graphical diagnostics to check model misspecification for the proportional odds regression model the cumulative logit or the proportional odds regression model is commonly used to study covariate effects on ordinal responses this paper provides some graphical and numerical methods for checking the adequacy of the proportional odds regression model the methods focus on evaluating functional misspecification for specific covariate effects but misspecification of the link function can also be dealt with under the same framework for the logistic regression model with binary responses arbogast and lin statist med 2005 24 229 247 developed similar graphical and numerical methods for assessing the adequacy of the model using the cumulative sums of residuals the paper generalizes their methods to ordinal responses and illustrates them using an example from the va normative aging study simulation studies comparing the performance of the different diagnostic methods indicate that some of the graphical methods are more powerful in detecting model misspecification than the hosmer lemeshow type goodness of fit statistics for the class of models studied

assessing proportionality in the proportional odds model for ordinal logistic regression the proportional odds model for ordinal logistic regression provides a useful extension of the binary logistic model to situations where the response variable takes on values in a set of ordered categories the model may be represented by a series of logistic regressions for dependent binary variables with common regression parameters reflecting the proportional odds assumption key to the valid application of the model is the assessment of the proportionality assumption an approach is described arising from comparisons of the separate correlated fits to the binary logistic models underlying the overall model based on asymptotic distributional results formal goodness of fit measures are constructed to supplement informal comparisons of the different fits a number of proposals including application of bootstrap simulation are discussed and illustrated with a data example

multinomial goodness of fit tests for logistic regression models we examine the properties of several tests for goodness of fit for multinomial logistic regression one test is based on a strategy of sorting the observations according to the complement of the estimated probability for the reference outcome category and then grouping the subjects into g equal sized groups a g x c contingency table where c is the number of values of the outcome variable is constructed the test statistic denoted as cg is obtained by calculating the pearson chi2 statistic where the estimated expected frequencies are the sum of the model based estimated logistic probabilities simulations compare the properties of cg with those of the ungrouped pearson chi2 test x2 and its normalized test z the null distribution of cg is well approximated by the chi2 distribution with g 2 x c 1 degrees of freedom the sampling distribution of x2 is compared with a chi2 distribution with n x c 1 degrees of freedom but shows erratic behavior with a few exceptions the sampling distribution of z adheres reasonably well to the standard normal distribution power simulations show that cg has low power for a sample of 100 observations but satisfactory power for a sample of 400 the tests are illustrated using data from a study of cytological criteria for the diagnosis of breast tumors

a system of one dimensional balls with gravity we introduce a hamiltonian system with many degrees of freedom for which the nonvanishing of some lyapunov exponents almost everywhere can be established analytically

discrimination and classification overview publisher summary this chapter discusses discrimination and classification the problem of discrimination and classification is mainly concerned with differentiating between g g 2 mutually exclusive populations and with classifying subjects or objects on the basis of multivariate observation the emphasis of the problem is on deriving a rule that can be used to optimally classify a new subject it is an important problem in statistics and applied mathematics as its applications are found in many diverse areas in practice the distributions are rarely known and if the form of the distributions are known or assumed the parameters are seldom known thus the problem for practical applications is to develop a sample based rule for discrimination and classification the earliest and most well known method is based on the linear discriminant function due to fisher

order restricted score parameters in association models for contingency tables abstract the row effects and column effects models for two way contingency tables have parameters for the row and column categories pertaining to the association between the variables for classifications having ordered categories it is often reasonable to assume that the association parameters have a corresponding ordering this article proposes order restricted estimates of the association parameters in these models the maximum likelihood solution can be determined by the solution of a simple isotonic regression of some of the model sufficient statistics the models are primarily log linear in form and can be expressed in terms of odds ratios for 2 2 subtables consisting of adjacent rows and adjacent columns for the order restricted solution these local log odds ratios have uniform sign goodness of fit statistics for this solution are related to corresponding statistics for collapsed tables and to statistics for testing equality of sets of the parameters the row effects model discussed in this art

diffusion in energy conserving coupled maps we consider a dynamical system consisting of subsystems indexed by a lattice each subsystem has one conserved degree of freedom energy the rest being uniformly hyperbolic the subsystems are weakly coupled together so that the sum of the subsystem energies remains conserved we prove that the subsystem energies satisfy the diffusion equation in a suitable scaling limit

hypothesis testing in the polychotomous logistic model with an application to detecting gastrointestinal cancer we discuss the use of the trichotomous logistic model to discriminate between patients with gastrointestinal gi cancer patients with benign gi disease and normal subjects using symptoms and the concentrations of some serum proteins that are potentially indicative of malignancy as covariates a parsimonious model can be obtained by invoking an indistinguishability hypothesis which is appropriate when a covariate is considered to have no predictive value between categories it is shown that the polychotomous model can be re parameterised under the null hypothesis to give a reduced form which can be fitted by maximum likelihood the validity of the use of the same methods for retrospective sampling is discussed the approach is illustrated by the development of a logistic model to identify symptomatic and asymptomatic subjects with a high risk of gi cancer

arma parameter estimation using only output cumulants several algorithms are developed to estimate the parameters of a causal nonminimum phase autoregressive moving average arma p q system which is excited by an unobservable independently identically distributed non gaussian process the output is contaminated by additive colored gaussian noise of unknown power spectral density a fundamental result is presented pertaining to the identifiability of ar parameters based on the yule walker equations drawn from a specific set of p 1 1 d slices of the kth k 2 order output cumulant several ma parameter estimation algorithms are developed one method uses q 1 d slices of the output cumulant a second method uses only two 1 d cumulant slices these methods do not involve computation of the residual i e ar compensated time series or polynomial factorization multidimensional versions of the various algorithms are presented a simulation study demonstrating the effectiveness of the algorithms is included

