual subject s rate of change the method adjusts for informative right censoring and provides estimates of the slopes of individual subjects as well as of the population under noninformative right censoring these estimators of the slopes are equivalent to bayes estimators fearn 1975 biometrika 62 89 100 the simulation study demonstrates that in cases where the censoring process is informative the proposed estimator is more efficient than either the unweighted or weighted estimator of slope the method is illustrated by the analysis of renal transplant data

k theorie bivariante pour les algebres de banach et conjecture de baum connes je suis parvenu dans ma these a construire une k theorie bivariante pour les algebres de banach cela m a permis de demontrer la conjecture de baum connes pour les groupes de lie semi simples et les groupes reductifs p adiques ainsi que pour les sous groupes discrets de type fini de sl 3 f avec f une extension finie de q p et pour les sous groupes discrets cocompacts de sp n 1 sl 3 r et sl 3 c j ai aussi demontre une conjecture analogue a la conjecture de baum connes pour les algebres l 1 et plus generalement toutes les bonnes completions des sous groupes fermes des groupes de lie semi simples et des groupes reductifs p adiques

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

complex hyperbolic hyperplane complements we study spaces obtained from a complete finite volume complex hyperbolic n manifold m by removing a compact totally geodesic complex n 1 submanifold s the main result is that the fundamental group of m setminus s is relatively hyperbolic relative to fundamental groups of the ends of m setminus s and m setminus s admits a complete finite volume a regular riemannian metric of negative sectional curvature it follows that for n 1 the fundamental group of m setminus s satisfies mostow type rigidity has solvable word and conjugacy problems has finite asymptotic dimension and rapid decay property satisfies borel and baum connes conjectures is co hopf and residually hyperbolic has no nontrivial subgroups with property t and has finite outer automorphism group furthermore if m is compact then the fundamental group of m setminus s is biautomatic and satisfies strong tits alternative

real hyperbolic hyperplane complements in the complex hyperbolic plane abstract this paper studies riemannian manifolds of the form m s where m 4 is a complete four dimensional riemannian manifold with finite volume whose metric is modeled on the complex hyperbolic plane c h 2 and s is a compact totally geodesic codimension two submanifold whose induced riemannian metric is modeled on the real hyperbolic plane h 2 in this paper we write the metric on c h 2 in polar coordinates about s compute formulas for the components of the curvature tensor in terms of arbitrary warping functions theorem 7 1 and prove that there exist warping functions that yield a complete finite volume riemannian metric on m s whose sectional curvature is bounded above by a negative constant theorem 1 1 1 the cases of m s modeled on h n h n 2 and c h n c h n 1 were studied by belegradek in 4 and 3 respectively one may consider this work as part 3 to this sequence of papers

nonparametric functional mapping of quantitative trait loci functional mapping is a useful tool for mapping quantitative trait loci qtl that control dynamic traits it incorporates mathematical aspects of biological processes into the mixture model based likelihood setting for qtl mapping thus increasing the power of qtl detection and the precision of parameter estimation however in many situations there is no obvious functional form and in such cases this strategy will not be optimal here we propose to use nonparametric function estimation typically implemented with b splines to estimate the underlying functional form of phenotypic trajectories and then construct a nonparametric test to find evidence of existing qtl using the representation of a nonparametric regression as a mixed model the final test statistic is a likelihood ratio test we consider two types of genetic maps dense maps and general maps and the power of nonparametric functional mapping is investigated through simulation studies and demonstrated by examples

