the standard error of cohen s kappa this paper gives a standard error for cohen s kappa conditional on the margins of the observed r x r table an explicit formula is given for the 2 x 2 table and a procedure for the more general situation a parsimonious log linear model is suggested for the general case and an approximate confidence interval for kappa is based on that model numerical illustrations are given comparing the standard results with exact conditional results and those of the present paper the conclusions are first that the standard error of kappa under the null hypothesis that kappa is zero should not be used except when the null is plausible secondly that the usual formula for the standard error appears appropriate except in the unusual circumstance of kappa taking large negative values and thirdly in small samples the distribution of the estimate of kappa appears very non symmetric and it is preferable to base confidence intervals on transformations of kappa a pc program with source code implementing the suggested procedures is available from the author

exact approaches for testing hypotheses based on the intra class kappa coefficient testing involving the intra class kappa coefficient is commonly performed in order to assess agreement involving categorical ratings a number of procedures have been proposed which make use of the limiting null distribution as the sample size goes to infinity in order to compute the observed significance as with many tests based on asymptotic null distributions these tests are associated with problematic type i error control for selected sample sizes and points in the parameter space we propose and study a collection of exact testing approaches for both the one sample and k sample scenarios for the one sample case p values are obtained using the exact distribution of the test statistic conditional on a sufficient statistic in addition unconditional approaches are considered on the basis of maximization across the nuisance parameter space numerical evaluation reveals advantages with the exact unconditional procedures copyright 2014 john wiley sons ltd

log linear non uniform association models for agreement between two ratings on an ordinal scale in agreement studies when objects are rated independently by two raters or twice by the same rater an association between their ratings on two categories arises reflecting the distinguishability of these two categories for these raters when ratings are performed on an ordinal scale this association between ratings on two categories increases when the distance between these categories increases on the ordinal scale goodman s log linear models derived for the analysis of agreement between two raters on an ordinal scale assume that distinguishabilities between adjacent categories are either constant or a priori fixed log non linear models that allow variations of the distinguishabilities between adjacent categories along the scale may lead to difficulties in parameter estimation this paper describes a new class of log linear non uniform association models these models extend the log linear uniform association model by allowing variations of distinguishability between adjacent categories along the scale these new models are used to analyse ordinal agreement between dermatologists when assessing the severity of different cutaneous signs of ageing on women faces

composite linear models for incomplete multinomial data a composite linear model clm is a matrix model for incomplete multinomial data a clm provides a unified approach for maximum likelihood inference which is applicable to a wide variety of problems involving incomplete multinomial data by formulating a model as a clm one can simplify computation of maximum likelihood estimates and asymptotic standard errors as an example we use clm to test marginal homogeneity for ordered categories subject to both ignorable and non ignorable missing data mechanisms

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

omitted variables in multilevel models statistical methodology for handling omitted variables is presented in a multilevel modeling framework in many nonexperimental studies the analyst may not have access to all requisite variables and this omission may lead to biased estimates of model parameters by exploiting the hierarchical nature of multilevel data a battery of statistical tools are developed to test various forms of model misspecification as well as to obtain estimators that are robust to the presence of omitted variables the methodology allows for tests of omitted effects at single and multiple levels the paper also introduces intermediate level tests these are tests for omitted effects at a single level regardless of the presence of omitted effects at a higher level a simulation study shows not surprisingly that the omission of variables yields bias in both regression coefficients and variance components it also suggests that omitted effects at lower levels may cause more severe bias than at higher levels important factors resulting in bias were found to be the level of an omitted variable its effect size and sample size a real data study illustrates that an omitted variable at one level may yield biased estimators at any level and in this study one cannot obtain reliable estimates for school level variables when omitted child effects exist however robust estimators may provide unbiased estimates for effects of interest even when the efficient estimators fail and the one degree of freedom test helps one to understand where the problem is located it is argued that multilevel data typically contain rich information to deal with omitted variables offering yet another appealing reason for the use of multilevel models in the social sciences

on similarity coefficients for 2 2 tables and correction for chance this paper studies correction for chance in coefficients that are linear functions of the observed proportion of agreement the paper unifies and extends various results on correction for chance in the literature a specific class of coefficients is used to illustrate the results derived in this paper coefficients in this class e g the simple matching coefficient and the dice sorenson coefficient become equivalent after correction for chance irrespective of what expectation is used the coefficients become either cohen s kappa scott s pi mak s rho goodman and kruskal s lambda or hamann s eta depending on what expectation is considered appropriate both a multicategorical generalization and a multivariate generalization are discussed

estimating kappa from binocular data a common error in statistical analysis of ophthalmic data is the lack of accounting for the positive correlation generally present between observations made in fellow eyes the alternative of data analysis from only one eye in each patient may lead to loss of power and unrealistically large confidence intervals this paper discusses a method to estimate kappa a measure of agreement between two graders when both graders rate the same set of pairs of eyes the method assumes that the true left eye and right eye kappa values are equal and makes use of the correlated binocular data to estimate confidence intervals for the common kappa simulations show that the new estimators are better than the estimator based on only one eye new confidence intervals had the correct coverage probability but were usually only about 70 per cent as wide as single eye intervals the general methodology described here applies to analysis of grader agreement in rating other paired body structures