ergodic systems of n balls in a billiard table we consider the motion ofn balls in billiard tables of a special form and we prove that the resulting dynamical systems are ergodic on a constant energy surface in fact they enjoy thek property these are the first systems of interacting particles proven to be ergodic for an arbitrary number of particles

a two stage regression model for epidemiological studies with multivariate disease classification data polytomous logistic regression is commonly used to analyze epidemiological data with disease subtype information in this approach effects of exposures on different disease subtypes are studied through separate exposure odds ratios comparing different case groups to the common control group this article considers the situation where disease subtypes can be defined using multiple characteristics of a disease for efficient analysis of such data a two stage modeling approach is proposed at the first stage a standard polytomous logistic regression model is considered for all possible distinct disease subtypes that can be defined by the cross classification of the different disease characteristics at the second stage the exposure odds ratio parameters for the first stage disease subtypes are further modeled in terms of the defining characteristics of the subtypes when the total number of first stage disease subtypes is small standard maximum likelihood methods can be used for inference in the proposed

decision combination in multiple classifier systems a multiple classifier system is a powerful solution to difficult pattern recognition problems involving large class sets and noisy input because it allows simultaneous use of arbitrary feature descriptors and classification procedures decisions by the classifiers can be represented as rankings of classifiers and different instances of a problem the rankings can be combined by methods that either reduce or rerank a given set of classes an intersection method and union method are proposed for class set reduction three methods based on the highest rank the borda count and logistic regression are proposed for class set reranking these methods have been tested in applications of degraded machine printed characters and works from large lexicons resulting in substantial improvement in overall correctness

steady state performance of spline adaptive filters recently a novel class of nonlinear adaptive filters called spline adaptive filters safs has been introduced and demonstrated to be very effective in many practical applications the learning rules of these architectures are based on the least mean square lms algorithm in order to provide theoretical foundation to the saf in this paper we provide a steady state performance evaluation in particular after the stochastic analysis of the mean behavior of the saf approach under the gaussian assumption the analytical derivation of the theoretical excess mean square error emse and the normalized misadjustment are derived and discussed the proposed analysis of emse and misadjustment is based on the energy conservation approach that has been extended to saf architecture the derived theoretical analysis allows to accurately predict the steady state performance therefore some properties for the correct choice of filter parameters are also provided experimental results demonstrate the effectiveness of the analysis results

retrospective prospective symmetry in the likelihood and bayesian analysis of case control studies summary prentice pyke 1979 established that the maximum likelihood estimate of an odds ratio in a case control study is the same as would be found by fitting a logistic regression in other words for this specific target the incorrect prospective model is inferentially equivalent to the correct retrospective model similar results have been obtained for other models and conditions have also been identified under which the corresponding bayesian property holds namely that the posterior distribution of the odds ratio is the same whether it is computed using the prospective or the retrospective likelihood in this article we demonstrate how these results follow directly from certain parameter independence properties of the models and priors and identify prior laws that support such reverse analysis for both standard and stratified designs somekeywords case controlstudy conditionalindependence hypermarkovlaw logisticregression retrospective likelihood

generalized estimating equations to estimate the ordered stereotype logit model for panel data by modeling the effects of predictor variables as a multiplicative function of regression parameters being invariant over categories and category specific scalar effects the ordered stereotype logit model is a flexible regression model for ordinal response variables in this article we propose a generalized estimating equations gee approach to estimate the ordered stereotype logit model for panel data based on working covariance matrices which are not required to be correctly specified a simulation study compares the performance of gee estimators based on various working correlation matrices and working covariance matrices using local odds ratios estimation of the model is illustrated using a real world dataset the results from the simulation study suggest that gee estimation of this model is feasible in medium sized and large samples and that estimators based on local odds ratios as realized in this study tend to be less efficient compared with estimators based on a working correlation matrix for low true correlations the efficiency gains seem to be rather small and if the working covariance structure is too flexible the corresponding estimator may even be less efficient compared with the gee estimator assuming independence like for gee estimators more generally if the true correlations over time are high then a working covariance structure which is close to the true structure can lead to considerable efficiency gains compared with assuming independence

hybrid nonlinear moments in array processing and spectrum analysis the aim of this paper is to provide a theoretical basis for the use of hybrid nonlinear hnl moments in array processing and spectrum analysis these moments are defined as the expected value of the product of one random variable times a nonlinear function of another random variable they generalize a class of twofold higher order moments and their additional flexibility can be exploited for optimization purposes or for computational convenience a number of properties beyond the classical bussgang s 1952 and price s 1958 theorems are found for hnl moments and matrices making these statistics suitable for harmonic analysis and bearing estimation covariance based and higher order moments based methods are extended to the hnl moments domain and a new class of gaussian noise rejecting statistics is added to cumulants the properties of some classes of matrices of hnl moments of practical interest are analyzed in detail

the ergodicity of a hamiltonian system of two particles in an external field abstract for a hamiltonian system of two one dimensional balls in an external field we prove the ergodicity if the total energy of the system is fixed our proof is an extension of that applied earlier to dispersing billiard systems

ergodicity in infinite hamiltonian systems with conservative noise we study the stationary measures of an infinite hamiltonian system of interacting particles in 3 subject to a stochastic local perturbation conserving energy and momentum we prove that the translation invariant measures that are stationary for the deterministic hamiltonian dynamics reversible for the stochastic dynamics and with finite entropy density are convex combination of gibbs states this result implies hydrodynamic behavior for the systems under consideration

a goodness of fit test for the ordered stereotype model this paper presents a new goodness of fit test for an ordered stereotype model used for an ordinal response variable the proposed test is based on the well known hosmer lemeshow test and its version for the proportional odds regression model the latter test statistic is calculated from a grouping scheme assuming that the levels of the ordinal response are equally spaced which might be not true one of the main advantages of the ordered stereotype model is that it allows us to determine a new uneven spacing of the ordinal response categories dictated by the data the proposed test takes the use of this new adjusted spacing to partition data a simulation study shows good performance of the proposed test under a variety of scenarios finally the results of the application in two examples are presented copyright 2016 john wiley sons ltd

