analysis of rank measures of association for ordinal data from longitudinal studies abstract consideration is given to longitudinal data settings in which an ordinal response variable is observed at two or more visits for each subject in two or more ordered groups with at least moderately large sample sizes e g overall n 40 rank measures of association between group and response are constructed at each time and the covariance matrix of these measures is also estimated variation among the rank measures of association is then analyzed through weighted least squares methods with weights based on the estimated covariance matrix such methods permit evaluation of group visit interaction as well as other effects of interest extensions of this type of analysis to situations with stratification or concomitant variables or missing data are also considered examples are provided for illustrative purposes

analysing repeated measurements with possibly missing observations by modelling marginal distributions suppose that subjects are observed repeatedly over a common set of time points with possibly time dependent covariates and possibly missing observations at each time point we model the marginal distribution of the response variable and the effect of the covariates on that distribution using a class of quasi likelihood models studied in mccullagh and nelder no parametric model of dependence of the repeated observations of the subject is assumed for large samples the quasi likelihood estimates of the time specific regression coefficients over the set of predetermined time points are shown to be approximately jointly normal this coupled with various inference procedures provides a global picture about the effects of the covariates on the response variable over the entire study period a lack of fit test for testing the adequacy of the assumed quasi likelihood model is also provided all the methods considered here are illustrated with real life examples

measuring renyi entanglement entropy in quantum monte carlo simulations we develop a quantum monte carlo procedure in the valence bond basis to measure the renyi entanglement entropy of a many body ground state as the expectation value of a unitary swap operator acting on two copies of the system an improved estimator involving the ratio of swap operators for different subregions enables convergence of the entropy in a simulation time polynomial in the system size we demonstrate convergence of the renyi entropy to exact results for a heisenberg chain finally we calculate the scaling of the renyi entropy in the two dimensional heisenberg model and confirm that the neel ground state obeys the expected area law for systems up to linear size l 32

a survey of models for repeated ordered categorical response data we survey models for analysing repeated observations on an ordered categorical response variable the models presented are univariate models that permit correlation among repeated measurements the models describe simultaneously the dependence of marginal response distributions on values of explanatory variables and on the occasion of response we present models for three transformations of the response distribution cumulative logits adjacent category logits and the mean for scores assigned to response categories we discuss three methods for fitting the models maximum likelihood weighted least squares and semi parametric weighted least squares is easily implemented with sas as illustrated with a study designed to compare a drug with a placebo for the treatment of insomnia

modelling transitional and joint marginal distributions in repeated categorical data when a repeated measures endpoint classifies people into several categories marginal and transitional models provide two distinct approaches for data analysis marginal models estimate the probabilities of being in different categories over time transitional models estimate the probability of changing between any two given states during follow up visits this paper develops transitional and marginal models and applies them to a clinical trial of treatments of opiate addiction the primary outcome was the presence or absence of opiates in a thrice weekly urine test administered for 17 weeks subjects frequently miss visits however and in effect respond in one of three ways to a visit missing opiates present or opiates absent thus we have three possible states our transitional model conditions on the current state and models the transition from state k to one of the other 0 k 1 states using a mutinomial logit model this model generalizes previous work of muenz and rubinstein significant covariates in this model are predictive of state changes our marginal model views the state at each time point rather than the transitions as the primary response here we model the probability of being in state k with a multinomial logit model correlation within individuals over visits can be handled by applying the approach of zeger and liang or the bootstrap significant covariates in this model can include more global summaries of a person such as extent of previous opiate use both marginal and transitional models are needed to provide a complete description of an individual s behaviour over time since global summaries might not affect transitions of particular substantive interest is how the opiate treatments affect both the marginal and transition probabilities

