bayesian analysis of dichotomous quantal response models abstract qualitative response models qrm s are analyzed from the bayesian point of view using diffuse and informative prior distributions exact finite sample bayesian and large sample bayesian and non bayesian estimation results are compared in addition the paper provides 1 plots and discussion of the properties of likelihood functions for qrm s 2 posterior distributions for logit models derivatives and elasticities 3 bayesian prediction procedures for qrm s 4 new estimates for the median and other fractiles of the logistic distribution 5 posterior odds ratios for model selection problems and 6 comparisons of two alternative monte carlo numerical integration procedures it is concluded that asymptotic approximations are not accurate for small to moderate sized samples even when only a single input variable is used and that operational bayesian methods are available for providing both exact small sample and large sample approximate inferences for drm s

artificial boundary conditions for two dimensional incompressible viscous flows around an obstacle abstract in this paper we consider numerical simulation of incompressible viscous flow around an obstacle in velocity pressure formulation two horizontal straight line artificial boundaries are introduced and the original flow is approximated by a flow in an infinite channel with slip boundary condition on the wall then two vertical segment artificial boundaries are introduced to limit the channel to a bounded computational domain in the region of the channel between the vertical boundaries and infinity the velocity of the flow is almost a constant vector in which the navier stokes equations can be linearised by oseen equations and thus a general solution can be derived by using separation of variables artificial boundary conditions on the vertical segments are then designed by imposing the continuity of velocity and normal stress therefore the original problem is reduced to a boundary value problem on a bounded computational domain numerical example shows that our artificial boundary conditions are very effective

a note on the existence of the posterior distribution for a class of mixed models for binomial responses summary necessary and sufficient conditions are given for the existence of the posterior distribution of the variance components in a class of mixed models for binomial responses the implications of our results are illustrated through an example

identifiability improper priors and gibbs sampling for generalized linear models markov chain monte carlo algorithms are widely used in the fitting of generalized linear models glms such model fitting is somewhat of an art form requiring suitable trickery and tuning to obtain results in which one can have confidence a wide range of practical issues arise the focus here is on parameter identifiability and posterior propriety in particular we clarify that nonidentifiability arises for usual glms and discuss its implications for simulation based model fitting because often some part of the prior specification is vague we consider whether the resulting posterior is proper providing rather general and easily checked results for glms we also show that if a gibbs sampler is run with an improper posterior then it may be possible to use the output to obtain meaningful inference for certain model unknowns

predictive variable selection in generalized linear models here we extend predictive method for model selection of laud and ibrahim to the generalized linear model this prescription avoids the need to directly specify prior probabilities of models and prior densities for the parameters instead a prior prediction for the response induces the required priors we propose normal and conjugate priors for generalized linear models each using a single prior prediction for the mean response to induce suitable priors for each variable subset model in this way an informative prior is used to select a subset of variables in addition to producing a ranking of models by size of the predictive criterion the standard deviation of the criterion is used as a calibration number to produce a set of equally good models a straightforward markov chain monte carlo algorithm is used to accomplish the necessary computations we illustrate this method with real and simulated datasets and compare results with the bayes factors and the akaike information and bayes information model

confidence intervals for multinomial logistic regression in sparse data logistic regression is one of the most widely used regression models in practice but alternatives to conventional maximum likelihood estimation methods may be more appropriate for small or sparse samples modification of the logistic regression score function to remove first order bias is equivalent to penalizing the likelihood by the jeffreys prior and yields penalized maximum likelihood estimates ples that always exist even in samples in which maximum likelihood estimates mles are infinite ples are an attractive alternative in small to moderate sized samples and are preferred to exact conditional mles when there are continuous covariates we present methods to construct confidence intervals ci in the penalized multinomial logistic regression model and compare ci coverage and length for the ple based methods to that of conventional mle based methods in trinomial logistic regressions with both binary and continuous covariates based on simulation studies in sparse data sets we recommend profile cis over asymptotic wald type intervals for the ples in all cases furthermore when finite sample bias and data separation are likely to occur we prefer ple profile cis over mle methods

on bayesian nonlinear regression with an enzyme example a nonlinear regression model is applied to several sets of enzyme kinetics data treating the entire regression vector as the parameter of interest the resulting marginal posterior distributions are presented alongside the usual student s t posterior distributions implicit in the usual linearized regression the prior distribution used is that which is minimally informative about the regression vector in the special case of the linear model this procedure leads to the prior density 1 thereby producing the classical inferences