merging multiple longitudinal studies with study specific missing covariates a joint estimating function approach merging multiple datasets collected from studies with identical or similar scientific objectives is often undertaken in practice to increase statistical power this article concerns the development of an effective statistical method that enables to merge multiple longitudinal datasets subject to various heterogeneous characteristics such as different follow up schedules and study specific missing covariates e g covariates observed in some studies but missing in other studies the presence of study specific missing covariates presents great statistical methodology challenge in data merging and analysis we propose a joint estimating function approach to addressing this challenge in which a novel nonparametric estimating function constructed via splines based sieve approximation is utilized to bridge estimating equations from studies with missing covariates to those with fully observed covariates under mild regularity conditions we show that the proposed estimator is consistent and asymptotically normal we evaluate finite sample performances of the proposed method through simulation studies in comparison to the conventional multiple imputation approach our method exhibits smaller estimation bias we provide an illustrative data analysis using longitudinal cohorts collected in mexico city to assess the effect of lead exposures on children s somatic growth

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 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

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

a random pattern mixture model for ordinal outcomes with informative dropouts we extend a random pattern mixture joint model for longitudinal ordinal outcomes and informative dropouts the patients are generalized to pattern groups based on known covariates that are potential surrogated for the severity of the underlying condition the random pattern effects are defined as the latent effects linking the dropout process and the ordinal longitudinal outcome conditional on the random pattern effects the longitudinal outcome and the dropout times are assumed independent estimates are obtained via the em algorithm we applied the model to the end stage renal disease esrd data anemia was found to be significantly affected by baseline iron treatment when the dropout information was adjusted via the study model as opposed to an independent or shared parameter model simulations were performed to evaluate the performance of the random pattern mixture model under various assumptions

on bundles that admit fiberwise hyperbolic dynamics this paper is devoted to rigidity of smooth bundles which are equipped with fiberwise geometric or dynamical structure we show that the fiberwise associated sphere bundle to a bundle whose leaves are equipped with continuously varying metrics of negative curvature is a topologically trivial bundle when either the base space is simply connected or more generally when the bundle is fiber homotopically trivial we present two very different proofs of this result a geometric proof and a dynamical proof we also establish a number of rigidity results for bundles which are equipped with fiberwise anosov dynamical systems finally we present several examples which show that our results are sharp in certain ways or illustrate necessity of various assumptions

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

higher symmetries of the conformal powers of the laplacian on conformally flat manifolds a r g gratefully acknowledges support from the royal society of new zealand via marsden grant nos 06 uoa 029 and 10 uoa 113 j s was supported by the max planck institute fur math ematik in bonn and by the grant agency of the czech republic under the grant no p201 12 g028

negatively curved manifolds with exotic smooth structures let m denote a compact real hyperbolic manifold with dimension m 5 and sectional curvature k 1 and let y be an exotic sphere of dimension m given any small number j 0 we show that there is a finite covering space m of m satisfying the following properties the connected sum m x is not diffeomorphic to m but it is homeomorphic to m m x supports a riemannian metric having all of its sectional curvature values in the interval 1 j 1 5 thus there are compact riemannian manifolds of strictly negative sectional curvature which are not diffeomorphic but whose fundamental groups are isomorphic this answers problem 12 of the list com piled by yau 22 i e it gives counterexamples to the lawson yau conjecture note that mostow s rigidity theorem 17 implies that m x does not support a riemannian metric whose sectional curvature is identically 1 in fact it is not diffeomorphic to any locally symmetric space thus the manifold m x supports a riemannian metric with sectional curvature arbitrarily close to 1 but it does not support a riemannian metric whose sectional curvature is iden tically 1 more complicated examples of manifolds satisfying the properties of the previous sentence were first constructed by gromov and thurston 11 department of mathematics columbia university new york new york 10027 department of mathematics suny at stony brook stony brook new york 11794 this content downloaded from 207 46 13 71 on wed 27 apr 2016 07 05 48 utc all use subject to http about jstor org terms

