bayesian residual analysis for binary response regression models summary in a binary response regression model classical residuals are difficult to define and interpret due to the discrete nature of the response variable in contrast bayesian residuals have continuous valued posterior distributions which can be graphed to learn about outlying observations two definitions of bayesian residuals are proposed for binary regression data plots of the posterior distributions of the basic observed fitted residuals can be helpful in outlier detection alternatively the notion of a tolerance random variable can be used to define latent data residuals that are functions of the tolerance random variables and the parameters in the probit setting these residuals are attractive in that a priori they are a sample from a standard normal distribution and therefore the corresponding posterior distributions are easy to interpret these residual definitions are illustrated in examples and contrasted with classical outlier detection methods for binary data

extended precise integration method for consolidation of transversely isotropic poroelastic layered media a new numerical method for calculating the consolidation behavior of the stratified transversely isotropic and poroelastic material is presented by combining the extended precise integration algorithm with the integral transformation techniques starting with the governing partial differential equations of a saturated medium with transversely isotropic skeleton and compressible fluid constituents an ordinary differential matrix equation is deduced with the aid of a laplace hankel transform an extended precise integration method for internal loading situations is proposed to solve the ordinary differential matrix equation in the transformed domain and the actual solution is recovered by a numerical inverse transformation numerical examples are also provided to prove the feasibility of this method

cross validation in survival analysis the predictive value of a statistical model is conceptually different from the explained variation in this paper we construct a measure of the predictive value of the cox proportional hazards model computed from the leave one out regression coefficients these coefficients can also be used to calculate a shrinkage factor which can be applied to improve the predictions and that can be used in r2 type measures of the proportion of explained variation our methods are illustrated by a study of chemotherapy for advanced ovarian cancer

logistic regression analysis of sample survey data summary standard chi squared x2 or likelihood ratio g2 test statistics for logistic regression analysis involving a binary response variable are adjusted to take account of the survey design these adjustments are based on certain generalized design effects logistic regression diagnostics to detect any outlying cell proportions in the table and influential points in the factor space are also developed taking account of the survey design finally the results are used to analyse some data from the october 1980 canadian labour force survey

adding a variable in generalized linear models the likelihood ratio statistic can be used to determine the significance of an explanatory variable in a generalized linear model in order to obtain such a statistic however we need two sets of iterations for two maximum likelihoods moreover the statistic is not directed to detect influential or outlying observations that affect the importance of the variable considered therefore we develop relatively simple procedures to help the analyst select an appropriate model and detect the effects of such observations on adding a variable into any model two examples are given for illustration

outliers and residual distributions in logistic regression abstract detection of outliers and other diagnostics based on residuals have gained widespread use in linear regression logistic regression has been both blessed and hindered by this development certainly logistic regression requires procedures to detect global and local model weaknesses thus the wealth of work done in linear regression provides guides and suggestions that may with care and ingenuity be applied to logistic regression several innovative attempts in this direction have been made by authors such as tsiatis 1980 pregibon 1979 1981 landwehr pregibon and shoemaker 1984 and cook and weisberg 1982 unfortunately the similarities that allow such techniques to adapt to logistic regression seem in addition to hide many of the differences two such differences are discussed here the first examines the effect of events of small probability on estimation in logistic regression the simple conclusion is that to estimate the probability of success p in an area in which p is small

diagnostics in linear discriminant analysis abstract some new diagnostic measures in discriminant analysis are proposed they can be expressed in terms of the two fundamental influence statistics in discriminant analysis d i 2 and i a theorem on the asymptotic distributions of the fundamental statistics is derived based on the theorem the proposed measures can be shown to be asymptotically distributed as functions of independent chi squared and standard normal random variables critical values and expected quantiles of the measures can then be constructed hence influential observations are detected using q q plots and significance tests two measures have analogous forms in regression the theorem is also useful for getting the asymptotic distributions of existing measures that are functions of d i 2 and i a comparison of the diagnostics in linear discriminant analysis linear regression and linear logistic regression discriminant analysis is made although discriminant coefficients can be determined under a regression model regress

binary regression models for contaminated data on etudie la nature des observations aberrantes dans le contexte des donnees de regression binaires on propose un modele simple qui permet qu un petit nombre de reponses binaires soit mal enregistre

generalized additive models some applications abstract generalized additive models have the form x fj x j where might be the regression function in a multiple regression or the logistic transformation of the posterior probability pr y 1 x in a logistic regression in fact these models generalize the whole family of generalized linear models x x where x g x is some transformation of the regression function we use the local scoring algorithm to estimate the functions fj xj nonparametrically using a scatterplot smoother as a building block we demonstrate the models in two different analyses a nonparametric analysis of covariance and a logistic regression the procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis a variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions these include confidence curves degrees of freedom estimates and approximat

deletion diagnostics for generalised estimating equations summary deletion diagnostics are proposed for generalised estimating equations the diagnostics consider leverage and residuals to measure the influence of a subset of observations on the estimated regression parameters and on the estimated values of the linear predictor computational formulae are provided which correspond to the influence of a single observation and of an entire cluster of correlated observations additionally diagnostics are given which approximate the effect of deletion of an arbitrary subset of observations under a model with general covariance structure and arbitrary link function extending proposition 3 of christensen pearson johnson 1992 the proposed measures are applied to medical data