on some entropy functionals derived from renyi information divergence we consider the maximum entropy problems associated with renyi q entropy subject to two kinds of constraints on expected values the constraints considered are a constraint on the standard expectation and a constraint on the generalized expectation as encountered in nonextensive statistics the optimum maximum entropy probability distributions which can exhibit a power law behaviour are derived and characterized the renyi entropy of the optimum distributions can be viewed as a function of the constraint this defines two families of entropy functionals in the space of possible expected values general properties of these functionals including nonnegativity minimum convexity are documented their relationships as well as numerical aspects are also discussed finally we work out some specific cases for the reference measure q x and recover in a limit case some well known entropies

modelling ordered categorical data recent advances and future challenges this article summarizes recent advances in the modelling of ordered categorical ordinal response variables we begin by reviewing some models for ordinal data introduced in the literature in the past 25 years we then survey recent extensions of these models and related methodology for special types of applications such as for repeated measurement and other forms of clustering we also survey other aspects of ordinal modelling such as small sample analyses power and sample size considerations and availability of software throughout we suggest problem areas for future research and we highlight challenges for statisticians who deal with ordinal data

a review of some statistical methods for covariance analysis of categorical data three general methods for covariance analysis of categorical data are reviewed and applied to an example from a clinical trial in rheumatoid arthritis the three methods considered are randomization model nonparametric procedures maximum likelihood logistic regression and weighted least squares analysis of correlated marginal functions a fourth heuristic approach the unweighted linear model analysis is an approximate procedure but it is easy to implement the assumptions and statistical issues for each method are discussed so as to emphasize philosophical differences between their rationales attention is given to computational differences but it is shown that the methods lead to similar results for analogous problems it is argued that the essential differences between the methods lie in their underlying assumptions and the generality of the conclusions which may be drawn

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

a comparison of likelihood based and marginal estimating equation methods for analysing repeated ordered categorical responses with missing data application to an intervention trial of vitamin prophylaxis for oesophageal dysplasia the purpose of this research was to develop appropriate methods for analysing repeated ordinal categorical data that arose in an intervention trial to prevent oesophageal cancer the measured response was the degree of oesophageal dysplasia at 2 5 and 6 years after randomization an important feature was that some response measurements were missing and the missingness was not completely at random mcar we show that standard likelihood based methods and standard methods based on marginal estimating equations yield biased results and we propose adaptations to both these approaches that yield valid inference under the weaker missing at random mar assumption on the basis of efficiency calculations simulation studies of finite sample properties ease of computation and flexibility for testing and exploring a range of treatment models we recommend the adapted likelihood based approach for problems of this type in which there are abundant data for estimating parameters

some general methods for the analysis of categorical data in longitudinal studies this paper is concerned with the analysis of multivariate categorical data from epidemiologic and clinical studies with longitudinal designs an expository discussion of pertinent hypotheses for such situations is provided within the context of two relevant data sets appropriate large sample tests of these hypotheses are developed through the application of weighted least squares to generate wald statistics these procedures are illustrated with extensive analyses of one of these data sets in some situations the resulting cross classification of the response variables leads to extremely sparse frequency data especially when the number of subjects is not large for such repeated measurement designs in which a single variable is measured repeatedly over time this paper considers the use of a generalized mantel haenszel strategy for tests of marginal homogeneity symmetry these randomization model methods are illustrated for data in which the repeated measurement variable is reported on an ordinal scale this paper also focuses on the available computing software to implement these methods within the version 5 release of the sas system the randomization model approach can be implemented within the freq procedure and a broad range of models and hypotheses can be investigated within the catmod procedure

relative entropies in conformal field theory relative entropy is a measure of distinguishability for quantum states and it plays a central role in quantum information theory the family of renyi entropies generalizes to renyi relative entropies that include as special cases most entropy measures used in quantum information theory we construct a euclidean path integral approach to renyi relative entropies in conformal field theory then compute the fidelity and the relative entropy of states in one spatial dimension at zero and finite temperature using a replica trick in contrast to the entanglement entropy the relative entropy is free of ultraviolet divergences and is obtained as a limit of certain correlation functions the relative entropy of two states provides an upper bound on their trace distance

