some aspects of polya tree distributions for statistical modelling the definition and elementary properties of polya tree distributions are reviewed two theorems are presented showing that polya trees can be constructed to concentrate arbitrarily closely about any desired pdf and that polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy thereby satisfying a consistency condition such theorems are false for dirichlet processes models are constructed combining partially specified polya trees with other information such as monotonicity or unimodality it is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications a numerical example is given a theorem of diaconis and freedman about dirichlet processes is generalized to polya trees allowing polya trees to be the models for errors in regression problems finally empirical bayes models using dirichlet processes are generalized to polya trees an example from berry and christensen is reanalyzed with a polya tree model

model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

a hybrid estimator in nonlinear and generalised linear mixed effects models summary a hybrid method that combines laplace s approximation and monte carlo simulations to evaluate integrals in the likelihood function is proposed for estimation of the parameters in nonlinear mixed effects models that assume a normal parametric family for the random effects simulations show that these parametric estimates of fixed effects are close to the nonparametric estimates even though the mixing distribution is far from the assumed normal parametric family an asymptotic theory of this hybrid method for parametric estimation without requiring the true mixing distribution to belong to the assumed parametric family is developed to explain these results this hybrid method and its asymptotic theory are also extended to generalised linear mixed effects models

nonlinear multilevel models with an application to discrete response data summary a procedure is proposed for the analysis of multilevel nonlinear models using a linearization the case of log linear models for discrete response data is studied in detail nonlinear models arise in a number of circumstances notably when modelling discrete data in this paper we consider the multilevel nonlinear model as in linear multilevel models we shall consider the general case where any of the model coefficients can be random at any level and where the random parameters may also be specified functions of the fixed parameter estimates discussed by h goldstein r prosser and j rasbash in an as yet unpublished report in the next two sections we set out the model and define notation this is followed by a section on estimation and then some examples

conjugate priors for exponential families having quadratic variance functions abstract consider a natural exponential family parameterized by it is well known that the standard conjugate prior on is characterized by a condition of posterior linearity for the expectation of the model mean parameter often however this family is not parameterized in terms of but rather in terms of a more usual parameter such at the mean the main question we address is under what conditions does a standard conjugate prior on induce a linear posterior expectation on itself we prove that essentially this happens iff the exponential family has quadratic variance function a consequence of this result is that the standard conjugate on coincides with the prior on induced by the standard conjugate on iff the variance function is quadratic the rest of the article covers more specific issues related to conjugate priors for exponential families in particular we analyze the monotonicity of the expected posterior variance for with respect to the sample size and the hyperparameter

estimation in generalized mixed models regression models containing fixed and random effects may have a response variable which is not normally distributed the generalized mixed model includes both discrete and continuous response variables and is developed here for problems in which the regression variables enter linearly into the model best linear unbiased predictor methods are extended to maximum likelihood and residual maximum likelihood estimation procedures applications in modelling discrete response variables and in survival analysis are discussed

multilevel time series models with applications to repeated measures data the analysis of repeated measures data can be conducted efficiently using a two level random coefficients model a standard assumption is that the within individual level 1 residuals are uncorrelated in some cases especially where measurements are made close together in time this may not be reasonable and this additional correlation structure should also be modelled a time series model for such data is proposed which consists of a standard multilevel model for repeated measures data augmented by an autocorrelation model for the level 1 residuals first and second order autoregressive models are considered in detail together with a seasonal component both discrete and continuous time are considered and it is shown how the autocorrelation parameters can themselves be structured in terms of further explanatory variables the models are fitted to a data set consisting of repeated height measurements on children

hierarchical poisson regression modeling abstract the poisson model and analyses here feature nonexchangeable gamma distributions although exchangeable following a scale transformation for individual parameters with standard deviations proportional to means a relatively uninformative prior distribution for the shrinkage values eliminates the ill behavior of maximum likelihood estimators of the variance components when tested in simulation studies the resulting procedure provides better coverage probabilities and smaller risk than several other published rules and thus works well from bayesian and frequentist perspectives alike the computations provide fast accurate density approximations to individual parameters and to structural regression coefficients the computer program is publicly available through statlib

on the precision of the conditionally autoregressive prior in spatial models summary bayesian analyses of spatial data often use a conditionally autoregressive car prior which can be written as the kernel of an improper density that depends on a precision parameter that is typically unknown to include in the bayesian analysis the kernel must be multiplied by k for some k this article rigorously derives k n i 2 for the l2 norm car prior also called a gaussian markov random field model and k n i for the l1 norm car prior where n is the number of regions and i the number of islands disconnected groups of regions in the spatial map since i 1 for a spatial structure defining a connected graph this supports knorr held s 2002 in highly structured stochastic systems 260 264 suggestion that k n 1 2 in the l2 norm case instead of the more common k n 2 we illustrate the practical significance of our results using a periodontal example