approximate case influence for the proportional hazards regression model with censored data a method is presented for approximating the influence of individual cases upon regression coefficient estimates obtained from the cox proportional hazards model observations can thus be identified which may greatly influence statistical inferences regarding the effects of prognostic factors upon survival time an example from a cancer clinical trial is given

left cells in the affine weyl group of typec4 we find a representative set of left cells of the affine weyl groupwaof typec4as well as its left cell graphs by applying an algorithm this algorithm was designed in my previous paper tohoku j math 46 1994 105 124 it is reformulated and improved in a more efficient form here these representatives of left cells are presented as the vertices of so called essential graphs so that the generalized invariants of left cells ofwaare actually described explicitly the latter almost characterize the left cells ofwa some comments and conjectures are proposed on cells of affine weyl groups mostly for the case of typecl l 2

a goodness of fit test for binary regression models based on smoothing methods a new global test statistic for models with continuous covariates and binary response is introduced the test statistic is based on nonparametric kernel methods explicit expressions are given for the mean and variance of the test statistic asymptotic properties are considered and approximate corrections due to parameter estimation are presented properties of the test statistic are studied by simulation the goodness of fit method is illustrated on data from a dutch follow up study on preterm infants recommendations for practitioners are given

transformations of the explanatory variables in the logistic regression model for binary data summary some results are presented on improving the fit of the logistic regression model for binary data by transforming the vector of explanatory variables the methods are based on consideration of the distributions of these variables conditional on outcome group the transformations required are the functions of the explanatory variables which appear in the log density ratio of the conditional distributions

residual analysis for spatial point processes with discussion summary we define residuals for point process models fitted to spatial point pattern data and we propose diagnostic plots based on them the residuals apply to any point process model that has a conditional intensity the model may exhibit spatial heterogeneity interpoint interaction and dependence on spatial covariates some existing ad hoc methods for model checking quadrat counts scan statistic kernel smoothed intensity and berman s diagnostic are recovered as special cases diagnostic tools are developed systematically by using an analogy between our spatial residuals and the usual residuals for non spatial generalized linear models the conditional intensity plays the role of the mean response this makes it possible to adapt existing knowledge about model validation for generalized linear models to the spatial point process context giving recommendations for diagnostic plots a plot of smoothed residuals against spatial location or against a spatial covariate is effective in diagnosing spatial trend or co variate effects q q plots of the residuals are effective in diagnosing interpoint interaction

comparison of case deletion diagnostic methods for cox regression case deletion diagnostics are a routine component of regression analysis since they identify unusual observations that substantially affect parameter estimates the exact approach is to compute the change in each regression parameter by dropping that individual and refitting the model repeating a cox regression for the removal of each individual is very time consuming and therefore not done in practice the two methods commonly used to approximate the exact case deletion change for cox regression are the empirical influence function approach and the covariate vector augmentation approach this paper reports the results of a simulation study on how well these methods estimate the exact change in a parameter estimate when deleting a known outlier or a known non outlier additionally we investigate how well these methods correctly identify outliers and non outliers the covariate augmentation approach clearly outperformed the influence function approach in these simulations

outlier tests for logistic regression a conditional approach summary we consider exact conditional methods for identifying outliers in logistic regression data tests for a single outlier and multiple outliers are developed assuming a logistic slippage model the p values for these tests are determined using an explicit enumeration of all possible responses consistent with the observed value of the sufficient statistic justifications are given for preferring this computationally intensive approach to standard methods based on asymptotic approximations the techniques are applied to two examples

influence diagnostics for the cross validated smoothing parameter in spline smoothing abstract this article addresses the problem of influence in estimating the smoothing parameter when fitting a univariate smoothing spline diagnostics are presented that can identify observations that locally influence the choice of smoothing parameter by generalized cross validation either through their case weights or observed responses generalized cross validation and the diagnostic methods are given a frequency interpretation and illustrated with examples

diagnostics in categorical data analysis diagnostics as measures of model deviations and of the influence of particular data sets are used extensively in modern regression analysis for contingency tables and more generally for the parametric multinomial distribution it is not the influence of individual observations which is of interest but rather the contribution to a lack of model fit or to the values of the parameter estimates from a single cell in the table which must be evaluated hence diagnostics for contingency tables take somewhat different forms

maximum likelihood estimation for cytogenetic dose response curves in vitro dose response curves are used to describe the relation between chromosome aberrations and radiation dose for human lymphocytes the lymphocytes are exposed to low let radiation and the resulting dicentric chromosome aberrations follow the poisson distribution the expected yield depends on both the magnitude and the temporal distribution of the dose a general dose response model that describes this relation has been presented by kellerer and rossi 1972 current topics on radiation research quarterly 8 85 158 1978 radiation research 75 471 488 using the theory of dual radiation action two special cases of practical interest are split dose and continuous exposure experiments and the resulting dose time response models are intrinsically nonlinear in the parameters a general purpose maximum likelihood estimation procedure is described and estimation for the nonlinear models is illustrated with numerical examples from both experimental designs poisson regression analysis is used for estimation hypothesis testing and regression diagnostics results are discussed in the context of exposure assessment procedures for both acute and chronic human radiation exposure

estimating heterogeneity in the probabilities of enumeration for dual system estimation abstract we show how conditional logistic regression can be used to estimate the probability of being enumerated in a census and apply the model to the 1990 post enumeration survey pes in the united states the estimates can be used in the estimation of population size and the estimation of correlation bias for example unlike the classical stratification approach the logistic approach permits the use of continuous explanatory variables model choice can be based on the standard techniques of the generalized linear models we discuss some special problems caused by the fact that the pes sample area is open to migration between the captures we also consider the effect of data errors in estimation we characterize hard to enumerate populations and give some tentative estimates of correlation bias