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

a goodness of fit approach to inference procedures for the kappa statistic confidence interval construction significance testing and sample size estimation we propose a new procedure for constructing a confidence interval about the kappa statistic in the case of two raters and a dichotomous outcome the procedure is based on a chi square goodness of fit test as applied to a model frequently used for clustered binary data the procedure provides coverage levels that are accurate in samples of smaller size than those required for other procedures the procedure also has use for significance testing and the planning of corresponding sample size requirements

a measure of 2 x 2 association with stable variance and approximately normal small sample distribution planning cost effective studies a measure of 2 x 2 association z is developed having the property that n z z has approximately a standard normal distribution even for quite small sample size examples of the use of this variance stabilizing and normalizing transformation in planning studies of 2 x 2 association are presented studies of the association between coronary artery disease and exercise stress test results are used to illustrate these methods

small sample inference for fixed effects from restricted maximum likelihood restricted maximum likelihood reml is now well established as a method for estimating the parameters of the general gaussian linear model with a structured covariance matrix in particular for mixed linear models conventionally estimates of precision and inference for fixed effects are based on their asymptotic distribution which is known to be inadequate for some small sample problems in this paper we present a scaled wald statistic together with an f approximation to its sampling distribution that is shown to perform well in a range of small sample settings the statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias this approach has the advantage that it reproduces both the statistics and f distributions in those settings where the latter is exact namely for hotelling t2 type statistics and for analysis of variance f ratios the performance of the modified statistics is assessed through simulation studies of four different reml analyses and the methods are illustrated using three examples

assessment on homogeneity tests for kappa statistics under equal prevalence across studies in reliability in this paper we assess the performance of homogeneity tests for two or more kappa statistics when prevalence rates across reliability studies are assumed to be equal the likelihood score method and the chi square goodness of fit gof test provide type 1 error rates that are satisfactorily close to the nominal level but a fleiss like test is not satisfactory for small or moderate sample sizes simulations show that the score test is more powerful than the chi square gof test and the approximate sample size required for a specific power of the former is substantially smaller than the latter in addition the score test is robust to deviations from the equal prevalence assumption while the gof test is highly sensitive and it may give a grossly misleading type 1 error rate when the assumption of equal prevalence is violated we conclude that the homogeneity score test is the preferred method

using replicate observations in observer agreement studies with binary assessments by introducing replicate observations into observer agreement studies one can obtain better measures of observer agreement than heretofore possible new methodology based on the analysis of latent variables allows a separation of within and between observer variation for binary measures of assessment among pairs of observers maximum likelihood estimation and hypothesis testing are discussed the methodology is illustrated using data on the assessment of dysplasia by pathologists

estimating equations for kappa statistics kappa statistics are often used for measuring agreement or association the original measure suggested by cohen for use with nominal data has since been generalized in a number of ways including the incorporation of weights adaptation for use with more than one rater and the application to bivariate data each modification requires a new approximation to the standard error and these have usually been derived by the delta method this paper illustrates the use of estimating equations for kappa which offer a unified approach to many previously considered situations and which is also applicable to more complex designs

maximum likelihood estimation of agreement in the constant predictive probability model and its relation to cohen s kappa the alpha agreement parameter is defined as the proportion of a population of items that are classified identically for cause by two classifiers the remaining items being classified at random the parameters of the corresponding constant predictive probability model are shown to be estimable by the method of maximum likelihood and a simulation study indicates applicability of the asymptotic results to finite samples the new estimator tends to be larger than cohen s kappa except in the case of uniform margins an application is made to the validity of cancer risk items included in a cancer registry

modelling patterns of agreement for nominal scales the measurement of agreement of repeat rating is the usual method of assessing the reliability of categorical scales measurement of agreement is also important in genetic twin studies based on categorical scales one of the most commonly used methods of analysis for both types of study is the kappa coefficient for scales with more than two categories one approach is to use a single summary kappa coefficient while this may be sufficient for many studies in some instances investigation of heterogeneity in the pattern of agreement may give additional insights as there may be greater agreement for some pairs of categories than for others in this paper kappa type coefficients are used to model heterogeneity in the pattern of agreement constraints are added to the heterogeneous model to obtain simplified models procedures for estimation confidence intervals and inference for these coefficients are described for the case of two ratings per subject for a single sample and for the comparison of two independent samples formulae for sample size and power calculation are derived using the non central chi squared distribution two simulation studies are carried out to check the empirical test size and power methods are illustrated by two examples involving nominal scales with three categories