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

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

a latent class mixture model for incomplete longitudinal gaussian data in the analyses of incomplete longitudinal clinical trial data there has been a shift away from simple methods that are valid only if the data are missing completely at random to more principled ignorable analyses which are valid under the less restrictive missing at random assumption the availability of the necessary standard statistical software nowadays allows for such analyses in practice while the possibility of data missing not at random mnar cannot be ruled out it is argued that analyses valid under mnar are not well suited for the primary analysis in clinical trials rather than either forgetting about or blindly shifting to an mnar framework the optimal place for mnar analyses is within a sensitivity analysis context one such route for sensitivity analysis is to consider next to selection models pattern mixture models or shared parameter models the latter can also be extended to a latent class mixture model the approach taken in this article the performance of the so obtained flexible model is assessed through simulations and the model is applied to data from a depression trial

pinching constants for hyperbolic manifolds we show in this paper that for everyn 4 there exists a closedn dimensional manifoldv which carries a riemannian metric with negative sectional curvaturek but which admits no metric with constant curvaturek 1 we also estimate the pinching constantsh for which our manifoldsv admit metrics with 1 k h

a copula model for repeated measurements with non ignorable non monotone missing outcome a normal copula based selection model is proposed for continuous longitudinal data with a non ignorable non monotone missing data process the normal copula is used to combine the distribution of the outcome of interest and that of the missing data indicators given the covariates parameters in the model are estimated by a pseudo likelihood method we first use the gee with a logistic link to estimate the parameters associated with the marginal distribution of the missing data indicator given the covariates assuming that covariates are always observed then we estimate other parameters by inserting the estimates from the first step into the full likelihood function a simulation study is conducted to assess the robustness of the assumed model under different missing data processes the proposed method is then applied to one example from a community cohort study to demonstrate its capability to reduce bias

on negatively curved bundles with hyperbolic fibers outside the igusa stable range we prove that the teichmuller space mathcal t dim m moreover some elements of infinite order in pi i b text diff m can be represented by bundles over s i with fiberwise negatively curved metrics

bayesian nonparametric analysis of longitudinal studies in the presence of informative missingness summaryin longitudinal clinical trials one often encounters missingness that is thought to be nonignorable such missingness introduces identifiability issues resulting in causal effects being nonparametrically unidentified it is then prudent to conduct a sensitivity analysis to assess how much of the inference is being driven by untestable assumptions needed to identify the effects of interest we introduce a bayesian nonparametric framework for conducting inference in the presence of nonignorable nonmonotone missingness this framework focuses on the specification of an auxiliary working prior on the space of complete data generating mechanisms this prior induces a prior on the observed data generating mechanism which is then used in conjunction with an identifying restriction to conduct inference advantages of this approach include a flexible modelling framework access to simple computational methods strong theoretical support straightforward sensitivity analysis and applicability to nonmonotone missingness

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

ricci flow einstein metrics and space forms the main results in this paper are 1 ricci pinched stable riemannian metrics can be deformed to einstein metrics through the ricci flow of r hamilton 2 suitably negatively pinched riemannian manifolds can be deformed to hyperbolic space forms through ricci flow and 3 l 2 pinched riemannian manifolds can be deformed to space forms through ricci flow

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

weighted semiparametric estimation in regression analysis with missing covariate data abstract this article investigates estimation of the regression coefficients in an assumed mean function when covariates on some subjects are missing we examine the performance of a horvitz and thompson 1952 type weighted estimator by using different estimates of the selection probabilities which may be treated as nuisance parameters or a nuisance function in particular we investigate the properties of the estimate of the regression parameters when the selection probabilities are estimated by kernel smoothers we present large sample theory for the new estimator and conduct simulation studies comparing the proposed estimator to the maximum likelihood estimator and multiple imputation under various model assumptions and different missingness mechanisms in addition we provide two real examples that motivate this investigation

efficient inference for random coefficient growth curve models with unbalanced data growth and dose response curve studies often result in incomplete or unbalanced data random effects models together with a variety of computer intensive iterative techniques have been suggested for the analysis of such data this paper is concerned with a noniterative method for estimating and comparing location parameters in random coefficient growth curve models consistent and asymptotically efficient estimators of the location parameters are obtained using estimated generalized least squares two criteria for testing multivariate general linear hypotheses are introduced and their asymptotic properties are investigated the results are applied to clinical data obtained on the blood ultrafiltration performance of hemodialyzers used in the treatment of patients with end stage renal disease