a precise numerical method for rayleigh waves in a stratified half space this paper presents theory and results for rayleigh waves propagating in a transversely isotropic stratified solid resting on an elastic semi infinite space it uses the precise integration method pim and the extended wittrick williams w w algorithm this problem can be reduced into the eigenvalue problem of ordinary differential equations odes in the frequency wavenumber domain the pim used here is a precise method for solving the odes with two point boundary conditions and the eigenvalue counting method of the extended w w algorithm is used to ensure that all eigenvalues are found without the possibility of any being missed hence the theory presented in this paper is exact in the sense that its precision is limited only by the precision of the computer used

consistency of deviance based m estimators summary in a general estimation problem the deviance function generates statistics that are similar to squared standardized residuals a deviance based m estimator dbme is defined as an adaptively weighted maximum likelihood estimator where the weights depend upon the deviances in both a single parameter and a regression setting we give some general conditions under which a dmbe is consistent for a suitable weighting scheme these conditions are satisfied in many continuous cramer rao regular families and in related linear or nonlinear regression cases the conditions fail and the estimator is inconsistent in most discrete families

detection of influential observation in linear regression a new measure based on confidence ellipsoids is developed for judging the contribution of each data point to the determination of the least squares estimate of the parameter vector in full rank linear regression models it is shown that the measure combines information from the studentized residuals and the variances of the residuals and predicted values two examples are presented

influence on confidence regions for regression coefficients in generalized linear models abstract influence on confidence regions has received less attention than the problem of influence on regression coefficients in generalized linear models diagnostics are given for examining influence on confidence intervals or the confidence region for all of the regression coefficients these diagnostics permit one to check whether the standard errors of the coefficients are sensitive to the exact specification of case weights or the observed values of the explanatory variables or the response the diagnostics are illustrated with an example and compared to standard diagnostic methods

a diagnostic for cox regression and general conditional likelihoods abstract diagnostics for changes in the maximum likelihood estimate of due to deletion of single observations in linear and logistic regression models are equivalent to diagnostics obtained from fitting augmented regression models for the cox proportional hazards model and in conditional models for arbitrarily matched or stratified data diagnostics derived from augmented regression models are again useful and easy to compute these diagnostics are applicable to different formulations for the multiplicative covariable effects over time or over strata an exact or approximate treatment of tied failure times or multiple case matching and time dependent covariates

assessing the adequacy of the logistic regression model for matched case control studies we apply results for the detection of outliers in logistic regression to the analysis of matched pairs data and illustrate the techniques with data from a matched pairs study of the relationship between iron binding proteins and mortality in the solomon islands although the study includes 90 matched pairs the conclusions change substantially with deletion of one or two pairs thus even with reasonably large data sets single observations may have a great impact on the results of an analysis the identification of influential observations should constitute part of the analysis of matched case control data

some diagnostics for binary logistic regression via smoothing summary the paper discusses some diagnostic tools for binary logistic regression which use smoothing techniques smoothed binary data are employed to devise a battery of diagnostic plots from simple procedures which indicate only the adequacy or inadequacy of a maximum likelihood logistic fit to partial residual plots and delta local deviance plots which reveal the type of model inadequacy suggesting perhaps the addition of nonlinear terms or interactions properties of the methods are studied by simulation the diagnostic tools are illustrated on a set of data in which the probability of having telephone service is related to the demographic variables income and education

on the use of nonparametric regression for model checking summary the use of nonparametric regression is explored to check the fit of a parametric regression model the principal aim is to check the validity of the regression curve rather than necessarily to detect outliers a pseudo likelihood ratio test is developed to provide a global assessment of fit and simulation bands are used to indicate the nature of departures from the model the types of data considered include discrete response variables where standard diagnostic techniques are often not appropriate and first order autoregressive series several numerical examples are given nonparametric regression can be used in an informal graphical way to assess the relationship between a response and an explanatory variable in this paper we aim to develop more formal methods of assessing the assumptions of a parametric model in particular when regression diagnostics of the type developed for normal linear models are not readily available the principal aim is to check the validity of the systematic part of the model by comparing a nonparametric estimate of the regression curve with a parametric one such a comparison may also identify outliers although the distinction between outliers and model inadequacy is not always easy two techniques are used to assess the fit of a parametric model in 2 confidence bands are constructed around the fitted regression curve by simulation a comparison of these with the nonparametric curve gives an indication of the nature of any departures from the model in 3 a pseudo likelihood ratio test is developed this provides a quantitative global assessment of fit in applying these ideas special emphasis is given to discrete data and notably logistic regression because of the difficulty in applying standard residual based model checking techniques to this type of response variable a poisson regression example is discussed in 4 however the underlying ideas have wider applications autoregressive time series of order 1 are discussed in 6 sections 5 and 7 discuss general issues we first discuss the context of binary regression with a single covariate and the difficulties caused by the discreteness of the response variable the observed data are assumed to be of the form xi yi ni where xi is a covariate value and yi has a binomial