generalized linear models with random effects salamander mating revisited in recent years much effort has been devoted to extending regression methodology to non gaussian data where responses are not independent these methods for dependent responses are suitable for data from longitudinal studies or nested designs however use of these methods for crossed designs seems to have serious limitations due to the intensive computations involved because of the intractable nature of the joint distribution in this paper we cast the problem in a bayesian framework and use a monte carlo method the gibbs sampler to avoid current computational limitations the flexibility of this approach is illustrated by analyzing the interesting salamander mating data reported by mccullagh and nelder 1989 generalized linear models 2nd edition london chapman and hall

homogeneity for open partitions of pairs of reals we prove a partition theorem for analytic sets of reals namely if a r is analytic and a 2 k 0 k 1 with k 0 relatively open then either there is a perfect 0 homogeneous subset or a is a countable union of 1 homogeneous subsets we also show that such a partition property for coanalytic sets is the same as that each uncountable coanalytic set contains a perfect subset a two person game for this partition property is also studied there are some applications of such partition properties

the autohomeomorphism group of the cech stone compactification of the integers it is shown to be consistent that there is a nontrivial autohomeomorphism of n n yet all such autohomeomorphisms are trivial on a dense p ideal furthermore the cardinality of the autohomeomorphism group of n n can be any regular cardinal between 2 n0 and 2 2n0 the model used is one due to velickovic in which coincidentally martin s axiom also holds

laplace s approximation for nonlinear mixed models summary an approximation to laplace s method for integrals is applied to marginal distributions of data arising from models in which both fixed and random effects enter nonlinearly the approach provides alternative derivations of some recent algorithms for fitting such models and it has direct ties with gaussian restricted maximum likelihood and the accompanying mixed model equations

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

a five element basis for the uncountable linear orders in this paper i will show that it is relatively consistent with the usual axioms of mathematics zfc together with a strong form of the axiom of infinity the existence of a supercompact cardinal that the class of uncountable linear orders has a five element basis in fact such a basis follows from the proper forcing axiom a strong form of the baire category theorem the elements are x o1 o 1 c c where x is any suborder of the reals of cardinality 1 and c is any countryman line this confirms a longstanding conjecture of shelah

diagnostics for mixed model analysis of variance we describe a new method for assessment of model inadequacy in maximum likelihood mixed model analysis of variance in particular we discuss its use in diagnosing perturbations from the usual assumption of constant error variance and from the assumption that each realization of a given random factor has been drawn from the same normal population computer implementation of the procedure is described and an example is presented involving the analysis of filter cartridges used with commercial respirators

strong homology is not additive using the continuum hypothesis ch we show that strong ho mology groups hp x do not satisfy milnor s additivity axiom moreover ch implies that strong homology does not have compact supports and that hp x need not vanish for p 0

hierarchical generalized linear models we consider hierarchical generalized linear models which allow extra error components in the linear predictors of generalized linear models the distribution of these components is not restricted to be normal this allows a broader class of models which includes generalized linear mixed models we use a generalization of henderson s joint likelihood called a hierarchical or h likelihood for inferences from hierarchical generalized linear models this avoids the integration that is necessary when marginal likelihood is used under appropriate conditions maximizing the h likelihood gives fixed effect estimators that are asymptotically equivalent to those obtained from the use of marginal likelihood at the same time we obtain the random effect estimates that are asymptotically best unbiased predictors an adjusted profile h likelihood is shown to give the required generalization of restricted maximum likelihood for the estimation of dispersion components a scaled deviance test for the goodness of fit a model selection criterion for choosing between various dispersion models and a graphical method for checking the distributional assumption of random effects are proposed the ideas of quasi likelihood and extended quasi likelihood are generalized to the new class we give examples of the poisson gamma binomial beta and gamma inverse gamma hierarchical generalized linear models a resolution is proposed for the apparent difference between population averaged and subject specific models a unified framework is provided for viewing and extending many existing methods

bayesian spatial modeling of disease risk in relation to multivariate environmental risk fields the relationship between exposure to environmental chemicals during pregnancy and early childhood development is an important issue which has a spatial risk component in this context we have examined mental retardation and developmental delay mrdd outcome measures for children in a medicaid population in south carolina and sampled measures of soil chemistry e g as hg etc on a network of sites which are misaligned to the outcome residential addresses during pregnancy the true chemical concentration at the residential addresses is not observed directly and must be interpolated from soil samples in this study we have developed a bayesian joint model which interpolates soil chemical fields and estimates the associated mrdd risk simultaneously having multiple spatial fields to interpolate we have considered a low rank kriging method for the interpolation which requires less computation than bayesian kriging we performed a sensitivity analysis for a bivariate smoothing changing the number of knots and the smoothing parameter these analyses show that a low rank kriging method can be used as an alternative to a full rank kriging reducing computational burden however the number of knots for the low rank kriging model need to be selected with caution as a bivariate surface estimation can be sensitive to the choice of the number of knots

