a composite likelihood approach to binary spatial data abstract conventional geostatistics addresses the problem of estimation and prediction for continuous observations but in many practical applications in public health environmental remediation or ecological research the most commonly available data are in the form of counts e g number of cases or indicator variables denoting above or below threshold values also in many situations it is less expensive to obtain an imprecise categorical observation than to obtain precise measurements of the variable of interest such as a contaminant this article proposes a computationally simple method for estimation and prediction using binary or indicator data in space the proposed method is based on pairwise likelihood contributions and the large sample properties of the estimators are obtained in a straightforward manner we illustrate the methodology through application to indicator data related to gypsy moth defoliation in massachusetts

comparing the performance of two indices for spatial model selection application to two mortality data the statistical analysis of spatially correlated data has become an important scientific research topic lately the analysis of the mortality or morbidity rates observed at different areas may help to decide if people living in certain locations are considered at higher risk than others once the statistical model for the data of interest has been chosen further effort can be devoted to identifying the areas under higher risks many scientists including statisticians have tried the conditional autoregressive car model to describe the spatial autocorrelation among the observed data this model has greater smoothing effect than the exchangeable models such as the poisson gamma model for spatial data this paper focuses on comparing the two types of models using the index lg the ratio of local to global variability two applications taiwan asthma mortality and scotland lip cancer are considered and the use of lg is illustrated the estimated values for both data sets are small implying a poisson gamma model may be favoured over the car model we discuss the implications for the two applications respectively to evaluate the performance of the index lg we also compute the bayes factor a bayesian model selection criterion to see which model is preferred for the two applications and simulation data to derive the value of lg we estimate its posterior mode based on samples derived from the bugs program while for bayes factor we use the double laplace metropolis method schwarz criterion and a modified harmonic mean for approximations the results of lg and bayes factor are consistent we conclude that lg is fairly accurate as an index for selection between poisson gamma and car model when easy and fast computation is of concern we recommend using lg as the first and less costly index

a regression method for spatial disease rates an estimating function approach abstract epidemiologists commonly study the geographical variation of disease rates to generate and refine testable hypotheses regarding etiology poisson regression is often used to estimate the parameters that characterize the effects of risk factors on disease hierarchical models have been proposed to utilize spatial locations and neighbors as surrogates for unknown or unmeasured risk factors in the analysis of disease rates although hierarchical models are useful in modeling spatial disease rates in a scientifically meaningful way the analytic tools for them are generally computationally intensive we overcome this issue by applying a conditional spatial modeling technique using unbiased estimating functions the resulting estimator of regression coefficients is consistent and asymptotically normally distributed under mild conditions a simulation study compares the performance of the proposed scheme with the approximate inference methods for hierarchical models we apply the proposed scheme to sco

on conditional and intrinsic autoregressions summary gaussian conditional autoregressions have been widely used in spatial statistics and bayesian image analysis where they are intended to describe interactions between random variables at fixed sites in euclidean space the main appeal of these distributions is in the markovian interpretation of their full conditionals intrinsic autoregressions are limiting forms that retain the markov property despite being improper they can have advantages over the standard autoregressions both conceptually and in practice for example they often avoid difficulties in parameter estimation without apparent loss or exhibit appealing invariances as in texture analysis however on small arrays and in nonlattice applications both forms of autoregression can lead to undesirable second order characteristics either in the variables themselves or in contrasts among them this paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using dempster s 1972 algorithm or an appropriate modification the approach represents a partial synthesis of standard geostatistical and gaussian markov random field formulations some nonspatial applications are also mentioned

higher order congruence relations on affine moment graphs the subgeneric case abstract we study the structure algebra z of the stable moment graph for the case of the affine root system a 1 the structure algebra z is an algebra over a symmetric algebra and in particular it is a module over a symmetric algebra we study this module structure on z and we construct a basis by setting c equal to zero in z we obtain the module z c 0 this module can be described in terms of the finite root system a 1 and we show that it is determined by a set of certain divisibility relations these relations can be regarded as a generalization of ordinary moment graph relations that define sections of sheaves on moment graphs and because of this we call them higher order congruence relations

modelling spatially correlated data via mixtures a bayesian approach the paper develops mixture models for spatially indexed data we confine attention to the case of finite typically irregular patterns of points or regions with prescribed spatial relationships and to problems where it is only the weights in the mixture that vary from one location to another our specific focus is on poisson distributed data and applications in disease mapping we work in a bayesian framework with the poisson parameters drawn from gamma priors and an unknown number of components we propose two alternative models for spatially dependent weights based on transformations of autoregressive gaussian processes in one the logistic normal model the mixture component labels are exchangeable in the other the grouped continuous model they are ordered reversible jump markov chain monte carlo algorithms for posterior inference are developed finally the performances of both of these formulations are examined on synthetic data and real data on mortality from a rare disease

spatial temporal analysis of mortality using splines a method for constructing contour maps in disease mapping using smoothing splines is presented smoothing errors are discussed and error maps suggested the approach is illustrated for mortality data on cardiovascular diseases in northwestern germany lower saxony and bremen 1970 1979

on parametric bootstrap methods for small area prediction summary the particularly wide range of applications of small area prediction e g in policy making decisions has meant that this topic has received substantial attention in recent years the problems of estimating mean squared predictive error of correcting that estimator for bias and of constructing prediction intervals have been addressed by various workers although existing methodology is still restricted to a narrow range of models to overcome this difficulty we develop new bootstrap based methods which are applicable in very general settings for constructing bias corrected estimators of mean squared error and for computing prediction regions unlike existing techniques which are based largely on taylor expansions our bias corrected mean squared error estimators do not require analytical calculation they also have the property that they are non negative our prediction intervals have a high degree of coverage accuracy o n 3 where n is the number of areas if double bootstrap methods are employed the techniques do not depend on the form of the small area estimator and are applicable to general two level small area models where the variables at either level can be discrete or continuous and in particular can be non normal most importantly the new methods are simple and easy to apply