regression diagnostics to detect nonrandom missingness in linear regression missing data is a common problem in regression analysis the usual estimation strategies require that the data values be missing completely at random mcar if this is not the case estimates can be severely biased in this article it is shown that tests can be constructed based on common regression diagnostics to detect non mcar behavior the construction of these tests and their properties when data are missing in one explanatory variable are detailed computer simulations indicate good power to detect various non mcar processes three examples are presented extensions to missing data in more than one explanatory variable and to arbitrary regression models are discussed

partial residual plots in generalized linear models abstract in this article we explore the structure and usefulness of partial residual plots as tools for visualizing curvature as a function of selected predictors x 2 in a generalized linear model glm where the vector of predictors x is partitioned as x t x t 1 x t 2 the glm extension of ceres plots is discussed but to a lesser extent the usefulness of these plots for obtaining a good visual impression of curvature may be limited by the specified glm the link function and the stochastic behavior of the predictors partial residual plots seem to work well when modeling is in a region where the conditional mean of the response given x stays well away from its extremes so that the link is essentially linear and e x 1 x 2 is linear in x 2 ceres plots however require only the first condition the behavior of these plots is contrasted with their behavior in additive error models

robust logistic discrimination summary logistic discrimination is a well established method for allocating observations to one of two or more populations in this paper we show how the logistic model can be adapted to make it robust against outlying observations from the populations the new model is successfully tested using simulated data against ordinary logistic discrimination and also against logistic discrimination where parameters are fitted using resistant methods the model is then used on data collected from the teeth of rats logistic discrimination is a partially parametric method for discrimination applicable in many situations and under several sampling schemes notable references to the method are day kerridge 1967 anderson 1972 1979 1982 and anderson blair 1982 the appeal of the logistic method for discrimination is that only the ratio of likelihoods has to be modelled and not the full distributional form of the underlying variables press wilson 1978 and byth mclachlan 1980 show that in many circumstances logistic discrimination is preferable to the usual linear discrimination based on either normal populations with equal covariance matrices or fisher s linear discriminant for the two group case suppose x is a vector of observations h1 and h2 are the two groups and a a parameter vector then the log likelihood ratio is modelled as

on continuity corrected residuals in logistic regression summary in choosing residuals for logistic regression the two primary candidates are pearson or linear residuals and deviance residuals an asymptotic bias correction can be used with the latter residuals can also be computed after the arc sine or the anscombe transformation and there are residuals which approximate an outlier score statistic williams 1984 recently pierce schafer 1986 proposed correcting for continuity before computing residuals they first studied the behaviour of residuals when calculated at the true parameter unknown in practice and then argued that when parameter estimates are close to the true values this same behaviour will be exhibited by residuals calculated at estimates our work shows that this two step approach can be quite misleading continuity corrections are detrimental to the approximate normality of residuals even in the simplest case study of graphical displays based on residuals in realistic problems indicates that the bias correction to the deviance is problematic for extreme probabilities the linear residuals as well as the unadjusted deviance residuals and the outlier test residuals perform well in moderate sized problems and in both null and contaminated situations

multi precision laplace transform inversion for the numerical inversion of laplace transforms we suggest to use multi precision computing with the level of precision determined by the algorithm we present two such procedures the gaver wynn rho gwr algorithm is based on a special sequence acceleration of the gaver functionals and requires the evaluation of the transform only on the real line the fixed talbot ft method is based on the deformation of the contour of the bromwich inversion integral and requires complex arithmetic both gwr and ft have only one free parameter m which is the number of terms in the summation both algorithms provide increasing accuracy as m increases and can be realized in a few lines using current computer algebra systems copyright 2004 john wiley sons ltd

covariance adjustment of relative risk estimates in matched studies the matched sample case control study is widely used by epidemiologists to estimate the relative incidence of disease among persons exposed to different levels of one or more risk factors an apparent limitation of such designs has been the necessity to control the effects of all potential confounding variables in the matching process conditional likelihood analyses based on the linear logistic equation enable one to model the effects of covariates while preserving the original matching this paper reviews the logical basis for this methodology and illustrates its value by application to a case control study of esophageal cancer

influence diagnostics for linear measurement error models summary influence functions are used to show that extreme observations affect linear measurement error model estimators in directions orthogonal to and along the fitted plane rather than vertically and horizontally as with least squares estimators influence diagnostics patterned after those for least squares and derived for generalized linear models by pregibon 1981 are adapted to assess the influence of extreme observations on linear measurement error model estimators perturbation arguments and asymptotic theory are used to support the use of these diagnostics it is well known that least squares estimators are influenced by extreme observations in the response and predictor variables geometrically these influences are in directions defined by the axis of the response variable and the plane defined by the predictors as shown in 3 measurement error model estimators are affected in directions that are perpendicular to and along the fitted plane thus least squares influence diagnostics that measure the effects of observations in the vertical and the horizontal directions may fail to accurately indicate the true effects of extreme observations on measurement error model estimators kelly 1984 derives an influence function for a structural measurement error model in which the error distribution has finite fourth moments and the error covariance matrix is known up to a multiple with the aid of this influence function diagnostics such as cook s d are defined by analogy to the corresponding least squares equivalents fuller 1987 defines a measurement error model estimator of the hat matrix using estimated predictor variable values he also proposes using diagnostics and plotting techniques analogous to those of least squares with estimated model errors and estimated leverage values from the hat matrix miller 1989 provides partial theoretical justification for the use of these least squares based techniques by showing that the estimated model errors asymptotically are a gaussian process apart from these efforts little research has been conducted on influence diagnostics for measurement error models the derivation of influence diagnostics for measurement error models in this paper proceeds as follows measurement error model estimators are briefly described in 2 a more comprehensive discussion of these estimators is given by fuller 1987 ch 2 in 3 influence function arguments are used to establish the need for measurement error model diagnostics measurement error model estimators are expressed as iteratively reweighted least squares estimators in 4 leading to the adaptation of diagnostics for