some properties of renyi entropy and renyi entropy rate in this paper we define the conditional renyi entropy and show that the so called chain rule holds for the renyi entropy then we introduce a relation for the rate of renyi entropy and use it to derive the rate of the renyi entropy for an irreducible aperiodic markov chain we also show that the bound for the renyi entropy rate is simply the shannon entropy rate

statistical mechanics based on renyi entropy abstract in this work we show that it is possible to obtain a generalized statistical mechanics thermostatistics based on renyi entropy to be maximized with adequate constraints the equilibrium probability distribution thus obtained has a very interesting property indeed it reminds us the statistical distribution proposed by tsallis known to conveniently describe a variety of phenomena in nonextensive systems moreover some examples are worked out in order to illustrate the main features of the herein introduced formalism

troublesome aspects of the renyi maxent treatment we study in great detail the possible existence of a renyi associated thermodynamics with negative results in particular we uncover a hidden relation in renyi s variational problem maxent this relation connects the two associated lagrange multipliers canonical ensemble with the mean energy u and the renyi parameter as a consequence of such relation we obtain anomalous renyi maxent thermodynamic results

simultaneously modeling joint and marginal distributions of multivariate categorical responses abstract we discuss model fitting methods for analyzing simultaneously the joint and marginal distributions of multivariate categorical responses the models are members of a broad class of generalized logit and loglinear models we fit them by improving a maximum likelihood algorithm that uses lagrange s method of undetermined multipliers and a newton raphson iterative scheme we also discuss goodness of fit tests and adjusted residuals and give asymptotic distributions of model parameter estimators for this class of models inferences are equivalent for poisson and multinomial sampling assumptions simultaneous models for joint and marginal distributions may be useful in a variety of applications including studies dealing with longitudinal data multiple indicators in opinion research cross over designs social mobility and inter rater agreement the models are illustrated for one such application using data from a recent general social survey regarding opinions about various types of government s

a test of the missing data mechanism for repeated categorical data due to the occurrence of missing observations longitudinal data are rarely balanced and complete weighted least squares analyses described by grizzle starmer and koch 1969 biometrics 25 489 504 have been developed for the analysis of incomplete longitudinal categorical data stanish gillings and koch 1978 biometrics 34 305 317 woolson and clarke 1984 journal of the royal statistical society series a 147 87 99 however all these analyses have assumed that missing observations are missing completely at random in the sense of rubin 1976 biometrika 63 581 592 when the occurrence of missing observations is related to the unobserved response values these analyses may result in biased results in this paper we develop a simple and practical test of the missing mechanism in incomplete repeated categorical data the proposed test is an extension of the test of little 1988 journal of the american statistical association 83 1198 1202 and uses a test criterion given in general form by wald the test is illustrated using data from a longitudinal investigation of obesity in school age children

fluctuations in the motions of mass and of patterns in one dimensional driven diffusive systems the stochastic spreading of mass fluctuations in systems described by a fluctuating burgers equation increases ast2 3 with time as a consequence the stochastic motion of a mass front a point through which no excess mass current is flowing is shown to increase ast1 3 the same is true for the stochastic displacement of mass points and shock fronts with respect to their average drift provided the initial configuration is fixed an additional average over the stationary distribution of the initial configuration yields stochastic displacements increasing with time ast1 2

semi parametric and non parametric methods for the analysis of repeated measurements with applications to clinical trials techniques applicable for the analysis of longitudinal data when the response variable is non normal are not nearly as comprehensive as for normally distributed outcomes however there have been several recent developments semi parametric and non parametric methodology for the analysis of repeated measurements is reviewed the commonly encountered design in which for each subject one assesses a univariate response variable at multiple fixed time points is considered the types of outcomes considered include binary ordered categorical and continuous but extremely non normal response variables all of the methods considered allow for incomplete data due to the occurrence of missing observations in addition discrete and or continuous covariates which may be time dependent are accommodated by some of the approaches the methods are demonstrated using data from three clinical trials

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