the grouped continuous model for multivariate ordered categorical variables and covariate adjustment the grouped continuous model for multivariate ordered categorical data is described this is based on partitioning an underlying multivariate normal distribution straightforward maximum likelihood estimation is really feasible only for one and two way tables we introduce an estimation system based on maximum likelihood estimation in the one and two way marginal tables of higher order tables this is computationally feasible and an example involving aspects of bird colouring is given the approach is extended to provide a regression model for multivariate ordered categorical data with an estimation scheme again based on the one and two way marginal tables the above example is developed to investigate the covariate effect of time the asymptotic efficiency of these sampling schemes is discussed it appears that they have high efficiency

proof of the boltzmann sinai ergodic hypothesis for typical hard disk systems we consider the system of n 2 hard disks of masses m 1 m n and radius r in the flat unit torus 2 we prove the ergodicity actually the b mixing property of such systems for almost every selection m 1 m n r of the outer geometric parameters

cumulant series expansion of hybrid nonlinear moments of et we will now present a simple example to illustrate the approach assume four receiver elements at 0 0 0 1 0 0 0 1 0 and 0 0 1 and a target at 2 0 0 by direct computation we find 6rl i 6r2 2 and 6ri 2 given that 6r i 6r2 6ri 2 h and that receivers are located at 0 0 0 i 0 0 0 i 0 and 0 0 i we will compute the coordinates of the target

regression and ordered categorical variables read before the royal statistical society by professor r l plackett on behalf of the late professor anderson at a meeting organized by the research section on wednesday october 5th 1983 professor j b copas in the chair summary a general approach to regression modelling for ordered categorical response variables y is given which is equally applicable to ordered and unordered y the regressor variables xt xi x may be continuous or categorical the method is based on the logistic family which contains a hierarchy of regression models ranging from ordered to unordered models ordered properties of the former the stereotype model are established the choice between models is made empirically on the basis of model fit this is particularly important for assessed ordered categorical response variables where it is not obvious a priori whether or not the ordering is relevant to the regression relationship model simplification is investigated in terms of whether or not the response categories are distinguishable with respect to x the models are fitted iteratively using the method of maximum likelihood examples are given

computing exact distributions for polytomous response data abstract this article presents an efficient method for computing exact conditional distributions of the sufficient statistics for the parameters of four polytomous response models for nominal response two baseline category logit models and for ordinal response two adjacent categories logit models are considered the method consists of recursive generation of the joint distribution of the sufficient statistics augmented by a technique to slice off portions of the evolving sample space so as to eventually yield the required conditional distribution two actual data sets are analyzed to illustrate the method

lyapunov modes in hard disk systems we consider simulations of a two dimensional gas of hard disks in a rectangular container and study the lyapunov spectrum near the vanishing lyapunov exponents to this spectrum are associated eigen directions called lyapunov modes we carefully analyze these modes and show how they are naturally associated with vector fields over the container we also show that the lyapunov exponents and the coupled dynamics of the modes where it exists follow linear laws whose coefficients only depend on the density of the gas but not on aspect ratio and very little on the boundary conditions

goodness of fit processes for logistic regression simulation results in this paper we use simulations to compare the performance of new goodness of fit tests based on weighted statistical processes to three currently available tests the hosmer lemeshow decile of risk test the pearson chi square and the unweighted sum of squares tests the simulations demonstrate that all tests have the correct size the power for all tests to detect lack of fit due to an omitted quadratic term with a sample of size 100 is close to or exceeds 50 per cent to detect moderate departures from linearity and is over 90 per cent for these same alternatives for sample size 500 all tests have low power with sample size 100 to detect lack of fit due to an omitted interaction between a dichotomous and continuous covariate while the power exceeds 80 per cent to detect extreme interaction with a sample size of 500 the power is low to detect any alternative link function with sample size 100 and for most alternative links for sample size 500 only in the case of sample size 500 and an extremely asymmetric link function is the power over 80 per cent the results from these simulations show that no single test new or current performs best in detecting lack of fit due to an omitted covariate or incorrect link function however one of the new weighted tests has power comparable to other tests in all settings simulated and had the highest power in the difficult case of an omitted interaction term we illustrate the tests within the context of a model for factors associated with abstinence from drug use in a randomized trial of residential treatment programmes we conclude the paper with a summary and specific recommendations for practice

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

a class of logistic regression models for multivariate binary time series abstract a logistic model for multivariate binary time series is proposed first we establish the equivalence between a log linear model for the marginal distribution of a multivariate binary random vector and logistic models for the conditional distributions of each component given the others the logistic formulation is used to describe a markov chain for each series which implies a markov model for the vector of time series a pseudolikelihood estimation procedure is presented the methods are illustrated with data on psychosomatic and psychological diagnoses for families in a health maintenance plan

on the existence of dynamical systems with exponentially decaying collision operators the paper is concerned with the time domain collision operator of the brussels school of non equilibrium statistical mechanics t plqexp tqlq qlp where l is the skew adjoint liouville operator of a dynamical system and p and q are complementary orthogonal projectors under the assumption that p is finite rank we prove that if is norm bounded by a decreasing exponential then l must have a certain spectral property and that conversely this spectral property guarantees the existence of a projector p for which the corresponding decays exponentially we use this characterization to show that k systems admit exponentially decaying collision operators we also show that this property is enjoyed by the collision operator of the pietenpol model for a large class of interactions this answers in the affirmative a question raised by coveney and penrose coveney p v and penrose o 1992 j phys a math gen 25 4947 66