assessing influence on regression coefficients in generalized linear models summary suggested diagnostics for influence on the estimated regression coefficients in a generalized linear model have generally approximated the effect of deleting a single case we apply the local influence method of cook 1986 to assess the effect of small perturbations of the data including the vector of responses case weights explanatory variables and the components of one case the resulting diagnostics allow one to check for different kinds of influence and may give insight into its workings two examples illustrate some of the diagnostics

data translated likelihood and jeifreys s rules according to the definition used by box tiao for a likelihood to be data translated it must have location form in terms of a sufficient statistic their definition is easily modified to cover cases such as the cauchy through the introduction of an ancillary statistic and their argument in favour of a uniform prior then becomes group theoretic their concept of approximate data translated likelihood may also be modified to produce a sharper local approximation

assessing the quadratic approximation to the log likelihood function in nonnormal linear models summary a general and simple technique for finding a bound for the error in the quadratic approximation to the log likelihood function is proposed this bound allows a likelihood interpretation for confidence regions constructed from the quadratic approximation applications are illustrated using logit and log linear models the use of confidence regions based on the drop in log likelihood has wide appeal the arguments supporting it include invariance good second order properties peers 1971 and other criteria of a more heuristic nature including the central role played by the likelihood function in the various theories of inference yet except in a few specific cases computational difficulties prevent the assessment of the precision of maximum like lihood estimates by means of the log likelihood function this is one of the main reasons for the extensive utilization of approximations to the drop in log likelihood in this presentation it will be assumed that the data consist of n independent observations y y y from a one parameter family and that the model being entertained expresses the parameter oi associated with the ith observation as a given

approximate marginal densities of nonlinear functions this paper presents an asymptotic approximation for the marginal density of a nonlinear function g 0 that is applicable when the joint density of 0 is dominated by a single mode and the jacobian of g is of full rank near that mode the approximation is based on laplace s method and its asymptotic properties are similar to those of the saddlepoint approximation the approximation is applied to the computation of a marginal posterior density a marginal sampling density and a marginal density based on a multivariate saddlepoint approximation to a joint density

risk reducing shrinkage estimation for generalized linear models to generalized linear models in a manner retaining the original spirit of shrinkage estimation which is to reduce risk the investigation identifies two classes of simple all purpose prior distributions which supplement such non informative priors as jeffreys s prior with mechanisms for risk reduction one new class of priors is motivated as optimizers of a core component of asymptotic risk the methodology is evaluated in a numerical exploration and application to an existing data set

on the complexity of logistic regression models we investigate the complexity of logistic regression models which is defined by counting the number of indistinguishable distributions that the model can represent balasubramanian 1997 we find

a new perspective on priors for generalized linear models abstract this article deals with specifications of informative prior distributions for generalized linear models our emphasis is on specifying distributions for selected points on the regression surface the prior distribution on regression coefficients is induced from this specification we believe that it is inherently easier to think about conditional means of observables given the regression variables than it is to think about model dependent regression coefficients previous use of conditional means priors seems to be restricted to logistic regression with one predictor variable and to normal theory regression we expand on the idea of conditional means priors and extend these to arbitrary generalized linear models we also consider data augmentation priors where the prior is of the same form as the likelihood we show that data augmentation priors are special cases of conditional means priors with current monte carlo methodology such as importance sampling and gibbs sampling our priors result in

bayesian inference for a generalized population attributable fraction the impact of early vitamin a levels on chronic lung disease in very low birthweight infants in this paper the population attributable fraction is studied using the potential responses framework of rubin s causal model this framework facilitates definition of a general measure of population attributable effect which can accommodate many valued and multivariate exposures as well as many valued responses inferential issues are considered from the bayesian perspective finite population inference is emphasized with inference in the case of a fully observed population given particular attention the key inferential issue concerns computation of the posterior distribution of unobserved potential responses given observed responses exposures and covariates a dependency on model parameters about which observed data are uninformative is highlighted and this reflects the unobservable nature of causal effects in an application to a small cohort study of respiratory problems in very low birthweight infants posterior inferences were found to be insensitive to assumptions concerning the joint distribution of potential response variables but sensitive to the assumption of weak ignorability a weaker form of the more familiar assumption of no confounding by omitted covariates in a model based set up the weak ignorability assumption is identified with setting a model parameter to zero and consequently uncertainty concerning this assumption can in principle be handled via the prior distribution for the model parameters