estimating a population of parameter values using bayes and empirical bayes methods abstract in standard bayes and empirical bayes component decision problems estimating inidividual parameters is the primary goal in multiple comparison problems and in comparisons of histograms of estimates however the primary goal is to produce parameter estimates that can be considered as an ensemble for example the histogram of estimates should be a good estimate of the histogram of parameters standard methods of estimating by the posterior expectation do minimize symmetric componentwise losses such as squared error but they produce ensembles of estimates with a sample variance smaller than the posterior expected sample variance for parameters in this article we propose new bayes and empirical bayes estimates that minimize a distance function between the empirical cdf of the estimates and the true parameters these estimators are weighted averages of the prior mean and the data with weight on the data being approximately the square root of that for the posterior expectation we give theoreti

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

effects of the choice of age adjustment method on maps of death rates maps of morbidity or mortality rates whether considered individually or as a layer in a geographic information system application invite multiple comparisons of area rates however comparisons of rates across different populations require standardization of the age specific rates to account for differences in population age structures the indirect standardization method or equivalently the standardized mortality ratio smr has been recommended for small areas where age specific rates can be quite variable although theoretically equivalent to directly adjusted rates under the assumption of independent age and area effects indirect summary measures are not comparable across areas when this assumption is violated we tested the validity of this assumption for the 10 most common causes of death in the united states during 1980 84 and examined the geographic clustering apparent when categorized death rates adjusted by different methods are presented as thematic maps although overall agreement between the methods was good rank correlation coefficient 82 per cent for each cause when the adjusted rates were classified into quintiles 18 per cent of the states fell into different categories depending on the method of adjustment using an internal standard for the indirect method reduced this discrepancy to 4 9 per cent however both traditional chi square tests and a generalized logistic spline model identified significant interactions between age and area for each cause of death a violation of the assumption required for equivalence of the methods potential variation in geographic inferences is illustrated by maps of direct and indirect rates and an empirical bayes posterior mean which is a function of these traditionally adjusted rates based on these results we recommend the direct age adjustment method for rate maps

posterior distribution of hierarchical models using car 1 distributions summary we examine properties of the conditional autoregressive model or car 1 model which is commonly used to represent regional effects in bayesian analyses of mortality rates we consider a bayesian hierarchical linear mixed model where the fixed effects have a vague prior such as a constant prior and the random effect follows a class of car 1 models including those whose joint prior distribution of the regional effects is improper we give sufficient conditions for the existence of the posterior distribution of the fixed and random effects and variance components we then prove the necessity of the conditions and give a one way analysis of variance example where the posterior may or may not exist finally we extend the result to the generalised linear mixed model which includes as a special case the poisson log linear model commonly used in disease mapping

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

testing for elevated disease rates using smoothed estimates a method of testing for elevated disease rates using smoothed estimates is introduced for counts aggregated to a specific administrative level a test statistic is developed based on an approximation to the smoothed estimate from a penalized multinomial likelihood and an approximate critical value is derived empirical investigation reveals that smoothing can increase the ability to detect elevated rates the methods are applied to data on gastric cancer in nova scotia canada

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

the category o for a general coxeter system abstract we study the category o for a general coxeter system using a formulation of fiebig the translation functors the zuckerman functors and the twisting functors are defined we prove the fundamental properties of these functors the duality of zuckerman functor and generalization of verma s result about homomorphisms between verma modules

the combinatorics of coxeter categories we present an alternative construction of soergel s category of bimodules associated to a reflection faithful representation of a coxeter system we show that its objects can be viewed as sheaves on the associated moment graph we introduce an exact structure and show that soergel s special bimodules are the projective objects then we construct the indecomposable projectives by both a global and a local method discuss a version of the kazhdan lusztig conjecture and prove it for universal coxeter systems

a bayesian space varying parameter model applied to estimating fertility schedules we propose a spatial generalized linear model glm to analyse the vital rates for small areas in each small area we have a response vector and covariates to explain its variability the statistical methodology is based on a spatial bayesian approach and it allows the covariates parameters of the generalized linear model to vary smoothly on space hence the effect of a covariate on the response varies depending on the random variables measurement location our model is an extension of disease mapping models allowing the space covariate interaction to be modelled in a natural way and giving space a position of intrinsic interest we introduce the model in the context of fertility curve estimation in each small area we have a curve describing the variation of fertility rates by age modelled by coale s fertility model which implies a glm in each area a simulation shows the advantages of our approach in addition the paper applies the procedure to census data used to study the diffusion of low fertility behaviour in brazil

identifiability and convergence issues for markov chain monte carlo fitting of spatial models the marked increase in popularity of bayesian methods in statistical practice over the last decade owes much to the simultaneous development of markov chain monte carlo mcmc methods for the evaluation of requisite posterior distributions however along with this increase in computing power has come the temptation to fit models larger than the data can readily support meaning that often the propriety of the posterior distributions for certain parameters depends on the propriety of the associated prior distributions an important example arises in spatial modelling wherein separate random effects for capturing unstructured heterogeneity and spatial clustering are of substantive interest even though only their sum is well identified by the data increasing the informative content of the associated prior distributions offers an obvious remedy but one that hampers parameter interpretability and may also significantly slow the convergence of the mcmc algorithm in this paper we investigate the relationship among identifiability bayesian learning and mcmc convergence rates for a common class of spatial models in order to provide guidance for prior selection and algorithm tuning we are able to elucidate the key issues with relatively simple examples and also illustrate the varying impacts of covariates outliers and algorithm starting values on the resulting algorithms and posterior distributions