model checking techniques for stratified case control studies we present graphical and numerical methods for assessing the adequacy of the logistic regression model for stratified case control data the proposed methods are derived from the cumulative sum of residuals over the covariate or linear predictor under the assumed model the cumulative residual process converges weakly to a zero mean gaussian process whose distribution can be approximated via monte carlo simulation the observed cumulative residual pattern can then be compared both visually and analytically to a number of simulated realizations from the approximate null distribution these comparisons enable one to examine the functional form of each covariate the logistic link function as well as the overall model adequacy simulation studies demonstrate that the proposed methods perform well in practical settings illustration with an oesophageal cancer study is provided

maximum likelihood and quasi likelihood for nonlinear exponential family regression models abstract linear and nonlinear exponential family and quasi likelihood regression models form a class of models with a structure that invites using one algorithmic framework to compute parameter estimates and regression diagnostics this framework extends our work on nonlinear least squares it includes iteratively reweighted least squares but also encompasses secant updates for part of the hessian matrix of the likelihood or quasi likelihood function along with tests for when to use this information the framework also provides basic machinery for computing leave one out style regression diagnostics we describe the framework discuss some implementation details and present some numerical experience

residual plots for log odds ratio regression models this article describes graphical diagnostic methods for log odds ratio regression models to study the effects of an additional covariate on log odds ratio regression analysis three types of residual plots based on weighted least squares wls are discussed i added variable plot partial regression plot ii partial residual plot and iii augmented partial residual plot these plots provide diagnostic procedures for identifying heterogeneity of error variances outliers or nonlinearity of the model they are especially useful for clarifying whether including a covariate as a linear term is appropriate or whether quadratic or other nonlinear transformations are preferable a well known data set for case control studies is analyzed to illustrate the residual plots

the relation lr on some elements of the affine weyl group c n abstract let w s be the affine weyl group of type c n with s its coxeter generator set let 2 n 1 be the set of all partitions 1 r of 2 n 1 such that j 1 2 k 1 j is odd for any k n with 2 k 1 r for any j s let w j be the longest element in the parabolic subgroup of w generated by j we define a map w j j s 2 n 1 and study the preorder lr on the set w j j s and its relation with the partial order on the set w j j s where iterating star operations and primitive pairs play an important role

regression by local fitting methods properties and computational algorithms abstract local regression is a procedure for estimating regression surfaces by the local fitting of linear or quadratic functions of the independent variables in a moving fashion that is analogous to how a moving average is computed for a time series the advantage of the methodology over the global fitting of parametric functions of the independent variables by least squares the current paradigm in regression studies is that a much wider class of regression functions can be estimated without distortion in this paper we discuss the methods their statistical properties and computational algorithms

consolidation of multilayered half space with anisotropic permeability and compressible constituents abstract this paper presents analytical solutions of consolidation for poroelastic and multilayered half space and anisotropy of the permeability and the compressibility of the pore fluid are considered state vector method together with laplace hankel transform techniques are used to solve the basic governing equations and obtain the transfer matrix in a clearly arranged way forward and backward transfer matrix techniques are utilized in the analytical formulation of solutions for the multilayered half space a numerical inversion scheme of crump s method is adopted to obtain time domain solution numerical results are presented for a single homogeneous soil layer and a multilayered half space and they illustrate the influences of the anisotropy of permeability and the compressibility of the pore fluid on the consolidation of the soils

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

exploring partial residual plots partial residual plots have a long history and judging from their prominence in the literature are frequently used in this article i explore the structure and usefulness of partial residual plots and augmented partial residual plots as basic tools for dealing with curvature as a function of selected covariates x 2 in regression problems in which the covariate vector x is partitioned as x t x 1 t x 2 t the usefulness of these plots for obtaining a good impression of curvature can depend on the behavior of the covariates through the conditional expectation e x 1 x 2 partial residual plots seem to perform best under linear conditional expectations augmented partial residual plots allow e x 1 x 2 to be a quadratic function of x 2 this development leads to a new class of plots called ceres plots that includes partial and augmented partial residual plots as special cases ceres plots may be useful for obtaining an impression of curvature as a function of x 2 when the conditional expectations

analysing the relationship between change in a risk factor and risk of disease there are numerous examples in the epidemiologic literature of analyses that relate the change in a risk factor such as serum cholesterol to the risk of an adverse outcome such as heart disease many of these analyses fit some type of regression model such as logistic regression or the cox model for survival time data that includes both the change in the risk factor and the baseline value as covariates we show that this method of adjusting for the baseline level can produce misleading results the problem occurs when the true value of the risk factor relates to the outcome and the measured value differs from the true value due to measurement error we may find the observed change in the risk factor significantly related to the outcome when there is in fact no relationship between the true change and the outcome if the question of interest is whether a person who lowers his level of the risk factor by means of drugs or lifestyle changes will thereby reduce his risk of disease then we should consider an association due solely to measurement error as spurious we present a method that adjusts for the measurement error in a linear regression analysis and show that an analogous adjustment applies asymptotically to logistic regression as in other errors in variables problems this analysis depends on knowledge of the relative variances of the random variation the true baseline value and the true change since the magnitudes of these variances are usually unknown and sometimes unknowable the distinction between true change and measurement error being ambiguous we recommend a sensitivity analysis that examines how the analysis results depend on the assumptions concerning the variances the commonly used analysis method corresponds to the extreme case in which there is no measurement error we use data from the framingham study and simulations to illustrate these points