empirical bayes estimates of age standardized relative risks for use in disease mapping there have been many attempts in recent years to map incidence and mortality from diseases such as cancer such maps usually display either relative rates in each district as measured by a standardized mortality ratio smr or some similar index or the statistical significance level for a test of the difference between the rates in that district and elsewhere neither of these approaches is fully satisfactory and we propose a new approach using empirical bayes estimation the resulting estimators represent a weighted compromise between the smr the overall mean relative rate and a local mean of the relative rate in nearby areas the compromise solution depends on the reliability of each individual smr and on estimates of the overall amount of dispersion of relative rates over different districts

filtrations of global equivariant k theory arone and lesh j reine angew math 604 73 136 2007 fund math 207 1 29 70 2010 constructed and studied spectrum level filtrations that interpolate between connective topological or algebraic k theory and the eilenberg maclane spectrum for the integers in this paper we consider global equivariant generalizations of these filtrations and another closely related class of filtrations the modified rank filtrations of the k theory spectra themselves we lift arone and lesh s description of the filtration subquotients to the equivariant context and apply it to compute algebraic filtrations on representation rings that arise on equivariant homotopy groups it turns out that these representation ring filtrations are considerably easier to express in a global equivariant context than over a fixed compact lie group furthermore they have formal similarities to the filtration on burnside rings induced by the symmetric products of spheres which was computed by schwede j am math soc 30 3 673 711 2017

estimation in generalized linear models with random effects summary a conceptually very simple but general algorithm for the estimation of the fixed effects random effects and components of dispersion in generalized linear models with random effects is proposed conditions are described under which the algorithm yields approximate maximum likelihood or quasi maximum likelihood estimates of the fixed effects and dispersion components and approximate empirical bayes estimates of the random effects the algorithm is applied to two data sets to illustrate the estimation of components of dispersion and the modelling of overdispersion

spatial regression and spillover effects in cluster randomized trials with count outcomes this paper describes methodology for analyzing data from cluster randomized trials with count outcomes taking indirect effects as well spatial effects into account indirect effects are modeled using a novel application of a measure of depth within the intervention arm both direct and indirect effects can be estimated accurately even when the proposed model is misspecified we use spatial regression models with gaussian random effects where the individual outcomes have distributions overdispersed with respect to the poisson and the corresponding direct and indirect effects have a marginal interpretation to avoid spatial confounding we use orthogonal regression in which random effects represent spatial dependence using a homoscedastic and dimensionally reduced modification of the intrinsic conditional autoregression model we illustrate the methodology using spatial data from a pair matched cluster randomized trial against the dengue mosquito vector aedes aegypti done in trujillo venezuela

automorphisms of corona algebras and group cohomology in 2007 phillips and weaver showed that assuming the continuum hypothesis there exists an outer automorphism of the calkin algebra the calkin algebra is the algebra of bounded operators on a separable complex hilbert space modulo the compact operators in this paper we establish that the analogous conclusion holds for a broad family of quotient algebras specifically we will show that assuming the continuum hypothesis if a is a separable algebra which is either simple or stable then the corona of a has nontrivial automorphisms we also discuss a connection with cohomology theory namely that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

residuals in generalized linear models abstract generalized linear models are regression type models for data not normally distributed appropriately fitted by maximum likelihood rather than least squares typical examples are models for binomial or poisson data with a linear regression model for a given ordinarily nonlinear function of the expected values of the observations use of such models has become very common in recent years and there is a clear need to study the issue of appropriate residuals to be used for diagnostic purposes several definitions of residuals are possible for generalized linear models the statistical package glim baker and nelder 1978 routinely prints out residuals where v is the function relating the variance to the mean of y and is the maximum likelihood estimate of the ith mean as fitted to the regression model these residuals are the signed square roots of the contributions to the pearson goodness of fit statistic another choice of residual is the signed square root of the contribution to the devia

the simulation smoother for time series models summary recently suggested procedures for simulating from the posterior density of states given a gaussian state space time series are refined and extended we introduce and study the simulation smoother which draws from the multivariate posterior distribution of the disturbances of the model so avoiding the degeneracies inherent in state samplers the technique is important in gibbs sampling with non gaussian time series models and for performing bayesian analysis of gaussian time series

random effects models for serial observations with binary response this paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants each subject s serial responses are assumed to arise from a logistic model but with regression coefficients that vary between subjects the logistic regression parameters are assumed to be normally distributed in the population inference is based upon maximum likelihood estimation of fixed effects and variance components and empirical bayes estimation of random effects exact solutions are analytically and computationally infeasible but an approximation based on the mode of the posterior distribution of the random parameters is proposed and is implemented by means of the em algorithm this approximate method is compared with a simpler two step method proposed by korn and whittemore using data from a panel study of asthmatics originally described in that paper one advantage of the estimation strategy described here is the ability to use all of the data including that from subjects with insufficient data to permit fitting of a separate logistic regression model as required by the korn and whittemore method however the new method is computationally intensive