tilting modules in category o and sheaves on moment graphs abstract we describe tilting modules of the deformed category o over a semisimple lie algebra as certain sheaves on a moment graph associated to the corresponding block of o we prove that they map to braden macpherson sheaves constructed along the reversed bruhat order under fiebig s localization functor by this means we get character formulas for tilting modules and explain how soergel s result about the andersen filtration gives a koszul dual proof of the semisimplicity of subquotients of the jantzen filtration

a latent process regression model for spatially correlated count data summary this paper proposes a regression model for spatially correlated count data that generalizes the work spatially correlated count data arise in a variety of settings frequently it is of interest to make marginal inferences about spatial trends and effects of explanatory variables on spatially correlated counts an example application described in this paper involves an examination of the distribution of neuronal cells in a laboratory culture well our motivation for examining the distribution of cells was due to the implications of the distribution for the choice of an efficient sampling method to estimate total cell count total neuronal cell counts in a culture well are ordinarily estimated using data from multiple sampled subregions viewed under the microscope or with the aid of an image analysis system because counting all cells in a culture well is prohibitively time consuming as routine practice these sampling methods typically involve recording from multiple randomly selected fields the area of the field and number of cells visible in the field a cell per unit area density is estimated from the data and the total cell count estimate for the well is obtained by multiplication by the total well area these methods implicitly assume that cells are distributed completely at random throughout the well experience would suggest that there could be trends in neuron density and clustering tendencies for example neurons can tend to a higher density in the center of a well due to a slight vortex that occurs when the cultures are moved before the neurons have attached themselves to the substratum neurons also may distribute in higher density around the edges of a culture well due to a meniscus and slightly greater depth of medium at the edge of a well spatial clustering may result if neural cell bodies migrate along the cell body s neuritic processes or if the tissue culture tray is

fixed width output analysis for markov chain monte carlo markov chain monte carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages a fundamental question is when sampling should stop that is at what point the ergodic averages are good estimates of the desired quantities we consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user specified value hence calculating a monte carlo standard error is a critical step in assessing the simulation output we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution we give sufficient conditions for the strong consistency of both methods and investigate their finite sample properties in various examples

hierarchical bayes estimation of hunting success rates with spatial correlations a bayesian hierarchical generalized linear model is used to estimate hunting success rates at the subarea level for postseason harvest surveys the model includes fixed week effects random geographic effects and spatial correlations between neighboring subareas the computation is done by gibbs sampling and adaptive rejection sampling techniques the method is illustrated using data from the missouri turkey hunting survey in the spring of 1996 bayesian model selection methods are used to demonstrate that there are significant week differences and spatial correlations of hunting success rates among counties the bayesian estimates are also shown to be quite robust in terms of changes of hyperparameters

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

jackknife estimation of mean squared error of small area predictors in nonlinear mixed models empirical bayes predictors of small area parameters of interest are often obtained under a linear mixed model for continuous response data or a generalized linear mixed model for binary responses or count data however estimation of the unconditional mean squared error of prediction is complicated particularly for a nonlinear mixed model jiang et al 2002 proposed a jackknife method for estimating the unconditional mean squared error and showed that the resulting estimator is nearly unbiased the leading term of this estimator does not depend on the area specific responses in the nonlinear case whereas the posterior variance of the small area parameter given the model parameters is area specific rao 2003 proposed an alternative method that leads to a computationally simpler jackknife estimator with an area specific leading term we show that a modification of rao s method leads to a nearly unbiased area specific jackknife estimator which is also nearly unbiased for the conditional mean squared error given the area specific responses we examine the relative performances of the jackknife estimators conditionally as well as unconditionally in a simulation study and apply the proposed method to estimate small area mean squared errors in disease mapping problems

a bayesian χ2 test for goodness of fit this article describes an extension of classical x 2 goodness of fit tests to bayesian model assessment the extension which essentially involves evaluating pearson s goodness of fit statistic at a parameter value drawn from its posterior distribution has the important property that it is asymptotically distributed as a x 2 random variable on k 1 degrees of freedom independently of the dimension of the underlying parameter vector by examining the posterior distribution of this statistic global goodness of fit diagnostics are obtained advantages of these diagnostics include ease of interpretation computational convenience and favorable power properties the proposed diagnostics can be used to assess the adequacy of a broad class of bayesian models essentially requiring only a finite dimensional parameter vector and conditionally independent observations

all maps of parameter estimates are misleading maps are frequently used to display spatial distributions of parameters of interest such as cancer rates or average pollutant concentrations by county it is well known that plotting observed rates can have serious drawbacks when sample sizes vary by area since very high and low observed rates are found disproportionately in poorly sampled areas unfortunately adjusting the observed rates to account for the effects of small sample noise can introduce an opposite effect in which the highest adjusted rates tend to be found disproportionately in well sampled areas in either case the maps can be difficult to interpret because the display of spatial variation in the underlying parameters of interest is confounded with spatial variation in sample sizes as a result spatial patterns occur in adjusted rates even if there is no spatial structure in the underlying parameters of interest and adjusted rates tend to look too uniform in areas with little data we introduce two models normal and poisson in which parameters of interest have no spatial patterns and demonstrate the existence of spatial artefacts in inference from these models we also discuss spatial models and the extent to which they are subject to the same artefacts we present examples from bayesian modelling but as we explain the artefacts occur generally

spatial modeling of regional variables abstract in this article accumulated sudden infant death syndrome sids data from 1974 1978 and 1979 1984 for the counties of north carolina are analyzed after a spatial exploratory data analysis markov random field models are fit to the data the spatial trend is meant to capture the large scale variation in the data and the variance and spatial dependence are meant to capture the small scale variation the trend could be a function of other explanatory variables or could simply be modeled as a function of spatial location both models are fit and compared the results give an excellent illustration of a phenomenon already well known in time series that autocorrelation in data can be due to an undiscovered explanatory variable indeed for 1974 1978 we confirm a dependence of sids rate on proportion of nonwhite babies born along with insignificant spatial correlation without this regressor variable however the spatial correlation is significant in 1979 1984 perhaps due to reporting bias o