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

ideal point discriminant analysis a new method of multiple discriminant analysis was developed that allows a mixture of continuous and discrete predictors the method can be justified under a wide class of distributional assumptions on the predictor variables the method can also handle three different sampling situations conditional joint and separate in this method both subjects cases or any other sampling units and criterion groups are represented as points in a multidimensional euclidean space the probability of a particular subject belonging to a particular criterion group is stated as a decreasing function of the distance between the corresponding points a maximum likelihood estimation procedure was developed and implemented in the form of a fortran program detailed analyses of two real data sets were reported to demonstrate various advantages of the proposed method these advantages mostly derive from model evaluation capabilities based on the akaike information criterion aic

a comparison of goodness of fit tests for the logistic regression model recent work has shown that there may be disadvantages in the use of the chi square like goodness of fit tests for the logistic regression model proposed by hosmer and lemeshow that use fixed groups of the estimated probabilities a particular concern with these grouping strategies based on estimated probabilities fitted values is that groups may contain subjects with widely different values of the covariates it is possible to demonstrate situations where one set of fixed groups shows the model fits while the test rejects fit using a different set of fixed groups we compare the performance by simulation of these tests to tests based on smoothed residuals proposed by le cessie and van houwelingen and royston a score test for an extended logistic regression model proposed by stukel the pearson chi square and the unweighted residual sum of squares these simulations demonstrate that all but one of royston s tests have the correct size an examination of the performance of the tests when the correct model has a quadratic term but a model containing only the linear term has been fit shows that the pearson chi square the unweighted sum of squares the hosmer lemeshow decile of risk the smoothed residual sum of squares and stukel s score test have power exceeding 50 per cent to detect moderate departures from linearity when the sample size is 100 and have power over 90 per cent for these same alternatives for samples of size 500 all tests had no power when the correct model had an interaction between a dichotomous and continuous covariate but only the continuous covariate model was fit power to detect an incorrectly specified link was poor for samples of size 100 for samples of size 500 stukel s score test had the best power but it only exceeded 50 per cent to detect an asymmetric link function the power of the unweighted sum of squares test to detect an incorrectly specified link function was slightly less than stukel s score test we illustrate the tests within the context of a model for factors associated with low birth weight

deletion diagnostics for the generalised linear mixed model with independent random effects the generalised linear mixed model glmm is widely used for modelling environmental data however such data are prone to influential observations which can distort the estimated exposure response curve particularly in regions of high exposure deletion diagnostics for iterative estimation schemes commonly derive the deleted estimates based on a single iteration of the full system holding certain pivotal quantities such as the information matrix to be constant in this paper we present an approximate formula for the deleted estimates and cook s distance for the glmm which does not assume that the estimates of variance parameters are unaffected by deletion the procedure allows the user to calculate standardised dfbetas for mean as well as variance parameters in certain cases such as when using the glmm as a device for smoothing such residuals for the variance parameters are interesting in their own right in general the procedure leads to deleted estimates of mean parameters which are corrected for the effect of deletion on variance components as estimation of the two sets of parameters is interdependent the probabilistic behaviour of these residuals is investigated and a simulation based procedure suggested for their standardisation the method is used to identify influential individuals in an occupational cohort exposed to silica the results show that failure to conduct post model fitting diagnostics for variance components can lead to erroneous conclusions about the fitted curve and unstable confidence intervals

conditionally unbiased bounded influence estimation in general regression models with applications to generalized linear models abstract in this article robust estimation in generalized linear models for the dependence of a response y on an explanatory variable x is studied a subclass of the class of m estimators is defined by imposing the restriction that the score function must be conditionally unbiased given x within this class of conditionally fisher consistent estimators optimal bounded influence estimators of regression parameters are identified and their asymptotic properties are studied the estimators studied in this article and the efficient bounded influence estimators studied by stefanski carroll and ruppert 1986 depend on an auxiliary centering constant and nuisance matrix the centering constant can be given explicitly for the conditionally fisher consistent estimators and thus they are easier to compute than the estimators studied by stefanski et al 1986 in addition estimation of the nuisance matrix has no effect on the asymptotic distribution of the conditionally fisher consistent estimators the sam

a diagnostic for cox regression with discrete failure time models summary changes in maximum likelihood parameter estimates due to deletion of individual observations are useful statistics both for regression diagnostics and for computing robust estimates of covariance for many likelihoods including those in the exponential family these delete one statistics can be approximated analytically from a one step newton raphson iteration on the full maximum likelihood solution but for general conditional likelihoods and the related cox partial likelihood the one step method does not reduce to an analytic solution for these likelihoods an alternative analytic approximation that relies on an appropriately augmented design matrix has been proposed in this paper we extend the augmentation approach to explicitly deal with discrete failure time models in these models an individual subject may contribute information at several time points thereby appearing in multiple risk sets before eventually experiencing a failure or being censored our extension also allows the covariates to be time dependent the new augmentation requires no additional computational resources while improving results