the system of one dimensional balls in an external field ii we modify the system introduced in w1 so that we can establish the nonvanishing ofall lyapunov exponents easily

a new residual for ordinal outcomes we propose a new residual for regression models of ordinal outcomes defined as e sign y y where y is the observed outcome and y is a random variable from the fitted distribution this new residual is a single value per subject irrespective of the number of categories of the ordinal outcome contains directional information between the observed value and the fitted distribution and does not require the assignment of arbitrary numbers to categories we study its properties describe its connections with other residuals ranks and ridits and demonstrate its use in model diagnostics

ordinal association in contingency tables some interpretive aspects abstract two families of models for ordered contingency tables goodman s association models and canonical correlation models are investigated and compared with respect to the interpretation of their parameters we show that the two families of models actually refer to different kinds of ordinal association stochastic order extremity for correlation models and stochastic order entropy for association models this difference is related to the way the two models scale interaction the scale difference is proven to be of substantial consequence especially under strong association

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

existence of nonequilibrium steady state for a simple model of heat conduction this paper contains rigorous results for a simple stochastic model of heat conduction similar to the kmp knipnis marchiori presutti model but with possibly energy dependent interaction rates we prove the existence and uniqueness of nonequilibrium steady states their relation to lebesgue measure and exponential convergence to steady states from suitable initial conditions

bayesian inference for the stereotype regression model application to a case control study of prostate cancer the stereotype regression model for categorical outcomes proposed by anderson j roy statist soc b 1984 46 1 30 is nested between the baseline category logits and adjacent category logits model with proportional odds structure the stereotype model is more parsimonious than the ordinary baseline category or multinomial logistic model due to a product representation of the log odds ratios in terms of a common parameter corresponding to each predictor and category specific scores the model could be used for both ordered and unordered outcomes for ordered outcomes the stereotype model allows more flexibility than the popular proportional odds model in capturing highly subjective ordinal scaling which does not result from categorization of a single latent variable but are inherently multi dimensional in nature as pointed out by greenland statist med 1994 13 1665 1677 an additional advantage of the stereotype model is that it provides unbiased and valid inference under outcome stratified sampling as in case control studies in addition for matched case control studies the stereotype model is amenable to classical conditional likelihood principle whereas there is no reduction due to sufficiency under the proportional odds model in spite of these attractive features the model has been applied less as there are issues with maximum likelihood estimation and likelihood based testing approaches due to non linearity and lack of identifiability of the parameters we present comprehensive bayesian inference and model comparison procedure for this class of models as an alternative to the classical frequentist approach we illustrate our methodology by analyzing data from the flint men s health study a case control study of prostate cancer in african american men aged 40 79 years we use clinical staging of prostate cancer in terms of tumors nodes and metastasis as the categorical response of interest

missing exposure data in stereotype regression model application to matched case control study with disease subclassification with advances in modern medicine and clinical diagnosis case control data with characterization of finer subtypes of cases are often available in matched case control studies missingness in exposure values often leads to deletion of entire stratum and thus entails a significant loss in information when subtypes of cases are treated as categorical outcomes the data are further stratified and deletion of observations becomes even more expensive in terms of precision of the category specific odds ratio parameters especially using the multinomial logit model the stereotype regression model for categorical responses lies intermediate between the proportional odds and the multinomial or baseline category logit model the use of this class of models has been limited as the structure of the model implies certain inferential challenges with nonidentifiability and nonlinearity in the parameters we illustrate how to handle missing data in matched case control studies with finer disease subclassification within the cases under a stereotype regression model we present both monte carlo based full bayesian approach and expectation conditional maximization algorithm for the estimation of model parameters in the presence of a completely general missingness mechanism we illustrate our methods by using data from an ongoing matched case control study of colorectal cancer simulation results are presented under various missing data mechanisms and departures from modeling assumptions

ergodicity of classical billiard balls abstract sinai in 1970 proved the ergodicity of two discs on the 2 torus further essential progress was only reached in 1987 when chernov and sinai established the ergodicity of two billiard balls on the v torus then by applying a strategy suggested by kramli simanyi and myself in 1989 we obtained the ergodicity of three and four balls on the torus 3 in the latter case in 1991 and 1992 while recently simanyi proved that of n balls on the torus whenever n after a survey of this progress the study of toric billiards with cylindric scatterers is initiated and the k property of a general class of these is claimed

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

the k property of orthogonal cylindric billiards toric billiards with cylindric scatterers briefly cylindric billards generalize the class of hamiltonian systems of elastic hard balls in this paper a class of cylindric billiards is considered where the cylinders are orthogonal or more exactly the constituent space of any cylindric scatterer is spanned by some of the of course orthogonal coordinate vectors adapted to the euclidean torus it is shown that the natural necessary condition for the k property of such billiards is also sufficient

the k property of four billiard balls a further step is achieved toward establishing the celebrated boltzmann sinai ergodic hypothesis for systems of four hard balls on the torus 2 it is shown that on the submanifold of the phase specified by the trivial conservation laws the system is ak flow all parts of our previous demonstration providing the analogous result for three hard balls are simplified and strengthened the main novelties are i a refinement of the geometric algebraic methods used earlier helps us to bound the codimension of the arising implicitly given set of degeneracies even if we can not calculate their exact dimension that was possible for three billiards as a matter of fact it is this part of our arguments where further understanding and new ideas are necessary before attacking the general ergodic problem ii in the pasting part of the proof which is a sophisticated version of hopf s classical device the arguments are so general that it is hoped they work in the general case too this is achieved for four balls in particular by a version of the transversal fundamental theorem which on one hand is simpler and more suitable for applications than the previous one and on the other hand as we have discovered earlier is the main tool to prove global ergodicity of semi dispersing billiards iii the verification of the chernov sinai ansatz is essentially simplified and the new idea of the proof also promises to work in the general case