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

semiparametric normal transformation models for spatially correlated survival data there is an emerging interest in modeling spatially correlated survival data in biomedical and epidemiologic studies in this article we propose a new class of semiparametric normal transformation models for right censored spatially correlated survival data this class of models assumes that survival outcomes marginally follow a cox proportional hazard model with unspecified baseline hazard and their joint distribution is obtained by transforming survival outcomes to normal random variables whose joint distribution is assumed to be multivariate normal with a spatial correlation structure a key feature of the class of semiparametric normal transformation models is that it provides a rich class of spatial survival models where regression coefficients have population average interpretation and the spatial dependence of survival times is conveniently modeled using the transformed variables by flexible normal random fields we study the relationship of the spatial correlation structure of the transformed no

non parametric maximum likelihood estimators for disease mapping a non parametric maximum likelihood approach to the estimation of relative risks in the context of disease mapping is discussed and a npml approximation to conditional autoregressive models is proposed npml estimates have been compared to other proposed solutions maximum likelihood via monte carlo scoring hierarchical bayesian models using real examples overall the npml autoregressive estimates with weighted term were closer to the bayesian estimates the exchangeable npml model ranked immediately after even if it implied a greater shrinkage while the truncated auto poisson showed inadequate for disease mapping the coefficients of the autoregressive term for the different mixtures have clear interpretations in the breast cancer example the larger cities in the region showed high rates and very low correlation with the neighbouring areas while the less populated rural areas with low rates were strongly positively correlated each other this pattern is expected since breast cancer is strongly correlated with parity and age at first birth and the female population of the rural areas experienced a decline in fertility much later than those living in the larger cities the leukemia example highlighted the failure of the poisson gamma model and other general overdispersion tests to detect high risk areas under specific conditions the npml approach in aitkin is very general simple and flexible however the user should be warned against the possibility of local maxima and the difficulty in detecting the optimal number of components special software such as caman or dismapwin had been developed and should be recommended mainly to not experienced users

computation of generalized equivariant cohomologies of kac moody flag varieties abstract in 1998 goresky kottwitz and macpherson showed that for certain projective varieties x equipped with an algebraic action of a complex torus t the equivariant cohomology ring h t x can be described by combinatorial data obtained from its orbit decomposition in this paper we generalize their theorem in three different ways first our group g need not be a torus second our space x is an equivariant stratified space along with some additional hypotheses on the attaching maps third and most important we allow for generalized equivariant cohomology theories e g instead of h t for these spaces we give a combinatorial description of e g x as a subring of e g f i where the f i are certain invariant subspaces of x our main examples are the flag varieties g p of kac moody groups g with the action of the torus of g in this context the f i are the t fixed points and e g is a t equivariant complex oriented cohomology theory such as h t k t or mu t we detail several explicit examples

a general maximum likelihood analysis of variance components in generalized linear models this paper describes an em algorithm for nonparametric maximum likelihood ml estimation in generalized linear models with variance component structure the algorithm provides an alternative analysis to approximate mql and pql analyses mcgilchrist and aisbett 1991 biometrical journal 33 131 141 breslow and clayton 1993 journal of the american statistical association 88 9 25 mcgilchrist 1994 journal of the royal statistical society series b 56 61 69 goldstein 1995 multilevel statistical models and to gee analyses liang and zeger 1986 biometrika 73 13 22 the algorithm first given by hinde and wood 1987 in longitudinal data analysis 110 126 is a generalization of that for random effect models for overdispersion in generalized linear models described in aitkin 1996 statistics and computing 6 251 262 the algorithm is initially derived as a form of gaussian quadrature assuming a normal mixing distribution but with only slight variation it can be used for a completely unknown mixing distribution giving a straightforward method for the fully nonparametric ml estimation of this distribution this is of value because the ml estimates of the glm parameters can be sensitive to the specification of a parametric form for the mixing distribution the nonparametric analysis can be extended straightforwardly to general random parameter models with full npml estimation of the joint distribution of the random parameters this can produce substantial computational saving compared with full numerical integration over a specified parametric distribution for the random parameters a simple method is described for obtaining correct standard errors for parameter estimates when using the em algorithm several examples are discussed involving simple variance component and longitudinal models and small area estimation

disease mapping using empirical bayes and bayes methods on mortality statistics in the netherlands data from the 64 public health service districts in the netherlands describing the health status of the dutch population were used to construct maps for several causes of deaths the choice and estimation of the relative risk measure is described the prior expected number of deaths was modelled using a poisson regression approach based on a model with main effects of district and age desirable properties of risk parameters for disease mapping are that they both reflect the level of risk and cope with the instability in the observed measure caused by the numbers at risk in each district different techniques for estimation of parameters were applied empirical bayes estimation eb using a non parametric prior and a gamma prior and a bayesian approach b with a uniform prior for the parametric eb also a constrained estimator was used the eb techniques studied in this paper shift or smooth the values of the risk parameter towards a global mean in the bayesian method applied here spatial dependence among districts can be modelled that is the estimates are smoothed towards a local mean the three eb estimates gave by and large similar results although the constrained eb estimate smoothed less as was expected the bayesian estimates smoothed the estimates more or less similarly to the constrained eb