leave k out diagnostics for time series abstract the authors propose diagnostics for arima model fitting for time series formed by deleting observations from the data and measuring the change in the estimates of the parameters the use of leave one out diagnostics is a well established tool in regression analysis this document demonstrates the efficacy of observation deletion based diagnostics for arima models addressing issues special to the time diagnostics based on the innovations variance it is shown that the dependency aspect of time series data gives rise to a smearing effect which confounds the diagnostics for the coefficients it is also shown that the diagnostics based on the innovations variance are much clearer and more sensitive than those for the coefficients a leave k out diagnostics approach is proposed to deal with patches of outliers and problems caused by masking are handled by use of iterative deletion an overall strategy for arima model fitting is given and applied to two data sets

diagnostic regression analysis and shifted power transformations diagnostic displays for outlying and influential observations are reviewed in some examples apparent outliers vanish after a power transformation of the data interpretation of the score statistic for transformations as regression on a constructed variable makes diagnostic methods available for detection of the influence of individual observations on the transformation methods for the power transformation are exemplified and extended to the power transformation after a shift in location for which there are two constructed variables the emphasis is on plots as diagnostic tools

ats methods nonparametric regression for non gaussian data abstract ats methods provide an approach to fitting curves and surfaces to data using nonparametric regression when distributions are not necessarily gaussian first a small amount of local averaging the a in ats is carried out then a variance stabilizing transformation is applied t and finally the result is smoothed s using a nonparametric regression procedure ats methods are quite broad in terms of applications in this article we show how they can be used for fitting a surface when the response is binary for estimating density and for estimating the spectrum of a time series we also present some theoretical investigations that give guidance on how to choose the amount of averaging and how efficient the methods are

exact stiffness method for quasi statics of a multi layered poroelastic medium abstract a method is presented to study the three dimensional quasi static response of a multi layered poroelastic half space with compressible constituents the system under consideration consists of n layers of different thickness and material properties overlying a homogeneous half space fourier expansion laplace transforms and hankel transforms with respect to the circumferential time and radial coordinates respectively are used in the formulation laplace hankel transforms of displacements and pore pressure at layer interfaces are considered as the basic unknowns exact stiffness matrices describing the relationship between generalized displacement and force vectors of a finite layer and a half space are derived explicitly in the transform space the global stiffness matrix of a layered system is assembled by considering the continuity of tractions and fluid flow at layer interfaces the time histories of displacements stresses and pore pressure are obtained by solving the stiffness equation system for discrete values of laplace and hankel transform parameters and using numerical quadrature schemes for laplace and hankel transform inversions selected numerical results for different layered systems are presented to portray the influence of layering and poroelastic material properties the advantage of the present method is that for an n layered system it yields a numerically stable symmetric stiffness matrix of order 4 n 4 n when compared to the unsymmetric and numerically unstable coefficient matrix of order 8 n 8 n associated with the conventional method based on the determination of layer arbitrary coefficients

the influence of variables in a logistic model summary the influence of explanatory variables in a logistic model is considered when the goal is to determine the predictive probability of a future observation in a bayesian approach we consider whether the predictive distribution will change greatly when subsets of the explanatory variables are omitted from the data set or from the future observation several authors have considered the problem of the influence of observations in a logistic regression model in particular pregibon 1981 cook weisberg 1982 ch 2 and johnson 1985 have developed measures of influence for identifying observations in the data which are influential relative to the estimation of the regression coefficients the determination of predictive probabilities and the classification of future observations our aim is to consider the influence of explanatory variables in a logistic regression model this has been studied extensively in regression analysis but usually for the estimation or testing of the regression parameters the novelty of the approach suggested here stems from the goal being to determine the effect on the predictive probability of a future observation of the omission of subsets of the explanatory variables

graphical methods for assessing logistic regression models abstract in ordinary linear regression graphical diagnostic displays can be very useful for detecting and examining anomalous features in the fit of a model to data for logistic regression models the discreteness of binary data makes it difficult to interpret such displays modifications and extensions of linear model displays lead to three methods for diagnostic checking of logistic regression models local mean deviance plots are useful for detecting overall lack of fit empirical probability plots help point out isolated departures from the fitted model partial residual plots when smoothed to show underlying structure help identify specific causes of lack of fit these methods are illustrated through the analyses of simulated and real data

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

the precise computation for wave propagation in stratified materials the problem of random wave travelling along stratified materials is important for earthquake engineering the soil is considered multi layered and located above the base rock whose material property is assumed to be much stiffer than the soil and the power spectrum density of the random excitation is given at the base rock the soils can be arbitrarily anisotropic for each layer the governing differential equations are derived in frequency and wavenumber domain and only a set of ordinary differential equations must then be solved the eigensolution expansion method is used to solve for the responses of the layers the precise integration algorithm in combination with the extended w w algorithm is applied to obtain all the eigensolutions of the ode ordinary differential equation thereafter the recently developed pseudo excitation method for structural random vibration is transplanted to the solution of the layered soil responses numerical results are given and compared with some classical results this shows the relevance of the presented method copyright 2004 john wiley sons ltd

adapting for the missing link we consider the fitting of generalized linear models in wich the link function is assumed to be unknown and propose the following computational method first estimate regression coefficients using the canonical link then estimate the link via a kernel smoother treating the direction in the predictor space determined by the regression coefficients as known then reestimate the direction using the estimated link and alternate between these two steps we show that under fairly general conditions n 1 2 consistent estimates of the direction are obtained a small monte carlo study is presented