constrained latent class analysis simultaneous classification and scaling of discrete choice data a reparameterization of a latent class model is presented to simultaneously classify and scale nominal and ordered categorical choice data latent class specific probabilities are constrained to be equal to the preference probabilities from a probabilistic ideal point or vector model that yields a graphical multidimensional representation of the classification results in addition background variables can be incorporated as an aid to interpreting the latent class specific response probabilities the analyses of synthetic and real data sets illustrate the proposed method

analysis of ordinal data in a study of endometrial cancer under a matched pairs case control design while estimating odds ratios ors in the context of dose levels of conjugated oestrogen exposure and development of endometrial cancer the categories formed by the levels of the exposure are ordinal in nature in the literature the binary logistic model is used for estimating or for each category relative to the baseline category we describe the use of two ordinal logistic models the cumulative logit and continuation ratio logit models to estimate the ors for the matched pairs case control data set of the los angeles endometrial cancer study a test for equality of the cumulative ors across the exposure levels is proposed the test statistic follows asymptotically the chi square distribution

bootstrap choice of estimators in parametric and semiparametric families an extension of eic ishiguro sakamoto et kitagawa 1997 ont propose le eic comme une extension du critere d akaike aic le eic est construit en corrigeant le biais de la log vraisemblance consideree comme un estimateur de l information de kullback leibler par la technique du bootstrap nous developpons ce critere dans un cadre d estimation semi parametrique multivarie et nous montrons qu il peut etre utilise pour choisir un estimateur parmi une famille d estimateurs parametriques et semi parametriques une etude de simulation basee sur un modele de regression montre que l eic est meilleur que les criteres usuels bien que la validation croisee appliquee a la vraisemblance donne des resulats tres proches sauf dans le cas d echantillons de faible taille nous avons applique le eic afin d estimer l evolution moyenne de la charge virale d enfants infectes par le vih

normal goodness of fit tests for multinomial models with large degrees of freedom abstract goodness of fit tests for independent multinomials with parameters to be estimated are usually based on pearson s x 2 or the likelihood ratio g 2 both are included in the family of power divergence statistics sd for increasing sample sizes each sd has an asymptotic x 2 distribution provided that the number of cells remains fixed we are dealing with an increasing cells approach where the number j of independent multinomials increases while the number of classes for each multinomial and the number of parameters remain fixed extending results on x 2 and g 2 the asymptotic normality of any sd is obtained for increasing cells the corresponding normal goodness of fit tests discussed here apply for models with large degrees of freedom with no restrictions imposed on the sizes n j of each multinomial allowing large as well as small expectations sparse data within each cell the asymptotic expectation and variance of sd are easy to compute for pearson s x 2 and simplify considerably for

the k property of three billiard balls sinai s strengthened version of the ergodic hypothesis is proved for three billiard balls on the v dimensional torus on connected components of the submanifold of the phase space specified by the trivial conservation laws of the energy and of the trajectory of the center of mass the system is a k flow to cope with the difficulty that in the isomorphic one particle billiard system the scatterers are not strictly convex geometric algebraic ergodic theoretic and topological methods are elaborated

nonequilibrium steady states of some simple 1 d mechanical chains we study nonequilibrium steady states of some 1 d mechanical models with n moving particles on a line segment connected to unequal heat baths for a system in which particles move freely exchanging energy as they collide with one another we prove that the mean energy along the chain is constant and equal to frac 1 2 sqrt t l t r where t l and t r are the temperatures of the two baths we then consider systems in which particles are trapped i e each confined to its designated interval in the phase space but these intervals overlap to permit interaction of neighbors for these systems we show numerically that the system has well defined local temperatures and obeys fourier s law with energy dependent conductivity provided we vary the masses randomly to enable the repartitioning of energy dynamical systems issues that arise in this study are discussed though their resolution is beyond reach

numerical simulation of polynomial speed convergence phenomenon we provide a hybrid method that captures the polynomial speed of convergence and polynomial speed of mixing for markov processes the hybrid method that we introduce is based on the coupling technique and renewal theory we propose to replace some estimates in classical results about the ergodicity of markov processes by numerical simulations when the corresponding analytical proof is difficult after that all remaining conclusions can be derived from rigorous analysis then we apply our results to seek numerical justification for the ergodicity of two 1d microscopic heat conduction models the mixing rate of these two models are expected to be polynomial but very difficult to prove in both examples our numerical results match the expected polynomial mixing rate well

standardized estimates from categorical regression models we consider the problem of interpreting categorical regression models such as the polytomous logistic model the continuation ratio model the stereotype model and the cumulative odds model we present a method to convert categorical regression coefficients into estimates of standardized fitted probabilities probability differences and probability ratios we use a delta method approach to estimate standard errors we then present a small simulation study to compare different transforms for setting confidence limits and provide an illustration of our approach in an observational study of drug therapy of polymyositis

cumulant series expansion of hybrid nonlinear moments of complex random variables a general theorem for zero memory nonlinear transformations of complex stochastic processes is presented it is shown that under general conditions the cross covariance between a stochastic process and a distorted version of another process can be represented by a series of cumulants the coefficients of this cumulant expansion are expressed by the expected values of the partial derivatives appropriately defined of the function describing the nonlinearity the theorem includes as a particular case the invariance property bussgang s 1952 theorem of gaussian processes while holding for any joint distribution of the processes the expansion in cumulants constitutes an effective means of analysis for higher order moment based estimation procedures involving non gaussian complex processes