sheaves on moment graphs and a localization of verma flags abstract to any moment graph g we assign a subcategory v of the category of sheaves on g together with an exact structure we show that in the case that the graph is associated to a non critical block of the equivariant category o over a symmetrizable kac moody algebra v is equivalent as an exact category to the subcategory of modules that admit a verma flag the projective modules correspond under this equivalence to the intersection cohomology sheaves on the graph and hence by a theorem of braden and macpherson to the equivariant intersection cohomologies of schubert varieties associated to kac moody groups

on state dependent delay partial neutral functional integro differential equations in this paper the existence of mild solutions for a class of abstract neutral integro differential equations with state dependent delay is studied

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

robust modeling for inference from generalized linear model classes generalized linear models glms are widely used for data analysis however their maximum likelihood estimators can be sensitive to outliers we propose new statistical models that allow robust inferences from the glm class of models including poisson and binomial glms and their extension to generalized linear mixed models the likelihood score equations from the new models give estimators with bounded influence so that the resulting estimators are robust against outliers while maintaining high efficiency in the absence of outliers

empirical bayes estimates of cancer mortality rates using spatial models we give results of empirical bayes eb estimation of mortality rates designed to smooth observed smr when random fluctuation of the observed deaths is important we have specially studied the case where the prior distributions of the eb method have a spatial structure the need for spatial modelling of cancer mortality rates in france is first shown with testing autocorrelation and fitting autoregressive spatial models conditional car or simultaneous sar a positive autocorrelation of the rates is shown for most cancer sites studied as expected eb estimates of mortality rates for common tumours are similar to smrs for rare tumours the eb method identifies the extreme rates more clearly than smrs by smoothing the smrs with large variances car or sar models are adequate prior distributions for autocorrelated rates and produce quite similar rate estimates

from moment graphs to intersection cohomology we describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori in terms of structure of the zero and one dimensional orbits the class of varieties to which our formula applies includes schubert varieties in flag varieties and affine flag varieties we also prove a monotonicity result on local intersection cohomology stalks

extension of the stein estimating procedure through the use of estimating functions abstract this article extends the stein method through the use of estimating functions godambe 1960 to address the simultaneous estimation of k population parameters 1 k in a mixed model setting the procedure generalizes the stein method by a allowing us to deal effectively with complications such as inequality of population variances that may arise in non gaussian mixed models b being appropriate for estimating i in populations of varying sizes and in particular populations of small sizes and c applying to situations where it cannot be assumed that the i s have unbiased estimators or even estimators of finite moment the focus of the article is on the quadratic variance function exponential family morris 1983b estimators for the parameters of the mixed model are developed in a regression model setting in which the i s are allowed to vary with a vector of covariates an application to incidence rates for the iceland breast cancer incidence data is presented for illustra

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

modelling complexity applications of gibbs sampling in medicine summary we review applications of gibbs sampling in medicine involving longitudinal spatial covariate measurement and survival models applications in immunology pharmacology transplantation cancer screening industrial epidemiology and genetic epidemiology are discussed

poisson gamma random field models for spatial statistics summary doubly stochastic bayesian hierarchical models are introduced to account for uncertainty and spatial variation in the underlying intensity measure for point process models inhomogeneous gamma process random fields and more generally markov random fields with infinitely divisible distributions are used to construct positively autocorrelated intensity measures for spatial poisson point processes these in turn are used to model the number and location of individual events a data augmentation scheme and markov chain monte carlo numerical methods are employed to generate samples from bayesian posterior and predictive distributions the methods are developed in both continuous and discrete settings and are applied to a problem in forest ecology

a bayesian semi parametric model for colorectal cancer incidences a bayesian semi parametric model is proposed to capture the interaction among demographic effects age and gender spatial effects county and temporal effects of colorectal cancer incidences simultaneously in particular an extension of multivariate conditionally autoregressive car processes to a partially informative gaussian demographic spatial temporal car dstcar process for a spatial temporal setting is proposed the precision matrix of the gaussian dstcar process is the kronecker product of several components the spatial component is modelled with a car prior a pth order intrinsic autoregressive prior iar p is implemented for the temporal component to estimate a smoothed and non parametric temporal trend the demographic component is modelled with a wishart prior data analysis shows significant spatial correlation only exists in the age group of 50 59 males and females in their 50s and 60s show fairly strong correlation the hypothesis testing based on bayes factor suggests that gender correlation cannot be ignored in this model

likelihood based clustering of meta analytic sroc curves meta analysis of diagnostic studies experience the common problem that different studies might not be comparable since they have been using a different cut off value for the continuous or ordered categorical diagnostic test value defining different regions for which the diagnostic test is defined to be positive hence specificities and sensitivities arising from different studies might vary just because the underlying cut off value had been different to cope with the cut off value problem interest is usually directed towards the receiver operating characteristic roc curve which consists of pairs of sensitivities and false positive rates 1 specificity in the context of meta analysis one pair represents one study and the associated diagram is called an sroc curve where the s stands for summary in meta analysis of diagnostic studies emphasis has traditionally been placed on modelling this sroc curve with the intention of providing a summary measure of the diagnostic accuracy by means of an estimate of the summary roc curve here we focus instead on finding sub groups or components in the data representing different diagnostic accuracies the paper will consider modelling sroc curves with the lehmann family which is characterised by one parameter only each single study can be represented by a specific value of that parameter hence we focus on the distribution of these parameter estimates and suggest modelling a potential heterogeneous or cluster structure by a mixture of specifically parameterised normal densities we point out that this mixture is completely nonparametric and the associated mixture likelihood is well defined and globally bounded we use the theory and algorithms of nonparametric mixture likelihood estimation to identify a potential cluster structure in the diagnostic accuracies of the collection of studies to be analysed several meta analytic applications on diagnostic studies including audit and audit c for detection of unhealthy alcohol use the mini mental state examination for cognitive disorders as well as diagnostic accuracy inspection data on metal fatigue of aircraft spare parts are discussed to illustrate the methodology