maximum likelihood estimation of the kappa coefficient from models of matched binary responses we present an estimate of the kappa coefficient of agreement between two methods of rating based on matched pairs of binary responses and show that the estimate depends on the common intraclass correlation coefficient between the pairs via monte carlo simulation we investigate power of the test of significance on kappa and the large sample bias and variance of its maximum likelihood estimator

aggregation vs global diffusive behavior in the higher dimensional keller segel model abstract we consider the classical parabolic parabolic keller segel system u t u u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n it is proved that in space dimension n 3 for each q n 2 and p n one can find e 0 0 such that if the initial data u 0 v 0 satisfy u 0 l q e and v 0 l p e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations in particular u v approaches the steady state m m as t where m is the total mass m u 0 of the population moreover we shall show that if is a ball then for arbitrary prescribed m 0 there exist unbounded solutions emanating from initial data u 0 v 0 having total mass u 0 m

statistical implications of the choice between a dichotomous or continuous trait in studies of interobserver agreement summary a frequently occurring question in the planning of studies is whether to measure the trait of interest on a continuous or dichotomous scale an issue potentially relevant to this question centers around the resulting reliability of the measure assuming the continuous trait has a normal distribution we quantify the effect of this decision on the overall number of subjects required to test whether a population reliability coefficient equals a specified criterion value two distinct cases are explored the choice between a continuous response and a truly dichotomous response and the case in which an inherently continuous response is dichotomized in both cases and particularly in the second the loss of power associated with measuring the trait on a dichotomous scale is shown to be always severe and frequently prohibitive

interval estimation for a difference between intraclass kappa statistics summary model based inference procedures for the kappa statistic have developed rapidly over the last decade however no method has yet been developed for constructing a confidence interval about a difference between independent kappa statistics that is valid in samples of small to moderate size in this article we propose and evaluate two such methods based on an idea proposed by newcombe 1998 statistics in medicine 17 873 890 for constructing a confidence interval for a difference between independent proportions the methods are shown to provide very satisfactory results in sample sizes as small as 25 subjects per group sample size requirements that achieve a prespecified expected width for a confidence interval about a difference of kappa statistic are also presented

fixed distance latent class models for the analysis of sets of two way contingency tables this paper develops and describes the application of modified latent class models for analyzing sets of two way contingency tables the proposed fixed distance models differ from traditional latent class models in that the assumption of local stochastic independence is superseded by allowing interactions of the manifest variables within each class which can be represented by a single association parameter as an example two data sets on eye color by hair color collected in caithness n1 5 387 and aberdeen n2 22 361 and fixed distance models with up to six classes three classes per data set are considered finally leading to satisfactory fit and rather simple interpretation

kappa coefficients in medical research kappa coefficients are measures of correlation between categorical variables often used as reliability or validity coefficients we recapitulate development and definitions of the k categories by m ratings kappas k x m discuss what they are well or ill designed to do and summarize where kappas now stand with regard to their application in medical research the 2 x m m 2 intraclass kappa seems the ideal measure of binary reliability a 2 x 2 weighted kappa is an excellent choice though not a unique one as a validity measure for both the intraclass and weighted kappas we address continuing problems with kappas there are serious problems with using the k x m intraclass k 2 or the various k x m weighted kappas for k 2 or m 2 in any context either because they convey incomplete and possibly misleading information or because other approaches are preferable to their use we illustrate the use of the recommended kappas with applications in medical research

quasisymmetric models for the analysis of square contingency tables summary a parametric model is presented for the analysis of square contingency tables where there is a one to one correspondence between the categories of the row variable and the categories of the column variable the model is applicable to cross classifications where the categories are unordered partially ordered or completely ordered but the interpretation of the model and associated parameters is most straightforward when the categories are at least partially ordered connections with the mover stayer model and the usual model of quasisymmetry are described two data sets are used to illustrate the utility of the model

classical test theory as a first order item response theory application to true score prediction from a possibly nonparallel test we give an account of classical test theory ctt in terms of the more fundamental ideas of item response theory irt this approach views classical test theory as a very general version of irt and the commonly used irt models as detailed elaborations of ctt for special purposes we then use this approach to ctt to derive some general results regarding the prediction of the true score of a test from an observed score on that test as well from an observed score on a different test this leads us to a new view of linking tests that were not developed to be linked to each other in addition we propose true score prediction analogues of the dorans and holland measures of the population sensitivity of test linking functions we illustrate the accuracy of the first order theory using simulated data from the rasch model and illustrate the effect of population differences using a set of real data