bias correction in generalised linear mixed models with a single component of dispersion summary general expressions are derived for the asymptotic biases in three approximate estimators of regression coefficients and variance component for small values of the variance component in generalised linear mixed models with canonical link function and a single source of extraneous variation the estimators involve first and second order laplace expansions of the integrated likelihood and a related procedure known as penalised quasilikelihood numerical studies of a series of matched pairs of binary outcomes show that the first order estimators of the variance component are seriously biased easily computed correction factors produce satisfactory estimators of small variance components comparable to those obtained with a second order laplace expansion and markedly improve the asymptotic performance for larger values for a series of matched pairs of binomial observations the variance correction factors rapidly approach one as the binomial denominators increase these results greatly extend the range of parameter values for which the approximate estimation procedures have satisfactory asymptotic properties

quasi likelihood for multiplicative random effects summary a quasi likelihood approach similar to that described by morton 1987 is found to be particularly simple in the case of measurements subject to two or more sources of multiplicative error still further simplification occurs when the effects of interest also are multiplicative the multiplicative analogue of the balanced split plot model is considered in detail an analysis of gamma deviance replaces the familiar analysis of variance the efficiency of the quasi likelihood method relative to a more conventional analysis of logarithms of the data is explored briefly

partitioning degrees of freedom in hierarchical and other richly parameterized models hodges and sargent 2001 have developed a measure of a hierarchical model s complexity degrees of freedom df that is consistent with definitions for scatterplot smoothers is interpretable in terms of simple models and enables control of a fit s complexity by means of a prior distribution on complexity but although df describes the complexity of the whole fitted model in general it remains unclear how to allocate df to individual effects here we present a new definition of df for arbitrary normal error linear hierarchical models consistent with that of hodges and sargent that naturally partitions the n observations into df for individual effects and for error the new conception of an effect s df is the ratio of the effect s modeled variance matrix to the total variance matrix this provides a way to describe the sizes of different parts of a model e g spatial clustering vs heterogeneity to place df based priors on smoothing parameters and to describe how a smoothed effect competes with

a filtration of spectra arising from families of subgroups of symmetric groups let ffn be a family of subgroups of enx which is closed under taking subgroups and conjugates such a family has a classifying space byfn and we showed in an earlier paper that a cormpatible choice of tfn for each n gives a simplicial monoid jjn btf which group completes to an infinite loop space in this paper we define a filtration of the associated spectrum whose filtration quotients given an extra condition on the families can be identified in terms of the classifying spaces of the families of subgroups that were chosen this gives a way to go from group theoretic data about the families to homotopy theoretic information about the associated spectrum we calculate two examples the first is related to elementary abelian p groups and the second gives a new expression for the desuspension of sp so spm i so as a suspension spectrum

effects of residual smoothing on the posterior of the fixed effects in disease mapping models disease mapping models for areal data often have fixed effects to measure the effect of spatially varying covariates and random effects with a conditionally autoregressive car prior to account for spatial clustering in such spatial regressions the objective may be to estimate the fixed effects while accounting for the spatial correlation but adding the car random effects can cause large changes in the posterior mean and variance of fixed effects compared to the nonspatial regression model this article explores the impact of adding spatial random effects on fixed effect estimates and posterior variance diagnostics are proposed to measure posterior variance inflation from collinearity between the fixed effect covariates and the car random effects and to measure each region s influence on the change in the fixed effect s estimates by adding the car random effects a new model that alleviates the collinearity between the fixed effect covariates and the car random effects is developed and extensions of these methods to point referenced data models are discussed

hierarchical likelihood approach for frailty models systematic likelihood inference for frailty models is possible using lee nelder s 1996 hierarchical likelihood we develop an inferential method for frailty models via hierarchical likelihood which gives a simple unified framework and a numerically efficient algorithm simulations and practical examples are presented to illustrate the new method

the beta geometric distribution applied to comparative fecundability studies a convenient measure of fecundability is time number of menstrual cycles required to achieve pregnancy couples attempting pregnancy are heterogeneous in their per cycle probability of success if success probabilities vary among couples according to a beta distribution then cycles to pregnancy will have a beta geometric distribution the likelihood ratio test can thus be employed in studies of exposures that may impair fecundability covariates are incorporated in a natural way the model is illustrated by applying it to data on cycles to pregnancy in smokers and nonsmokers with adjustment for covariates summary in fre excerpt

an extended quasi likelihood function summary wedderburn s original definition of quasi likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions the relationship between generalized linear models and the use of transformations of the response variable is explored and the ideas are illustrated by three examples generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined in making this extension wedderburn widened the scope of generalized linear models in a way very similar to that of gauss when he replaced the assumption of normality in classical linear models by that of equal variance for generalized linear models with distributions in the exponential family likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions these methods are also applicable with wedderburn s form of quasi likelihood however neither of these methods is suitable for the comparison of different variance functions in this paper we introduce an extended quasi likelihood function which allows for the comparison of various forms of all the components of a generalized linear model i e the linear predictor the link function and the variance function we then apply the ideas to the analysis of several sets of data