unimodal models to relate species to environment to assess the impact of environmental change on biological communities knowledge about species environment relationships is indispensable ecologists attempt to uncover the relationships between species and environment from data obtained from field surveys in the survey species are scored on their presence or their abundance at each of several sampling sites and environmental variables that ecologists believe to be important are measured the research that led to this thesis aimed to unravel the assumptions required for the application of statistical methods that are popular among ecologists to analyse such data from a statistical point of view species data are difficult to analyse there are many species involved 10 500 many species occur at a few sites only so the data contain numerous zeroes relations between species and environmental variables are not linear but unimodal a plant for example preferably grows under for that species optimal moisture conditions and is encountered less frequently at drier or wetter sites a mathematical model for a unimodal relationship is the gaussian response model standard statistical methods such as linear regression principal components analysis and canonical correlation analysis are often inappropriate for analysing species data because they are based on linear relationships one of the methods that ecologists use instead is correspondence analysis this thesis contributes to the understanding of the underlying response model with correspondence analysis species and sites are arranged to discover the structure in the data ordination and the arrangement is subsequently related to environmental variables it is an indirect method to detect relations between species and environment hence r h whittaker s term indirect gradient analysis correspondence analysis has been invented around 1935 but did not receive interest from ecologists before 1973 when m o hill derived the technique once more as the repeated application of weighted averaging a method that was familiar to ecologists ever since 1930 weighted averaging has the advantage of being simple to apply the method can be used for two different aims 1 to estimate the optimum of a species for an environmental variable and 2 to estimate the value of an environmental variable at a site from known optima of the species present calibration in chapter 2 estimating optima by weighted averaging is compared with the results of non linear regression on the basis of the gaussian response model under particular conditions both methods agree precisely in other cases weighted averaging gives a biased estimate of the optimum and non linear regression is the method to be preferred an additional advantage of non linear regression is that it can also be used to fit response models with more than one environmental variable in chapter 3 weighted averaging to estimate the value of an environmental variable is compared with calibration on the basis of the gaussian response model also in this context the techniques are sometimes equivalent chapter 4 deals with correspondence analysis it is shown that under particular conditions correspondence analysis approximates ordination on the basis of the gaussian response model which is computationally much more complicated to detect relations indirect methods have an important disadvantage the impact of some environmental variables on the species composition can be so large that the impact of other interesting environmental variables may fail to be detected this problem can be overcome by using non linear regression but with many species and environmental variables this is laborious in chapter 5 a simpler direct method is proposed canonical correspondence analysis in chapter 6 canonical correspondence analysis turns out to be a multivariate extension of weighted averaging the results can be displayed graphically in chapter 7 an extension with covariables is discussed which leads to partial canonical correspondence analysis chapter 7 also shows that gaussian models and hence canonical correspondence analysis are relevant to the analysis of contingency tables chapter 8 describes a study to estimate ecological amplitudes of plant species with respect to ellenberg s moisture scale from species data alone the question that is addressed as well is how consequent ellenberg s moisture indicator values are finally chapter 9 cross tabulates various gradient analysis techniques by the type of problem regression calibration ordination etc and the response model linear or unimodal furthermore improvements are proposed for detrended correspondence analysis a computer program named mom is written which can perform most of the methods discussed

two goodness of fit tests for logistic regression models with continuous covariates assessing goodness of fit in logistic regression models can be problematic in that commonly used deviance or pearson chi square statistics do not have approximate chi square distributions under the null hypothesis of no lack of fit when continuous covariates are modelled we present two easy to implement test statistics similar to the deviance and pearson chi square tests that are appropriate when continuous covariates are present the methodology uses an approach similar to that incorporated by the hosmer and lemeshow goodness of fit test in that observations are classified into distinct groups according to fitted probabilities allowing sufficient cell sizes for chi square testing the major difference is that the proposed tests perform this grouping within the cross classification of all categorical covariates in the model and in some situations allow for a more powerful assessment of where model predicted and observed counts may differ a variety of simulations are performed comparing the proposed tests to the hosmer lemeshow test

a theorem on ergodicity of two dimensional hyperbolic billiards ergodicity of two dimensional billiards which satisfy some general conditions is proved this theorem is applied to one concrete class of billiards that contains in particular billiards in the stadium

classification efficiency of multinomial logistic regression relative to ordinal logistic regression abstract classification procedures are useful for the prediction of a response or outcome as a result of knowledge of the levels of one or more independent or predictor variables the procedure is said to classify the possibly multivariate observation to a level of the response variable an example might be the prediction of whether an individual will be well suffer a nonfatal heart attack or suffer a fatal heart attack this prediction might be made on the basis of the levels of various independent variables such as weight blood pressure and serum cholesterol to name a few the three response categories of the aforementioned example are ordinal an example of three nonordered response categories might be as follows well death from heart attack and death from cancer there is some recent interest in ordinal classification procedures it is reasonable to assume that when the response variable is ordinal inclusion of ordinality in the classification model to be estimated should improve mode

a note on regression analysis of ordinal data with variability of classification summary the effect of observation variability in the classification of an ordinal response is examined within the framework of the proportional hazards regression model for ordinal data a model is proposed which incorporates this type of variability and the behaviour of the proportional hazards model when this new model is appropriate is examined four data sets are used to illustrate the practical application of the model

standardizing the power of the hosmer lemeshow goodness of fit test in large data sets the hosmer lemeshow test is a commonly used procedure for assessing goodness of fit in logistic regression it has for example been widely used for evaluation of risk scoring models as with any statistical test the power increases with sample size this can be undesirable for goodness of fit tests because in very large data sets small departures from the proposed model will be considered significant by considering the dependence of power on the number of groups used in the hosmer lemeshow test we show how the power may be standardized across different sample sizes in a wide range of models we provide and confirm mathematical derivations through simulation and analysis of data on 31 713 children from the collaborative perinatal project we make recommendations on how to choose the number of groups in the hosmer lemeshow test based on sample size and provide example applications of the recommendations