coupled fixed points of multivalued operators and first order odes with state dependent deviating arguments abstract we establish a coupled fixed point theorem for a meaningful class of mixed monotone multivalued operators and then we use it to derive some results on the existence of quasisolutions and unique solutions to first order functional differential equations with state dependent deviating arguments our results are very general and can be applied to functional equations featuring discontinuities with respect to all of their arguments but we emphasize that they are new even for differential equations with continuously state dependent delays

positive periodic solutions for a class of higher dimensional state dependent delay functional differential equations with feedback control we use the krasnosel skii s fixed point theorem for cones to study the existence of positive periodic solutions of the following a class of higher dimensional state dependent delay functional differential equations with feedback controlx t a t x t f t x t x t t t x t u t a t u t b t u t c t x h t x t moreover we apply our results obtained in this paper to some population model

existence and multiplicity of positive periodic solutions for a class of higher dimension functional differential equations with impulses this paper deals with the existence of multiple periodic solutions for n dimensional functional differential equations with impulses by employing the krasnoselskii fixed point theorem we obtain some easily verifiable sufficient criteria which extend previous results

positive periodic solutions of a class of delay differential system with feedback control with the help of continuation theorem based on gaines and mawhin s coincidence degree sufficient and realistic conditions are obtained for the global existence of positive periodic solutions for the delay differential system with feedback control when these results are applied to some special delay population models some new results are obtained and some known results are generalized

parametric spatial cure rate models for interval censored time to relapse data summary several recent papers e g chen ibrahim and sinha 1999 journal of the american statistical association94 909 919 ibrahim chen and sinha 2001a biometrics57 383 388 have described statistical methods for use with time to event data featuring a surviving fraction i e a proportion of the population that never experiences the event such cure rate models and their multivariate generalizations are quite useful in studies of multiple diseases to which an individual may never succumb or from which an individual may reasonably be expected to recover following treatment e g various types of cancer in this article we extend these models to allow for spatial correlation estimable via zip code identifiers for the subjects as well as interval censoring our approach is bayesian where posterior summaries are obtained via a hybrid markov chain monte carlo algorithm we compare across a broad collection of rather high dimensional hierarchical models using the deviance information criterion a tool recently developed for just this purpose we apply our approach to the analysis of a smoking cessation study where the subjects reside in 53 southeastern minnesota zip codes in addition to the usual posterior estimates our approach yields smoothed zip code level maps of model parameters related to the relapse rates over time and the ultimate proportion of quitters the cure rates

disease mapping via negative binomial regression m quantiles we introduce a semi parametric approach to ecological regression for disease mapping based on modelling the regression m quantiles of a negative binomial variable the proposed method is robust to outliers in the model covariates including those due to measurement error and can account for both spatial heterogeneity and spatial clustering a simulation experiment based on the well known scottish lip cancer data set is used to compare the m quantile modelling approach with a disease mapping approach based on a random effects model this suggests that the m quantile approach leads to predicted relative risks with smaller root mean square error the paper concludes with an illustrative application of the m quantile approach mapping low birth weight incidence data for english local authority districts for the years 2005 2010 copyright 2014 john wiley sons ltd

modelling spatial disease rates using maximum likelihood this paper concerns maximum likelihood estimation for a generalized linear mixed model glmm useful for modelling spatial disease rates the model allows for log linear covariate adjustment and local smoothing of rates through estimation of spatially correlated random effects the covariance structure of the random effects is based on a recently proposed model which parameterizes spatial dependence through the inverse covariance matrix a markov chain monte carlo algorithm for performing maximum likelihood estimation for this model is described results of a computer simulation study that compared maximum likelihood ml and penalized quasi likelihood pql estimators are presented compared with pql ml produced less biased estimates of the intercept but the ml estimates were slightly more variable estimates of the other regression coefficients were unbiased and nearly identical for the two methods ml estimators of the random effects standard deviation and spatial correlation were more biased than the corresponding pql estimators the conclusion is that ml estimators for glmms cannot be expected to perform better than pql for small samples copyright 2000 john wiley sons ltd

multiple positive periodic solutions for a class of state dependent delay functional differential equations with feedback control by employing krasnoselskii fixed point theorem we investigate the existence of multiple positive periodic solutions for a class of state dependent delay functional differential equations with feedback control the system considered in this paper is more general and incorporates as special cases various problems which have been studied extensively in the literature moreover some easily verifiable sufficient criteria are established

a latent model to detect multiple clusters of varying sizes this article develops a latent model and likelihood based inference to detect temporal clustering of events the model mimics typical processes generating the observed data we apply model selection techniques to determine the number of clusters and develop likelihood inference and a monte carlo expectation maximization algorithm to estimate model parameters detect clusters and identify cluster locations our method differs from the classical scan statistic in that we can simultaneously detect multiple clusters of varying sizes we illustrate the methodology with two real data applications and evaluate its efficiency through simulation studies for the typical data generating process our methodology is more efficient than a competing procedure that relies on least squares