bayesian semiparametric regression for longitudinal binary processes with missing data longitudinal studies with binary repeated measures are widespread in biomedical research marginal regression approaches for balanced binary data are well developed whereas for binary process data where measurement times are irregular and may differ by individuals likelihood based methods for marginal regression analysis are less well developed in this article we develop a bayesian regression model for analyzing longitudinal binary process data with emphasis on dealing with missingness we focus on the settings where data are missing at random mar which require a correctly specified joint distribution for the repeated measures in order to draw valid likelihood based inference about the marginal mean to provide maximum flexibility the proposed model specifies both the marginal mean and serial dependence structures using nonparametric smooth functions serial dependence is allowed to depend on the time lag between adjacent outcomes as well as other relevant covariates inference is fully bayesian using simulations we show that adequate modeling of the serial dependence structure is necessary for valid inference of the marginal mean when the binary process data are mar longitudinal viral load data from the hiv epidemiology research study are analyzed for illustration

a test of missing completely at random for multivariate data with missing values abstract a common concern when faced with multivariate data with missing values is whether the missing data are missing completely at random mcar that is whether missingness depends on the variables in the data set one way of assessing this is to compare the means of recorded values of each variable between groups defined by whether other variables in the data set are missing or not although informative this procedure yields potentially many correlated statistics for testing mcar resulting in multiple comparison problems this article proposes a single global test statistic for mcar that uses all of the available data the asymptotic null distribution is given and the small sample null distribution is derived for multivariate normal data with a monotone pattern of missing data the test reduces to a standard t test when the data are bivariate with missing data confined to a single variable a limited simulation study of empirical sizes for the test applied to normal and nonnormal data suggests th

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

unbalanced repeated measures models with structured covariance matrices the question of how to analyze unbalanced or incomplete repeated measures data is a common problem facing analysts we address this problem through maximum likelihood analysis using a general linear model for expected responses and arbitrary structural models for the within subject covariances models that can be fit include standard univariate and multivariate models with incomplete data random effects models and models with time series and factor analytic error structures we describe newton raphson and fisher scoring algorithms for computing maximum likelihood estimates and generalized em algorithms for computing restricted and unrestricted maximum likelihood estimates an example fitting several models to a set of growth data is included

dimension and rigidity of quasi fuchsian representations let fo c so n 1 n 2 be a cocompact lattice and p fo f be an injective representation into a convex cocompact discrete isometric subgroup of a noncompact rank 1 symmetric space the hausdorff dimension 6 f of the limit set of f p fo satisfies 6 f 6 fo n 1 we prove that equality holds if and only if p is a fuchsian representation i e f preserves a totally geodesic copy of hri in hr this generalizes the result of 2 and settles a question raised by tukia 43 p 428 actually we prove a more general result in the context of variable negative curvature strikingly there are no quasi fuchsian representations at least for the lower codimensional case in complex hyperbolic geometry that is for a cocompact lattice fo c su n 1 n 2 and an injective representation p fo su m 1 n m 2n 1 with f p fo convex cocompact we prove that one always has 6 f 6 fo and moreover f must stabilize a totally geodesic copy of h in h this can be viewed as a global generalization of goldman and millson s local rigidity theorem see 20 another global generalization was obtained by k corlette 5 various other related rigidity results are also obtained