penalized maximum likelihood estimation in logistic regression and discrimination summary maximum likelihood estimation of the parameters of the binary logistic regression model for pr h i x is discussed with separate discussion of sampling from i the conditional distribution of h given x ii the joint distribution of h and x and iii the conditional distribution of x given h difficulties associated with continuous x in the latter sampling scheme are discussed to avoid these penalized maximum likelihood estimates are introduced which give estimates of the logistic parameters and a nonparametric spline estimate of the marginal distribution of x extensions to multinomial logistic regression are outlined

prediction of ordinal outcomes when the association between predictors and outcome differs between outcome levels there are a number of regression models which are widely used to predict ordinal outcomes the commonly used models assume that all predictor variables have a similar effect at all levels of the outcome variable if this is not the case for example if some variables predict susceptibility to a disease and others predict the severity of the disease then a more complex model is required one possibility is the multinomial logistic regression model which assumes that the predictor variables have different effects at all levels of the outcome variable an alternative is to use the stereotype family of regression models a one dimensional stereotype model makes the assumption that the effect of each predictor is the same at all outcome levels however it is possible to fit stereotype models with more than one dimension up to a maximum of min k 1 p where k is the number of outcome categories and p is the number of predictor variables a stereotype model of this maximum dimension is equivalent to a multinomial logistic regression model in that it will produce the same predicted values and log likelihood if there are sufficient outcome levels and or predictor variables there may be a number of stereotype models of differing dimension the method is illustrated with an example of prediction of damage to joints in rheumatoid arthritis

properties of ideal point classification models for bivariate binary data the ideal point classification ipc model was originally proposed for analysing multinomial data in the presence of predictors in this paper we studied properties of the ipc model for analysing bivariate binary data with a specific focus on three quantities 1 the marginal probabilities 2 the association structure between the two binary responses and 3 the joint probabilities we found that the ipc model with a specific class point configuration represents either the marginal probabilities or the association structure however the ipc model is not able to represent both quantities at the same time we then derived a new parametrization of the model the bivariate ipc bipc model which is able to represent both the marginal probabilities and the association structure like the standard ipc model the results of the bipc model can be displayed in a biplot from which the effects of predictors on the binary responses and on their association can be read we will illustrate our findings with a psychological example relating personality traits to depression and anxiety disorders

bayesian model comparison for the order restricted rc association model association models constitute an attractive alternative to the usual log linear models for modeling the dependence between classification variables they impose special structure on the underlying association by assigning scores on the levels of each classification variable which can be fixed or parametric under the general row column rc association model both row and column scores are unknown parameters without any restriction concerning their ordinality however when the classification variables are ordinal order restrictions on the scores arise naturally under such restrictions we adopt an alternative parameterization and draw inferences about the equality of adjacent scores using the bayesian approach to achieve that we have constructed a reversible jump markov chain monte carlo algorithm for moving across models of different dimension and estimate accurately the posterior model probabilities which can be used either for model comparison or for model averaging the proposed methodology is evaluated through a simulation study and illustrated using actual datasets

reduction from cost sensitive ordinal ranking to weighted binary classification we present a reduction framework from ordinal ranking to binary classification the framework consists of three steps extracting extended examples from the original examples learning a binary classifier on the extended examples with any binary classification algorithm and constructing a ranker from the binary classifier based on the framework we show that a weighted 0 1 loss of the binary classifier upper bounds the mislabeling cost of the ranker both error wise and regret wise our framework allows not only the design of good ordinal ranking algorithms based on well tuned binary classification approaches but also the derivation of new generalization bounds for ordinal ranking from known bounds for binary classification in addition our framework unifies many existing ordinal ranking algorithms such as perceptron ranking and support vector ordinal regression when compared empirically on benchmark data sets some of our newly designed algorithms enjoy advantages in terms of both training speed and generalization performance over existing algorithms in addition the newly designed algorithms lead to better cost sensitive ordinal ranking performance as well as improved listwise ranking performance

a goodness of fit test for the proportional odds regression model we examine goodness of fit tests for the proportional odds logistic regression model the most commonly used regression model for an ordinal response variable we derive a test statistic based on the hosmer lemeshow test for binary logistic regression using a simulation study we investigate the distribution and power properties of this test and compare these with those of three other goodness of fit tests the new test has lower power than the existing tests however it was able to detect a greater number of the different types of lack of fit considered in this study moreover the test allows for the results to be summarized in a contingency table of observed and estimated frequencies which is a useful supplementary tool to assess model fit we illustrate the ability of the tests to detect lack of fit using a study of aftercare decisions for psychiatrically hospitalized adolescents the test proposed in this paper is similar to a recently developed goodness of fit test for multinomial logistic regression a unified approach for testing goodness of fit is now available for binary multinomial and ordinal logistic regression models copyright 2012 john wiley sons ltd

methods for estimating the autocorrelation function of complex gaussian stationary processes new methods for estimating the autocorrelation function acf of a complex gaussian stationary process are presented these methods are based on a general invariance property for the autocorrelation of a common class of the above processes this property suggests estimation procedures based on magnitude hard limiting and phase quantization the procedures are an extension of the relay estimator currently employed for real processes the computational cost and the general properties of the methods are discussed in particular some estimators especially suited for very simple implementations are considered the performance of the complex hybrid sign estimator is evaluated and compared to that of the classical direct estimator the proposed methods are attractive for many applications in the field of digital signal processing

generalized dynamical thermostating technique we demonstrate that the nose method for constant temperature molecular dynamics simulation mol phys 52 255 1984 can be substantially generalized by the addition of auxiliary variables to encompass an infinite variety of hamiltonian thermostats such thermostats can be used to enhance ergodicity in systems such as the one dimensional harmonic oscillator or certain molecular systems for which the standard nose hoover methods fail to reproduce converged canonical distributions in this respect the method is similar in spirit to the method of nose hoover chains but is both more general and hamiltonian in structure which allows for the use of efficient symplectic integration schemes in particular we show that within the generalized nose formalism outlined herein any hamiltonian system can be thermostated with any other including a copy of itself this gives one an enormous flexibility in choosing the form of the thermostating bath numerical experiments are included in which a harmonic oscillator is thermostated with a collection of noninteracting harmonic oscillators as well as by a soft billiard system