recent developments in computer assisted analysis of mixtures this paper reviews recent developments in the area of computer assisted analysis of mixture distributions c a man given a biometric situation of interest in which under homogeneity assumptions a certain parametric density occurs such as the poisson the binomial the geometric the normal and so forth then it is argued that this situation can easily be enlarged to allow a variation of the scalar parameter in the population this situation is called unobserved heterogeneity this naturally leads to a specific form of nonparametric mixture distribution that can then be assumed to be the standard model in the biometric application of interest since it also incorporates the homogeneous situations as a special case besides developments in theory and algorithms the work focuses on developments in biometric applications such as meta analysis fertility studies estimation of prevalence under clustering and estimation of the distribution function of survival time under interval censoring the approach is nonparametric for the mixing distribution including leaving the number of components subpopulations of the mixing distribution unknown

using a mixed effects model to estimate geographic variation in cancer rates commonly used methods for depicting geographic variation in cancer rates are based on rankings they identify where the rates are high and low but do not indicate the magnitude of the rates nor their variability yet such measures of variability may be useful in suggesting which types of cancer warrant further analytic studies of localized risk factors we consider a mixed effects model in which the logarithm of the mean poisson rate is additive in fixed stratum effects e g age effects and in logarithms of random relative risk effects associated with geographic areas these random effects are assumed to follow a gamma distribution with unit mean and variance 1 alpha similar to clayton and kaldor 1987 biometrics 43 671 681 we present maximum likelihood and method of moments estimates with standard errors for inference on alpha 1 2 the relative risk standard deviation rrsd the moment estimates rely on only the first two moments of the poisson and gamma distributions but have larger standard errors than the maximum likelihood estimates we compare these estimates with other measures of variability several examples suggest that the rrsd estimates have advantages compared to other measures of variability

a generalized estimating equations approach for spatially correlated binary data applications to the analysis of neuroimaging data this paper proposes a generalized estimating equations approach for the analysis of spatially correlated binary data when there are large numbers of spatially correlated observations on a moderate number of subjects this approach is useful when the scientific focus is on modeling the marginal mean structure proper modeling of the spatial correlation structure is shown to provide large efficiency gains along with precise standard error estimates for inference on mean structure parameters generalized estimating equations for estimating the parameters of both the mean and spatial correlation structure are proposed the use of semivariogram models for parameterizing the correlation structure is discussed and estimation of the sample semivariogram is proposed as a technique for choosing parametric models and starting values for generalized estimating equations estimation the methodology is illustrated with neuroimaging data collected as part of the national institute of neurological disorders and stroke ninds stroke data bank a simulation study demonstrates the importance of accurate modeling of the spatial correlation structure in data with large numbers of spatially correlated observations such as those found in neuroimaging studies

monotone empirical bayes estimators based on more informative samples abstract an empirical bayes estimator is constructed for the case of nonidentical components coming from an exponential family of distributions that is closed under convolution examples are the normal poisson and binomial distributions the construction is based on robbins approach to empirical bayes estimation and generalizes results of o bryan and stijnen and van houwelingen the method is elaborated for the normal and poisson case and examples are given that show that this approach gives results comparable to those from the nonparametric maximum likelihood approach of laird

non parametric small area estimation using penalized spline regression the paper proposes a small area estimation approach that combines small area random effects with a smooth non parametrically specified trend by using penalized splines as the representation for the non parametric trend it is possible to express the non parametric small area estimation problem as a mixed effect model regression the resulting model is readily fitted by using existing model fitting approaches such as restricted maximum likelihood we present theoretical results on the prediction mean squared error of the estimator proposed and on likelihood ratio tests for random effects and we propose a simple non parametric bootstrap approach for model inference and estimation of the small area prediction mean squared error the applicability of the method is demonstrated on a survey of lakes in north eastern usa

periodicity and stability for a lotka volterra type competition system with feedback controls and deviating arguments abstract this paper deals with the general periodic lotka volterra type competition systems with feedback controls and deviating arguments by employing fixed point index theory on cone an explicit necessary and sufficient condition for the global existence of the positive periodic solution of the systems is proved by constructing a suitable lyapunov functional a set of easily verifiable sufficient conditions for the global asymptotic stability of the positive periodic solution of the systems is given

bayesian disease mapping using product partition models our objective is to develop a model to estimate the relative risk of disease in each area a i i 1 n of a region and to identify areas of unusually high or low risk we use a product partition model ppm in which we assume that the true relative risks can be partitioned into a number of components or sets of areas where the relative risks are equal the ppm allows the data to weight those partitions likely to hold and inference about particular parameters may be made by first conditioning on the partition and then averaging over all partitions we develop markov chain monte carlo mcmc techniques to approximate the posterior distributions of the partitions and the parameters we first test the method in a simulation study and then apply it to data for two separate groups of different types of cancer in the mid western health board region in ireland the results are compared with those obtained using the standardized mortality ratio method an empirical bayes method a spatial scan method and a nonparametric bayesian method

a constrained empirical bayes estimator for incidence rates in areas with small populations maps that show the geographic distribution of incidence rates can be useful tools for analysing spatial variation in mortality and morbidity to attain the necessary geographic resolution however production of such maps often requires estimation of incidence in areas with small populations where the observed rates may be highly unstable manton et al have presented an empirical bayes stabilization procedure in which the observed rate is combined with an area specific estimate of the underlying incidence the approach allows for the mapping of outcomes with varied and possibly unknown etiologies without necessitating covariate dependent modelling of the expected rate the empirical distribution of a collection of these estimates however may not provide an adequate description of the dispersion among the true rates as a result decisions based on the histogram of the empirical bayes estimates may be suspect we propose a modified version of the approach in which the mean and sample variance of the ensemble of estimates are constrained to equal the appropriate moments of the posterior distribution the resulting collection of constrained empirical bayes estimators has nearly the stability of the unconstrained approach and provides an improved estimator of the true rate distribution we illustrate use of the estimator by producing stabilized county level maps of u s fire and burn related mortality rates and validate the analytic results using a simulation analysis