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 summary measures analysis of the linear mixed effects model for repeated measures when data are not missing completely at random subjects often drop out of longitudinal studies prematurely yielding unbalanced data with unequal numbers of measures for each subject a simple and convenient approach to analysis is to develop summary measures for each individual and then regress the summary measures on between subject covariates we examine properties of this approach in the context of the linear mixed effects model when the data are not missing completely at random in the sense that drop out depends on the values of the repeated measures after conditioning on fixed covariates the approach is compared with likelihood based approaches that model the vector of repeated measures for each individual methods are compared by simulation for the case where repeated measures over time are linear and can be summarized by a slope and intercept for each individual our simulations suggest that summary measures analysis based on the slopes alone is comparable to full maximum likelihood when the data are missing completely at random but is markedly inferior when the data are not missing completely at random analysis discarding the incomplete cases is even worse with large biases and very poor confidence coverage

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

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

an autoregressive linear mixed effects model for the analysis of longitudinal data which include dropouts and show profiles approaching asymptotes we are interested in longitudinal data of a continuous response that show profiles with an initial sharp change and approaching asymptotes for each patient and many patients drop out with a reason related to the response in this paper we focus on a model that assumes a dropout process is missing at random mar in this dropout process we can obtain consistent maximum likelihood estimators as long as both the mean and covariance structures are correctly specified however parsimonious covariance structures for the profiles approaching asymptotes are unclear an autoregressive linear mixed effects model can express the profile with random individual asymptotes we show that this model provides a new parsimonious covariance structure the covariance structure at steady state is compound symmetry and the other elements of the covariance depend on the measurement points in simulation studies the estimate of the asymptote is unbiased in mar dropouts but biased in non ignorable dropouts we also applied this model to actual schizophrenia trial data

discussion of double sampling for survival analysis design in various types of biomedical studies subjects drop out for reasons related to failure for example less healthy patients may seek treatment elsewhere or healthier subjects may be discharged from hospital prior to the end of the study typically inference requires strong assumptions about the dropout mechanism baker wax and patterson 1993 bwp the primary motivation of double sampling is to provide information to relax the distributional assumptions there are two basic designs dropout double sampling at the time of dropout a subject is randomly assigned to follow up or no follow up this is the design in frangakis and rubin fr where dropout was loss to follow up failure time was time from surgery to failure of prosthesis and administrative censoring varied depending on time of entry initial double sampling at the start of the study such as time of surgery or randomization to treatment assignment subjects are randomly assigned to either a partial follow up pf or full follow up ff group if a subject in the ff group drops out of the study he is followed until failure or administrative censoring if a subject in the partial followup group drops out he is not followed after dropout this is the design in bwp where dropout was discharge from the hospital failure time was time of infection following surgery and administrative censoring was fixed at 30 days initial double sampling is a special case of dropout double sampling imagine that as a result of initial double sampling each subject is randomly assigned a label either pf or ff and when dropout occurs subjects are randomly assigned to follow up or not based on the pf and ff labels the only difference with dropout double sampling is the additional random assignment to pf and ff among subjects who did not drop out this extra information is not used in either the fr or bwp analysis

estimation and comparison of changes in the presence of informative right censoring conditional linear model a general linear regression model for the usual least squares estimated rate of change slope on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups two noniterative estimators for the group slope means the linear minimum variance unbiased lmvub estimator and the linear minimum mean squared error lmmse estimator are proposed under this conditional model in realistic situations we illustrate that the lmvub and lmmse estimators derived under a simple linear regression model are quite competitive compared to the pseudo maximum likelihood estimator pmle derived by modeling the censoring probabilities generalizations to polynomial response curves and general linear models are also described

the double of a hyperbolic manifold and non positively curved exotic pl structures we give examples of non compact finite volume real hyperbolic manifolds of dimension greater than five such that their doubles admit at least three non equivalent smoothable pl structures two of which admit a riemannian metric of non positive curvature while the third does not we also prove that the doubles of non compact finite volume real hyperbolic manifolds of dimension greater than four are differentiably rigid