potentials on the two torus for which the hamiltonian flow is ergodic we consider the motion of a test particle in a compound central potential field on a two dimensional torus we discuss three different classes of potentials attracting repelling and mixed that lead to hamiltonian systems which have positive lyapunov exponent almost everywhere and are ergodic included among the mixed potentials are smooth potentials without singularities

multiple group logistic discrimination among the many possible approaches suggested for statistical discrimination the logistic method can be classified midway between fully distributional solutions of which the assumption of multivariate normality is a classical example 16l and the distribution free techniques using for instance kernel or nearest neighbor methods l 32 therefore it is often called a partially parametric or partially distributional method l l this central position of the logistic model makes it one of the most attractive and widely used tools for solving regression and discrimination problems indeed since there are fewer distributional assumptions than for fully parametric models the logistic method is applicable to a larger family of multivariate distributions involving both discrete and continuous variables moreover in spite of its wide applicability and generality the method remains feasible and easy to use in contrast with nonparametric methods this paper is intended to review the basic ideas and principles of logistic discrimination 415 22 25 and also to bring additional results to some of the queries raised by professor anderson before his death we restrict our attention to discrimination between qualitatively distinct groups and do not envisage the case where groups are quantitatively distinct or ordered 3 12 14 we dedicate this paper to our friend and mentor the late professor j a anderson for his fundamental contribution to discriminant analysis and for his continuous support of our research efforts

the k property of 4d billiards with nonorthogonal cylindric scatterers the k property of cylindric billiards given on the 4 torus is established these billiards are neither orthogonal where general necessary and sufficient conditions were obtained by d szasz nor isomorphic to hard ball systems where the connecting path formula of n simanyi is at hand

are ordinal models useful for classification there is recent interest in classification procedures intended for use only when the response is ordinal ordinal response however is evident in the parameters estimated by either multinomial logistic or normal discriminant analyses both of which classify either ordinal or non ordinal responses further there may be harm in applying ordinal models inappropriately and ample opportunity to assume mistakenly ordinality in real data sets therefore it becomes important to ascertain whether there is benefit obtained in the appropriate application of ordinal models this paper presents the results of a simulation study designed to compare classification accuracy of various models we show that ordinal models classify less accurately than the multinomial logistic and normal discriminant procedures under a variety of circumstances until further studies become available we presently conclude that ordinal models confer no advantage when the main purpose of the analysis is classification

a new model for ordinal pain data from a pharmaceutical study for the pain data analysed previously by cox and chuang this paper proposes a new model that assumes monotone scores for ordered response categories this proposed model possesses several attractive features and allows a stochastic ordering of the drugs under comparison such a model also provides insight regarding the ordinal scale used to classify response estimation of the parameters in the model is obtained by use of bmdp3r

reversibility in infinite hamiltonian systems with conservative noise the set of stationary measures of an infinite hamiltonian system with noise is investigated the model consists of particles moving in r 3 with bounded velocities and subject to a noise that does not violate the classical laws of conservation see ovy following lo we assume that the noise has also a finite radius of interaction and prove that translation invariant stationary states of finite specific entropy are reversible with respect to the stochastic component of the evolution therefore the results of lo imply that such invariant measures are superpositions of gibbs states

calculation of polychotomous logistic regression parameters using individualized regressions summary the use of individualized logistic regression in which a series of separate simple logistic regression analyses are performed as a replacement for polychotomous logistic regression is studied the asymptotic relative efficiencies of the individual parameter estimates are observed to be generally high as are the efficiencies of predicted probability estimates and to a somewhat lesser extent joint tests of parameters from different regressions

goodness of fit tests for ordinal response regression models it is well documented that the commonly used pearson chi square and deviance statistics are not adequate for assessing goodness of fit in logistic regression models when continuous covariates are modelled in recent years several methods have been proposed which address this shortcoming in the binary logistic regression setting or assess model fit differently however these techniques have typically not been extended to the ordinal response setting and few techniques exist to assess model fit in that case we present the modified pearson chi square and deviance tests that are appropriate for assessing goodness of fit in ordinal response models when both categorical and continuous covariates are present the methods have good power to detect omitted interaction terms and reasonable power to detect failure of the proportional odds assumption or modelling the wrong functional form of a continuous covariate these tests also provide immediate information as to where a model may not fit well in addition the methods are simple to understand and implement and are non specific that is they do not require prespecification of a type of lack of fit to detect

discriminant analysis when all variables are ordered determination of the equation that relates an ordered dependent variable to ordered independent variables is sought one solution non parametric discriminant analysis npd involves obtaining the best monotonic step function by means of a computer search procedure although one can use alternative selection criteria in obtaining the equation the illustrative examples use absolute distance this paper compares the prediction procedures obtained from npd with those from linear discriminant analysis linear regression with and without transformed variables and logistic regression we show that npd is analogous to regression tree analysis with incorporation of ordered variables and monotonicity we use various prediction functions to predict the example data the data using the leave one out technique and a verification set consistently non parametric discriminant analysis performs as good as or better than the tested alternatives

a log linear model for ordinal data to characterize differential change among treatments we propose a family of log linear models for ordinal data that contain parameters reflecting change patterns to compare treatments relative to change from baseline under the most general model rates of change can depend not only upon the direction of change but also upon the level of the baseline classification we describe methods for selection of a parsimonious model and for tests of hypotheses concerning treatment differences interpretation of treatment differences in the follow up response profiles within baseline strata employs the concept of stochastic ordering data from two clinical trials illustrate the proposed procedure