computational methods using a bayesian hierarchical generalized linear model abstract suppose we observe y 1 yn where yi has the exponential density f yi i exp i yi i b i c yi i the parameters of interest are not the canonical parameters i but the means i b i in the usual generalized linear model glm setup suppose the means 1 n are believed to satisfy a specific p dimensional glm g i xt i where the link function g and the regression coefficients xi are known and the regression vector is unknown two problems of interest are the assessment of the goodness of fit of the glm and the estimation of the means i the approach to these problems is by the use of a bayesian two stage prior distribution a generalization of a model used by lindley and smith 1972 in the normal mean estimation problem at the first stage of the model we assign independent conjugate distributions to 1 n where the prior means of the i satisfy the glm there are p 1 unknown hyperparameters in this specification the elements of the regression

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

on bayesian analysis of generalized linear models using jeffreys s prior abstract generalized linear models glm s have proved suitable for modeling various kinds of data consisting of exponential family response variables with covariates bayesian analysis of such data requires specification of a prior for the regression parameters in the model used uniform priors are very commonly used as conventional noninformative priors we show however that uniform priors for glm s can lead to improper posterior distributions thus making them undesirable alternative reference priors may be constructed from jeffreys s rule in this article we give two theorems that support the use of jeffreys s priors for glm s with intrinsically fixed or known scale parameters these theorems provide i sufficient and ii necessary and sufficient conditions for the propriety of the i posterior and ii prior distributions as well as for the existence of moments implications of these theorems for some commonly used glm s are discussed finally an illustrative example is given for the binomial

fully exponential laplace approximations to expectations and variances of nonpositive functions abstract tierney and kadane 1986 presented a simple second order approximation for posterior expectations of positive functions they used laplace s method for asymptotic evaluation of integrals in which the integrand is written as f exp nh and the function h is approximated by a quadratic the form in which they applied laplace s method however was fully exponential the integrand was written instead as exp nh log f this allowed first order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second order expansion for the ratio other second order expansions hartigan 1965 johnson 1970 lindley 1961 1980 mosteller and wallace 1964 require computation of more derivatives of the log likelihood function in this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions to obtain a second order approximation to an expectation e g we use the fully exponential method to approximate

bayesian analysis of proportional hazards models built from monotone functions we consider the usual proportional hazards model in the case where the baseline hazard the covariate link and the covariate coefficients are all unknown both the baseline hazard and the covariate link are monotone functions and thus are characterized using a dense class of such functions which arises upon transformation as a mixture of beta distribution functions we take a bayesian approach for fitting such a model since interest focuses more upon the likelihood we consider vague prior specifications including jeffreys s prior computations are carried out using sampling based methods model criticism is also discussed finally a data set studying survival of a sample of lung cancer patients is analyzed

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

logistic regression diagnostics a maximum likelihood fit of a logistic regression model and other similar models is extremely sensitive to outlying responses and extreme points in the design space we develop diagnostic measures to aid the analyst in detecting such observations and in quantifying their effect on various aspects of the maximum likelihood fit the elements of the fitting process which constitute the usual output parameter estimates standard errors residuals etc will be used for this purpose with a properly designed computing package for fitting the usual maximum likelihood model the diagnostics are essentially free for the asking in particular good data analysis for logistic regression models need not be expensive or time consuming

jeffreys prior for logit models abstract this paper investigates jeffreys prior for logit models with covariates in conditional logit models it is not recommended and in multinomial logit models it has properties similar to a neutral natural conjugate prior but none of the computational advantages

propriety of the posterior distribution and existence of the mle for regression models with covariates missing at random characterizing model identifiability in the presence of missing covariate data is a very important issue in missing data problems in this article we characterize the propriety of the posterior distribution of the regression coefficients for some general classes of regression models including the class of generalized linear models glm s and parametric survival models with right censored data toward this goal we derive some very general and easy to check conditions for the matrix of covariates we also derive sufficient conditions for the existence of the maximum likelihood estimates and establish novel results for checking propriety of the posterior when the sample size is large several theorems are given to establish propriety of the posterior and the existence of the maximum likelihood estimator the conditions reduce to solving a system of linear equations which can be carried out using software such as maple imsl or sas we assume that the missing covariates are missing at random and assume