isomorphism conjectures in algebraic k theory duff in this paper we are concerned with the four functors y y3 if and which map from the category of topological spaces x to the category diff of q spectra the functor 3 or if maps the space x to the qspectrum of stable topological or smooth pseudoisotopies of x the spaces diff dif 3i x id9 x i 0 are hatcher s deloopings of 970 x yd x cf 29 the functor j maps a path connected space x to the algebraic ktheoretic s2 spectrum for the integral group ring z7r1x the spaces x x i 0 are the gersten wagoner deloopings of z o x cf 27 52 5 x is also defined on a nonpath connected space x to be x1 where xi i e i are the path components of x and ii xi indicates the direct limit of all finite products of the i xi i e i if x is path connected then the functor maps x to the l surgery classifying spaces for oriented surgery problems with fundamental group r i1x identified by their fourfold periodicity k 90 x k 4j x cf 35 39 44 47 in addition if x is not path connected then we set y77 x tiei5 fj xi where xi i e i are the path components of x results obtained by the authors over the past five years cf 15 25 and in particular 19 16 a 1 a 18 together with many earlier results and conjectures which are reviewed in 1 6 1 1 6 6 suggest that for any connected cwcomplex x and for 5 denoting any of the above functors the q spectrum x should be computable in a simple way from the values taken by 9 on covering spaces of x which have very simple fundamental groups in more detail we let 9 x denote the collection of all subgroups h c ir ix which are either finite or virtually infinite cyclic a group h is virtually infinite cyclic if there is a short exact sequence of groups 1 z h f 1 with f equal to a finite group and z equal to the infinite cyclic group for each such group we let xh x denote the covering space projection corresponding to h then the q spectrum 5 x should be computable in a simple way from the q spectra 5 xh h e 9 x the purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any x

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

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

robust estimation in mixed linear models with non monotone missingness we introduce a model to account for abrupt changes among repeated measures with non monotone missingness development of likelihood inferences for such models is hard because it involves intractable integration to obtain the marginal likelihood we use hierarchical likelihood to overcome such difficulty abrupt changes among repeated measures can be well described by introducing random effects in the dispersion a simulation study shows that the resulting estimator is efficient robust against misspecification of fatness of tails for illustration we use a schizophrenic behaviour data presented by rubin and wu

spheres with more than 7 vector fields all the fault of spin 9 we give an interpretation of the maximal number of linearly independent vector fields on spheres in terms of the spin 9 representation on r16 this casts an insight on the role of spin 9 as a subgroup of so 16 on the existence of vector fields on spheres parallel to the one played by complex quaternionic and octonionic structures on r2 r4 and r8 respectively

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

higher symmetries of the laplacian we identify the symmetry algebra of the laplacian on euclidean space as an explicit quotient of the universal enveloping algebra of the lie algebra of conformal motions we construct analogues of these symmetries on a general conformal manifold

the teichmuller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible for a smooth manifold m we define the teichmuller space 2t m of all riemannian metrics on m and the teichmuller space 2t m of pinched negatively curved metrics on m where 0 oo we prove that if m is hyperbolic the natural inclusion f m 2t m is in general not homotopically trivial in particular 2t m is in general not contractible

the analysis of longitudinal ordinal data with nonrandom drop out a model is proposed for longitudinal ordinal data with nonrandom drop out which combines the multivariate dale model for longitudinal ordinal data with a logistic regression model for drop out since response and drop out are modelled as conditionally independent given complete data the resulting likelihood can be maximised relatively simply using the em algorithm which with acceleration is acceptably fast and with appropriate additions can produce estimates of precision the approach is illustrated with an example such modelling of nonrandom drop out requires caution because the interpretation of the fitted models depends on assumptions that are unexaminable in a fundamental sense and the conclusions cannot be regarded as necessarily robust the main role of such modelling may be as a component of a sensitivity analysis

fischer decomposition for the symplectic group we prove the fischer decomposition for the space of spinor valued polynomials defined on euclidean space of four fold dimension in terms of irreducible modules for the symplectic group consisting of so called osp 4 2 monogenics