posterior mode estimation by extended kalman filtering for multivariate dynamic generalized linear models abstract a family of multivariate dynamic generalized linear models is introduced as a general framework for the analysis of time series with observations from the exponential family besides common conditionally gaussian models this article deals with univariate models for counted and binary data and as the most interesting multivariate case models for nonstationary multicategorical time series for univariate responses a related yet different class of models has been introduced in a bayesian setting by west harrison and migon assuming conjugate prior posterior distributions for the natural parameter of the exponential family they derive an approximate filter for estimation of time varying states or parameters however their method raises some problems in particular in extending it to the multivariate case a different approach to filtering and smoothing is chosen in this article to avoid a full bayesian analysis based on numerical integration which becomes computationally critical for higher

dynamic linear model diagnostics summary in time series analysis using dynamic linear models retrospective analysis involves the calculation of filtered or smoothed distributions for state parameters in the past we develop and illustrate novel results that are useful in retrospective assessment of the influence of individual observations on such distributions in particular new and computationally simple filtering equations are derived for past state parameters based on leaving out one observation at a time providing dynamic model based versions of methods currently used in standard static regression diagnostics

spatial analyses of periodontal data using conditionally autoregressive priors having two classes of neighbor relations attachment loss the extent of a tooth s root in millimeters that is no longer attached to surrounding bone by periodontal ligament is often used to measure the current state of a patient s periodontal disease and monitor disease progression attachment loss data can be analyzed using a conditionally autoregressive car prior distribution that smooths fitted values toward neighboring values however it may be desirable to have more than one class of neighbor relation in the spatial structure so the different classes of neighbor relations can induce different degrees of smoothing for example we may wish to allow smoothing of neighbor pairs bridging the gap between teeth to differ from smoothing of pairs that do not bridge such gaps adequately modeling the spatial structure may improve the monitoring of periodontal disease progression this article develops a two neighbor relation car model to handle this situation and presents associated theory to help explain the sometimes unusual posterior distr

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

on the consistency strength of the proper forcing axiom abstract in recent work the second author extended combinatorial principles due to jech and magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals for inaccessible cardinals these modifications have no effect and the resulting principles still give the same characterization of large cardinals we prove that the proper forcing axiom pfa implies these principles hold for 2 using this we argue to show that any of the known methods for forcing models of pfa from a large cardinal assumption requires a strongly compact cardinal if one forces pfa using a proper forcing then we get the optimal result that a supercompact cardinal is necessary

marginal regression analysis of a multivariate binary response we propose the use of the mean parameter for regression analysis of a multivariate binary response we model the association using dependence ratios defined in terms of the mean parameter the components of which are the joint success probabilities of all orders this permits flexible modelling of higher order associations using maximum likelihood estimation we reanalyse two data sets one with variable cluster size and the other a longitudinal data set with constant cluster size

the commutant of l h in its ultrapower may or may not be trivial kirchberg asked in 2004 whether the commutant of l h in its norm ultrapower is trivial assuming the continuum hypothesis we prove that the answer depends on the choice of the ultrafilter

a simple derivation of deletion diagnostic results for the general linear model with correlated errors a general simple and intuitive derivation is provided for diagnostics associated with the deletion of arbitrary subsets for the linear model with general covariance structure these are seen to be most simply expressed even for the well studied case of independent and identically distributed data in terms of a residual known variously as the conditional residual the deletion prediction residual and the cross validation residual particularly simple specializations arise when the subsets are of size 1 and of size 2 but the method is easy to apply for all subsets and to conditional deletions

some covariance models for longitudinal count data with overdispersion a family of covariance models for longitudinal counts with predictive covariates is presented these models account for overdispersion heteroscedasticity and dependence among repeated observations the approach is a quasi likelihood regression similar to the formulation given by liang and zeger 1986 biometrika 73 13 22 generalized estimating equations for both the covariate parameters and the variance covariance parameters are presented large sample properties of the parameter estimates are derived the proposed methods are illustrated by an analysis of epileptic seizure count data arising from a study of progabide as an adjuvant therapy for partial seizures

bayesian modeling of multivariate spatial binary data with applications to dental caries dental research gives rise to data with potentially complex correlation structure assessments of dental caries yield a binary outcome indicating the presence or absence of caries experience for each surface of each tooth in a subject s mouth in addition to this nesting caries outcome exhibit spatial structure among neighboring teeth we develop a bayesian multivariate model for spatial binary data using random effects autologistic regression that controls for the correlation within tooth surfaces and spatial correlation among neighboring teeth using a sample from a clinical study conducted at the medical university of south carolina we compare this autologistic model with covariates to alternative models to demonstrate the improvement in predictions and also to assess the effects of covariates on caries experience