fractional derivative modeling for axisymmetric consolidation of multilayered cross anisotropic viscoelastic porous media abstract this paper presents a fractional derivative modeling for axisymmetric consolidation of multilayered cross anisotropic viscoelastic porous media the elastic solution for multilayered porous media is acquired by the extended precise integration method the flexibility coefficient of the fractional merchant viscoelastic model is deduced in the laplace transformed domain and the viscoelastic solution of the problem is further obtained by the elastic viscoelastic correspondence principle the numerical results are compared with those of published literatures to verify the proposed method and the influences of fractional derivative order viscoelasticity of solid skeleton and layering of materials on the axisymmetric consolidation of multilayered viscoelastic porous media are investigated by several examples

resistant fits for some commonly used logistic models with medical application logistic regression type models are used in many applications some examples include the classical dose response experiment prospective and retrospective studies of disease incidence with and without matching and the analysis of ordinal data in most instances the model is fitted by the method of maximum likelihood which like least squares is sensitive to atypical observations an alternative to maximum likelihood is proposed and illustrated by examples

assessing partial influence in generalized linear models summary a general technique for assessing partial influence in generalized linear models is presented the method consists of a graphical diagnostic display to identify observations that are either influential on a subset of parameters or substantially contributing to a particular model departure application of the method is illustrated with an example in logistic regression this paper is concerned with assessing partial influence for the class of generalized linear models glms described by nelder and wedderburn 1972 we present a unified approach for the detection and assessment of observations that are influential only on one parameter or a subset of parameters throughout the emphasis is on supplementing formal significance tests with effective graphical methods in linear regression the added variable plot and the partial residual plot are two wellknown diagnostic displays that have been suggested for studying the influence of an observation on a single regression parameter explanatory variable see chatterjee and hadi 1986 for a review for glms the corresponding generalizations are given by wang 1985 and landwehr pregibon and shoemaker 1984 the latter proposed a partial residual plot just for binary logistic regression but its glm version can be constructed analogously both types of plots should appear linear if the parameter of interest is significant furthermore it is expected that the plots will give an overall impression of the scatter of the data points individual points that are separated from the rest of the data provide heuristic information about potentially influential observations and may suggest cases for further study however as shown in cook and wang 1983 these plots can fail if important information about the variable arises from observations at leverage points i e points remote in the design space this is because leverage points usually produce relatively small residuals and such points may make only a small contribution to the plot even if they are influential for that variable another difficulty occurs in the situation where a subset of regression parameters are of interest simultaneously since the above plots can handle only one variable parameter at a time the aim of this paper is therefore to derive a reliable diagnostic display for revealing individual observations that are influential on one or more parameters within the framework of glms in section 2 a new diagnostic display called a partial influence plot is described the plot provides additional insight in identifying leverage and influential observations and moreover has general applicability throughout the model checking procedure an example illustrating the usefulness of the partial influence plot is given in section 3 the last section contains further comments on related work on partial influence

a general method for local sensitivity analysis with application to regression models and other optimization problems this article introduces a method for sensitivity analysis of general applicability the method is based on the well known duality property of mathematical programming which states that the partial derivatives of the primal objective function with respect to the constraints on the right side parameters are the negative of the optimal values of the dual problem variables for the parameters or data for which sensitivities are sought to appear on the right side they are converted into artificial variables and set to their actual values thus obtaining the desired constraints the method is applicable to linear and nonlinear models to normal and nonnormal models and to least squares and other methods of estimation in addition to its general applicability the method is also computationally inexpensive because the necessary information becomes available without extra calculations the theoretical basis for the method is given and illustrated by its application to least squares least absolute value an

predictive value of statistical models a review is given of different ways of estimating the error rate of a prediction rule based on a statistical model a distinction is drawn between apparent optimum and actual error rates moreover it is shown how cross validation can be used to obtain an adjusted predictor with smaller error rate a detailed discussion is given for ordinary least squares logistic regression and cox regression in survival analysis finally the splitsample approach is discussed and demonstrated on two data sets

analysis of contingency tables by ideal point discriminant analysis cross classified data are frequently encountered in behavioral and social science research the loglinear model and dual scaling correspondence analysis are two representative methods of analyzing such data an alternative method based on ideal point discriminant analysis da is proposed for analysis of contingency tables which in a certain sense encompasses the two existing methods a variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and or as direct constraints on model parameters this along with maximum likelihood estimation of the model parameters allows interesting model comparisons this is illustrated by the analysis of several data sets

analyzing repeated measures on generalized linear models via the bootstrap the analysis of longitudinal data for which the response variables have nonnormal error distributions previously has been complex and or dependent on restrictive assumptions in this paper simple methods are introduced for the class of generalized linear models glms regressions are fit to the data at each observation time functions of the resulting coefficients may be bootstrapped or the coefficients combined through closed form estimation of their covariances application is made to a data set on xerophthalmia in indonesian children

a counter example to a conjecture of lusztig abstract in lusztig 2003 7 lusztig defined two functions a and a on a coxeter group w and conjectured that they are the same in the present paper we give a counter example to lusztig s conjecture then we propose a new conjecture for the description of two sided cells of w which can be regarded as an improvement of lusztig s conjecture some examples are given to support our conjecture