using empirical bayes methods in biopharmaceutical research a compound sampling model where a unit specific parameter is sampled from a prior distribution and then observed are generated by a sampling distribution depending on the parameter underlies a wide variety of biopharmaceutical data for example in a multi centre clinical trial the true treatment effect varies from centre to centre observed treatment effects deviate from these true effects through sampling variation knowledge of the prior distribution allows use of bayesian analysis to compute the posterior distribution of clinic specific treatment effects frequently summarized by the posterior mean and variance more commonly with the prior not completely specified observed data can be used to estimate the prior and use it to produce the posterior distribution an empirical bayes or variance component analysis in the empirical bayes model the estimated prior mean gives the typical treatment effect and the estimated prior standard deviation indicates the heterogeneity of treatment effects in both the bayes and empirical bayes approaches estimated clinic effects are shrunken towards a common value from estimates based on single clinics this shrinkage produces more efficient estimates in addition the compound model helps structure approaches to ranking and selection provides adjustments for multiplicity allows estimation of the histogram of clinic specific effects and structures incorporation of external information this paper outlines the empirical bayes approach coverage will include development and comparison of approaches based on parametric priors for example a gaussian prior with unknown mean and variance and non parametric priors discussion of the importance of accounting for uncertainty in the estimated prior comparison of the output and interpretation of fixed and random effects approaches to estimating population values estimating histograms and identification of key considerations in the use and interpretation of empirical bayes methods

schubert calculus and torsion explosion the author observes that certain numbers occurring in schubert calculus for sl also occur as entries in intersection forms controlling decompositions of soergel bimodules in higher rank these numbers grow exponentially this observation gives many counter examples to the expected bounds in lusztig s conjecture on the characters of simple rational modules for sl over fields of positive characteristic the examples also give counter examples to the james conjecture on decomposition numbers for symmetric groups

modelling categorical covariates in bayesian disease mapping by partition structures we consider the problem of mapping the risk from a disease using a series of regional counts of observed and expected cases and information on potential risk factors to analyse this problem from a bayesian viewpoint we propose a methodology which extends a spatial partition model by including categorical covariate information such an extension allows detection of clusters in the residual variation reflecting further possibly unobserved covariates the methodology is implemented by means of reversible jump markov chain monte carlo sampling an application is presented in order to illustrate and compare our proposed extensions with a purely spatial partition model here we analyse a well known data set on lip cancer incidence in scotland

analyzing spatially distributed binary data using independent block estimating equations summary we estimate the relation between binary responses and corresponding covariate vectors both observed over a large spatial lattice we assume a hierarchical generalized linear model with probit link function partition the lattice into blocks and adopt the working assumption of independence between the blocks to obtain an easily solved estimating equation standard errors are obtained using the sandwich estimator together with window subsampling sherman 1996 journal of the royal statistical society series b 58 509 523 we apply this to a large data set describing long term vegetation growth together with two other approximate likelihood approaches pairwise composite likelihood cl and estimation under a working assumption of independence the independence and cl methods give similar point estimates and standard errors while the independent block approach gives considerably smaller standard errors as well as more easily interpretable point estimates we present numerical evidence suggesting this increased efficiency may hold more generally

generalized linear models for small area estimation abstract bayesian methods have been used quite extensively in recent years for solving small area estimation problems particularly effective in this regard has been the hierarchical or empirical bayes approach which is especially suitable for a systematic connection of local areas through models however the development to date has mainly concentrated on continuous valued variates often the survey data are discrete or categorical so that hierarchical or empirical bayes techniques designed for continuous variates are inappropriate this article considers hierarchical bayes generalized linear models for a unified analysis of both discrete and continuous data a general theorem is provided that ensures the propriety of posteriors under diffuse priors this result is then extended to the case of spatial generalized linear models the hierarchical bayes procedure is implemented via markov chain monte carlo integration techniques two examples one featuring spatial correlation structure are given to illu

estimability and likelihood inference for generalized linear mixed models using data cloning maximum likelihood estimation for generalized linear mixed models glmm an important class of statistical models with substantial applications in epidemiology medical statistics and many other fields poses significant computational difficulties in this article we use data cloning a simple computational method that exploits advances in bayesian computation in particular the markov chain monte carlo method to obtain maximum likelihood estimators of the parameters in these models this method also leads to a simple estimator of the asymptotic variance of the maximum likelihood estimators determining estimability of the parameters in a mixed model is in general a very difficult problem data cloning provides a simple graphical test to not only check if the full set of parameters is estimable but also and perhaps more importantly if a specified function of the parameters is estimable one of the goals of mixed models is to predict random effects we suggest a frequentist method to obtain predict

kazhdan lusztig combinatorics in the moment graph setting abstract motivated by a question on the graded rank of the stalks of the canonical sheaf on a bruhat graph we lift some equalities concerning parabolic kazhdan lusztig polynomials to this moment graph setting our proofs hold also in positive characteristic under some technical assumptions

bayesian estimates of disease maps how important are priors in the fully bayesian fb approach to disease mapping the choice of the hyperprior distribution of the dispersion parameter is a key issue in this context we investigated the sensitivity of the rate ratio estimates to the choice of the hyperprior via a simulation study we also compared the performance of the fb approach to mapping disease risk to the conventional approach of mapping maximum likelihood ml estimates and p values the study was modelled on the incidence data of insulin dependent diabetes mellitus iddm as observed in the communes of sardinia