hierarchical generalized linear models and frailty models with bayesian nonparametric mixing summary this paper proposes bayesian nonparametric mixing for some well known and popular models the distribution of the observations is assumed to contain an unknown mixed effects term which includes a fixed effects term a function of the observed covariates and an additive or multiplicative random effects term typically these random effects are assumed to be independent of the observed covariates and independent and identically distributed from a distribution from some known parametric family this assumption may be suspect if either there is interaction between observed covariates and unobserved covariates or the fixed effects predictor of observed covariates is misspecified another cause for concern might be simply that the covariates affect more than just the location of the mixed effects distribution as a consequence the distribution of the random effects could be highly irregular in modality and skewness leaving parametric families unable to model the distribution adequately this paper therefore proposes a bayesian nonparametric prior for the random effects to capture possible deviances in modality and skewness and to explore the observed covariates effect on the distribution of the mixed effects

likelihood quasi likelihood and pseudolikelihood some comparisons summary there is considerable interest in the fitting of models jointly to the mean and dispersion of a response for the mean parameter the wedderburn estimating equations are widely accepted however there is some controversy about estimating the dispersion parameters finite sampling properties of several dispersion estimators are investigated for three models by simulation we compare the maximum extended quasi likelihood estimator the maximum pseudolikelihood estimator and the maximum likelihood estimator if it exists of these estimators the maximum extended quasi likelihood estimator is usually superior in minimizing the mean squared error

prediction diagnostics for spatial linear models case deletion diagnostics are developed for detecting observations that are influential for prediction in linear models with a general covariance matrix a primary application of such results is in universal kriging and the related methodologies of ordinary kriging and intrinsic random function kriging in these applications the linear model and the covariance matrix of the data are determined by the locations at which observations are taken computational formulae are given that make the procedures feasible the diagnostics are illustrated using an example

analysis of variance why it is more important than ever analysis of variance anova is an extremely important method in exploratory and confirmatory data analysis unfortunately in complex problems e g split plot designs it is not always easy to set up an appropriate anova we propose a hierarchical analysis that automatically gives the correct anova comparisons even in complex scenarios the inferences for all means and variances are performed under a model with a separate batch of effects for each row of the anova table we connect to classical anova by working with finite sample variance components fixed and random effects models are characterized by inferences about existing levels of a factor and new levels respectively we also introduce a new graphical display showing inferences about the standard deviations of each batch of effects we illustrate with two examples from our applied data analysis first illustrating the usefulness of our hierarchical computations and displays and second showing how the ideas of anova are helpful in understanding a previously fit hierarchical model

laplace approximation of high dimensional integrals summary it is shown that the usual laplace approximation is not a valid asymptotic approximation when the dimension of the integral is comparable with the limiting parameter n the formal laplace expansion for multidimensional integrals is given and used to construct asymptotic approximations for high dimensional integrals one example is considered in which the dimension of the integral is o n 1 2 and the relative error of the unmodified laplace approximation is 0 1 nevertheless it is possible to construct a valid asymptotic expansion by regrouping terms in the formal expansion according to asymptotic order in n

the analysis of binomial data by a generalized linear mixed model summary methods for generalized linear models are extended to provide estimates of location and variance parameters for mixed models fitted to binomial data formed by classifying samples from an underlying normal distribution the method estimates the parameters directly on the underlying scale for a balanced one way random effects model the variance estimator simplifies to the usual analysis of variance one the estimation of variances and the prediction of random effects for binomial traits is required by animal breeders the predictors given are analogous to best linear unbiased predictors henderson 1973 but differ from those presented by harville mee 1984

generalized linear models with random effects a gibbs sampling approach abstract generalized linear models have unified the approach to regression for a wide variety of discrete continuous and censored response variables that can be assumed to be independent across experimental units in applications such as longitudinal studies genetic studies of families and survey sampling observations may be obtained in clusters responses from the same cluster cannot be assumed to be independent with linear models correlation has been effectively modeled by assuming there are cluster specific random effects that derive from an underlying mixing distribution extensions of generalized linear models to include random effects has thus far been hampered by the need for numerical integration to evaluate likelihoods in this article we cast the generalized linear random effects model in a bayesian framework and use a monte carlo method the gibbs sampler to overcome the current computational limitations the resulting algorithm is flexible to easily accommodate changes in the number