asymptotic bias in the linear mixed effects model under non ignorable missing data mechanisms summary in longitudinal studies missingness of data is often an unavoidable problem estimators from the linear mixed effects model assume that missing data are missing at random however estimators are biased when this assumption is not met in the paper theoretical results for the asymptotic bias are established under non ignorable drop out drop in and other missing data patterns the asymptotic bias is large when the drop out subjects have only one or no observation especially for slope related parameters of the linear mixed effects model in the drop in case intercept related parameter estimators show substantial asymptotic bias when subjects enter late in the study eight other missing data patterns are considered and these produce asymptotic biases of a variety of magnitudes

an approximation to maximum likelihood estimates in reduced models summary an approximation to the maximum likelihood estimates of the parameters in a model can be obtained from the corresponding estimates and information matrices in an extended model i e a model with additional parameters the approximation is close provided that the data are consistent with the first model applications are described to log linear models for discrete data to models for multivariate normal distributions with special covariance matrices and to mixed discrete continuous models

semiparametric regression models for repeated measures of mortal cohorts with non monotone missing outcomes and time dependent covariates we propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non response to estimate regression parameters interpreted as conditioned on being alive our proposed method accommodates outcomes and time dependent covariates that are missing not at random with non monotone missingness patterns via inverse probability weighting missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse probability weighted estimating equation thus we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates the approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition inflammation and age

a simulation study of estimators for rates of change in longitudinal studies with attrition many longitudinal studies and clinical trials are designed to compare rates of change over time in one or more outcome variables in several groups most such studies have incomplete data because some patients drop out before completing the study the missing data may induce bias and inefficiency in naive estimates of important parameters this paper uses monte carlo methods to compare the bias and efficiency of several two stage estimators of the effect of treatment on the mean rate of change when the missing data arise from one of four processes we also study the validity of confidence intervals and the power of hypothesis tests based on these estimates and their standard errors in general the weighted least squares estimator does relatively well as does an analysis of covariance type estimator proposed by wu et al the best estimates of variance components are based on complete cases or maximum likelihood

bayesian quantile regression for longitudinal studies with nonignorable missing data summary we study quantile regression qr for longitudinal measurements with nonignorable intermittent missing data and dropout compared to conventional mean regression quantile regression can characterize the entire conditional distribution of the outcome variable and is more robust to outliers and misspecification of the error distribution we account for the within subject correlation by introducing a l2 penalty in the usual qr check function to shrink the subject specific intercepts and slopes toward the common population values the informative missing data are assumed to be related to the longitudinal outcome process through the shared latent random effects we assess the performance of the proposed method using simulation studies and illustrate it with data from a pediatric aids clinical trial

reparameterizing the pattern mixture model for sensitivity analyses under informative dropout summary pattern mixture models are frequently used to analyze longitudinal data where missingness is induced by dropout for measured responses it is typical to model the complete data as a mixture of multivariate normal distributions where mixing is done over the dropout distribution fully parameterized pattern mixture models are not identified by incomplete data little 1993 journal of the american statistical association88 125 134 has characterized several identifying restrictions that can be used for model fitting we propose a reparameterization of the pattern mixture model that allows investigation of sensitivity to assumptions about nonidentified parameters in both the mean and variance allows consideration of a wide range of nonignorable missing data mechanisms and has intuitive appeal for eliciting plausible missing data mechanisms the parameterization makes clear an advantage of pattern mixture models over parametric selection models namely that the missing data mechanism can be varied without affecting the marginal distribution of the observed data to illustrate the utility of the new parameterization we analyze data from a recent clinical trial of growth hormone for maintaining muscle strength in the elderly dropout occurs at a high rate and is potentially informative we undertake a detailed sensitivity analysis to understand the impact of missing data assumptions on the inference about the effects of growth hormone on muscle strength