two graphical displays for outlying and influential observations in regression summary the paper describes two procedures for detecting observations with outlying values either in the response variable or in the explanatory variables in multiple regression these procedures are presented as half normal plots with envelopes derived from simulation in order to avoid overinterpretation of the data analysis of a well known data set leads to the use of a data transformation a simple test for which is commended and to some comments on the relationship with robust regression the widespread availability of sophisticated computer software has made the fitting of multiple regression equations painless but the very ease of these procedures may cause insufficient care to be given to checking and scrutiny of the data transcription punching and data manipulation errors may lead to bad values both of the observations y and of the independent or explanatory variables x it is the purpose of the present paper to describe and exemplify two plots which provide checks against such bad values it is argued that these plots and a test for transformations should accompany any thorough regression analysis bad values of the response or outliers have long been detected by a variety of plots of residuals bad values of the explanatory variables are less easily detected even when they lead to an extreme point in the design space since the fitted equation will pass close to such an influential observation the residual may not be especially large even after allowance has been made for the small variance of the fitted value at such a point to detect such behaviour a quantity is needed which exhibits the dependence of the fitted model on each point or group of points discussions of such quantities are given by hoaglin welsch 1978 and more recently by belsley kuh welsch 1980 2 1 by cook weisberg 1980 and by pregibon 1981 the quantity used here is a modification of a statistic proposed by cook 1977 which is derived in 2 two half normal plots are presented in 3 as an aid to interpretation envelopes to the plots are generated by simulation examples of the plots are given in 4 both for simulated data and for the stack loss data given by brownlee 1965 p 454 transformations of this data set are discussed in 5 the paper concludes with some comments on the relationship with robust

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

saturation and elementary equivalence of c algebras we study the saturation properties of several classes of c algebras saturation has been shown by farah and hart to unify the proofs of several properties of coronas of unital c algebras we extend their results by showing that some coronas of non unital c algebras are countably degree 1 saturated we then relate saturation of the abelian c algebra c x where x is 0 dimensional to topological properties of x particularly the saturation of cl x we also characterize elementary equivalence of the algebras c x in terms of cl x when x is 0 dimensional and show that elementary equivalence of the generalized calkin algebras of densities and implies elementary equivalence of the ordinals and

using hierarchical likelihood for missing data problems most statistical solutions to the problem of statistical inference with missing data involve integration or expectation this can be done in many ways directly or indirectly analytically or numerically deterministically or stochastically missing data problems can be formulated in terms of latent random variables so that hierarchical likelihood methods of lee nelder 1996 can be applied to missing value problems to provide one solution to the problem of integration of the likelihood the resulting methods effectively use a laplace approximation to the marginal likelihood with an additional adjustment to the measures of precision to accommodate the estimation of the fixed effects parameters we first consider missing at random cases where problems are simpler to handle because the integration does not need to involve the missing value mechanism and then consider missing not at random cases we also study tobit regression and refit the missing not at random selection model to the antidepressant trial data analyzed in diggle kenward 1994

case deletion diagnostics for mixed models mixed linear models arise in many areas of application standard estimation methods for mixed models are sensitive to bizarre observations such influential observations can completely distort an analysis and lead to inappropriate actions and conclusions we develop case deletion diagnostics for detecting influential observations in mixed linear models diagnostics for both fixed effects and variance components are proposed computational formulas are given that make the procedures feasible the methods are illustrated using examples

equivariant properties of symmetric products sp n x x n n we denote by sp n fsp n s m gm 0 the spectrum whose terms are the n th symmetric products of spheres a celebrated theorem of dold and thom 7 asserts that sp 1 s m is an eilenberg mac lane space of type z m for m 1 so sp 1 is an eilenberg mac lane spectrum for the group z the resulting ltration

a generalized linear model with nested strata of extra poisson variation on etend la definition de quasivraisemblance pour tenir compte de plusieurs strates de variation emboitees dans un modele lineaire generalise

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

likelihood inference in a correlated probit regression model correlated binary observations arise in a variety of applications for example in animal studies the term litter effect is used to describe the greater alikeness of responses within a litter as compared to that between litters at a given set of experimental conditions in a setting that motivated this work cells on individual atomic bomb survivors were scored as to the presence or absence of chromosomal aberrations the number of aberrant cells varied among subjects at a specified radiation exposure estimate age at exposure city of exposure and sex in a manner that substantially exceeds that consistent with a binomial error structure otake prentice 1984 this extra binomial variation or overdispersion reflects the fact that the binary response of cells from a given survivor tend to be more alike than are the responses of cells from distinct survivors having the same estimated radiation exposure level and other covariate values such overdispersion may result from individual differences in susceptibility to radiation damage from omitted covariates or most plausibly in this setting from substantial random errors in the estimated radiation exposure levels failure to acknowledge overdispersion may lead to serious underestimation of the standard errors associated with regression parameter estimates and to unduly precise inferences

restricted spatial regression methods implications for inference the issue of spatial confounding between the spatial random effect and the fixed effects in regression analyses has been identified as a concern in the statistical literature multiple authors have

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

dimension reduction and alleviation of confounding for spatial generalized linear mixed models summary non gaussian spatial data are very common in many disciplines for instance count data are common in disease mapping and binary data are common in ecology when fitting spatial regressions for such data one needs to account for dependence to ensure reliable inference for the regression coefficients the spatial generalized linear mixed model offers a very popular and flexible approach to modelling such data but this model suffers from two major shortcomings variance inflation due to spatial confounding and high dimensional spatial random effects that make fully bayesian inference for such models computationally challenging we propose a new parameterization of the spatial generalized linear mixed model that alleviates spatial confounding and speeds computation by greatly reducing the dimension of the spatial random effects we illustrate the application of our approach to simulated binary count and gaussian spatial data sets and to a large infant mortality data set

analysis of covariance and standardization as instances of prediction in this paper prediction provides the basis for unifying the procedures of covariances adjustment and standardization analysis of covariance is a method of forming predictions from a linear model it is used when qualitative effects are to be studied and the effects of continuous variables are to be adjusted for an essential feature is the division into effects of interest and effects for which adjustment is required covariates may also be qualitative as such they are used implicitly in experimental designs with blocks where treatment effects are adjusted for the effect of blocks the technique of standardization is well known in epidemiology and demography as a method of adjusting explicitly for qualitative effects the same division of effects applies when an analysis that uses generalized linear models is summarized two distinct types of prediction which give identical results in classical linear models are available prediction may be conditional on a fixed value of a covariate or marginal on a distribution of values such as the distribution in the set of data being analysed prediction methods are illustrated by the analysis of a table of proportions by use of a logit model

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

bias correction in generalized linear mixed models with multiple components of dispersion abstract general formulas are derived for the asymptotic bias in regression coefficients and variance components estimated by penalized quasi likelihood pql in generalized linear mixed models with canonical link function and multiple sets of independent random effects easily computed correction matrices result in variance component estimates that have satisfactory asymptotic behavior for small values of the variance components and significantly reduce bias for larger values both first order and second order correction procedures are developed for regression coefficients estimated by pql the methods are illustrated through an analysis of an experiment on salamander matings involving crossed male and female random effects and their properties are evaluated in a simulation study

c algebras with and without increasing approximate units abstract for elements a b of a c algebra we denote a a b by a b we show that all 1 unital c algebras have increasing approximate units extending a classical result for unital c algebras we also construct in zfc the first examples of c algebras with no increasing approximate unit one of these examples is a c subalgebra of b l 2 1 these examples are by necessity not approximately finite dimensional af but some of them are still scattered and so locally finite dimensional lf in the sense of farah and katsura it follows that there are scattered c algebras which are not af we further show that the existence of a c subalgebra of b l 2 with no increasing approximate unit or the existence of an lf but not af c subalgebra of b l 2 are independent of zfc our examples also show that an extension of an af algebra by an af algebra does not have to be an af algebra in the nonseparable case

localization and completion theorems for mu module spectra let g be a finite extension of a torus working with highly structured ring and module spectra let m be any module over mu examples include all of the standard homotopical mu modules such as the brown peterson and morava k theory spectra we shall prove localization and completion theorems for the computation of m bg and m bg the g spectrum mug that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum sg and there is an mug module mg whose underlying mu module is m this allows the use of topological analogues of constructions in commutative algebra the computation ofm bg andm bg is expressed in terms of spectral sequences whose respective e2 terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring mu and its module m the central feature of the proof is a new norm map in equivariant stable homotopy theory the construction of which involves the new concept of a global i functor with smash product

all automorphisms of the calkin algebra are inner we prove that it is relatively consistent with the usual ax ioms of mathematics that all automorphisms of the calkin algebra are inner together with a 2006 phillips weaver construction of an outer automorphism using the continuum hypothesis this gives a complete solution to a 1977 problem of brown douglas fillmore we also give a simpler and self contained proof of the phillips weaver result

robust empirical bayes analyses of event rates a collection of i similar items generates point event histories for example machines experience failures or operators make mistakes suppose the intervals between events are modeled as iid exponential i or the counts as poisson i t i for the ith item furthermore so as to represent between item variability each individual rate parameter i is presumed drawn from a fixed super population with density g being a vector parameter a parametric empirical bayes peb setup for g specified alternatively as log student t n or gamma we exhibit the results of numerical procedures for estimating superpopulation parameters ll and for describing pooled estimates of the individual rates i obtained via bayes s formula three data sets are analyzed and convenient explicit approximate formulas are furnished for i estimates in the student t case the individual estimates are seen to have a robust quality

heterogeneity in mantel haenszel type models summary the mantel haenszel problem involves inferences about a common odds ratio in a set of 2 x 2 tables although it is a fairly standard practice to test whether the odds ratios are indeed constant there is remarkably little methodology available for proceeding when there is evidence of some heterogeneity our interest is in models where the log odds ratios ok for tables k 1 2 k are thought of as a sample from a population with mean 0 and standard deviation a and inferences are desired regarding the parameters 0 a by mantel haenszel type models we mean to include the generalization involving pairs of poisson observations rather than 2 x 2 tables where the ok are logarithms of ratios of the poisson means within pairs direct computation of the likelihood function for 0 a in these settings involves numerical integration and the main point here is a simple approximation to this likelihood the approximation is based on laplace s method and is very accurate for practical applications inference regarding the parameter 0 of primary interest may be made from the profile likelihood function an alternative approach to likelihood methods based on approximations to the marginal means and variances is also considered the methods explored here can be readily generalized to settings where the parameters ok depend on covariables as well