an importance sampling algorithm for exact conditional tests in log linear models summary a simple but quite general simulation method for conducting exact conditional lackof fit tests in log linear models is proposed our monte carlo approximation utilises an importance sampling method motivated by the crude normal approximation to the poisson distribution examples considered include tests of quasi symmetry and related models for square tables and tests concerning higher order interactions in multi way tables the method is competitive with direct simulation from the exact conditional distribution when this is feasible and outperforms alternative monte carlo procedures when direct simulation is infeasible provided the number of degrees of freedom of the test is not too large extension of the method to tests against non saturated alternatives is straightforward and is briefly discussed and illustrated

multilevel modeling with correlated effects when there exist omitted effects measurement error and or simultaneity in multilevel models explanatory variables may be correlated with random components and standard estimation methods do not provide consistent estimates of model parameters this paper introduces estimators that are consistent under such conditions by employing generalized method of moments gmm estimation techniques in multilevel modeling the authors present a series of estimators along a robust to efficient continuum this continuum depends on the assumptions that the analyst makes regarding the extent of the correlated effects it is shown that the gmm approach provides an overarching framework that encompasses well known estimators such as fixed and random effects estimators and also provides more options these gmm estimators can be expressed as instrumental variable iv estimators which enhances their interpretability moreover by exploiting the hierarchical structure of the data the current technique does not require additional variables unlike traditional iv methods further statistical tests are developed to compare the different estimators a simulation study examines the finite sample properties of the estimators and tests and confirms the theoretical order of the estimators with respect to their robustness and efficiency it further shows that not only are regression coefficients biased but variance components may be severely underestimated in the presence of correlated effects empirical standard errors are employed as they are less sensitive to correlated effects when compared to model based standard errors an example using student achievement data shows that gmm estimators can be effectively used in a search for the most efficient among unbiased estimators

general observer agreement measures on individual subjects and groups of subjects general chance corrected measures of agreement on individual subjects for several observers using nominal or ordinal categories are developed the subject specific measures can be used to identify subjects whom the observers find difficult to rate the relationship of the subject specific measures to a general chance corrected measure of agreement for a group of subjects is demonstrated by suitable choices of disagreement functions the measure of agreement for a group of subjects is shown to include as special cases many of the kappa like statistics also it is asymptotically equivalent to various intraclass correlation coefficients the measures do not require that the observers all use the classification scale in the same way the asymptotic null and non null variances obtained by taylorseries approximations for the statistics are presented the application of the measures is illustrated by data obtained when seven pathologists classified slides on a five point ordinal scale for the diagnosis of carcinoma in situ of the uterine cervix

a general goodness of fit approach for inference procedures concerning the kappa statistic the kappa statistic is frequently used as a measure of agreement among two or more raters although considerable research on statistical inferences for this statistic has been published for the case of two raters and a binary outcome relatively little work has appeared on inference problems for the case of multiple raters and or polytomous nominal outcome categories in this paper we propose a new procedure for constructing inferences for the kappa statistic that may be applied to this general case the procedure is based on a chi square goodness of fit test as applied to the dirichlet multinomial model and is a natural extension of previously proposed procedures that apply to more restricted cases a simulation study shows that the new procedure provides confidence interval coverage levels and type i error rates close to nominal over a wide range of parameter combinations we also present a sample size formula which may be used to determine the required number of subjects and raters for a given number of outcome categories

measurement of interrater agreement with adjustment for covariates summary the kappa coefficient measures chance corrected agreement between two observers in the dichotomous classification of subjects the marginal probability of classification by each rater may depend on one or more confounding variables however failure to account for these confounders may lead to inflated estimates of agreement a multinomial model is used that assumes both raters have the same marginal probability of classification but this probability may depend on one or more covariates the model may be fit using software for conditional logistic regression additionally likelihood based confidence intervals for the parameter representing agreement may be computed a simple example is discussed to illustrate model fitting and application of the technique

higher order kappa type statistics for a dichotomous attribute in multiple ratings we introduce a sequence of parameters called canonical moments to characterize the predictive value of the pattern of consensus among kappa 1 ratings to an additional rating the usual kappa coefficient of agreement among ratings two or more is just the second canonical moment by using the higher canonical moments we can evaluate the consensus of multiple binary ratings

regression analysis for categorical variables with outcome subject to nonignorable nonresponse abstract we develop a log linear model for categorical response subject to nonignorable nonresponse the paper differs from fay 1986 in its focus on estimation and hypothesis testing in a regression setting as opposed to imputation in a multivariate setting we present several new results concerning the existence of solutions on the boundary of the parameter space and the construction of confidence intervals for estimates we illustrate the method by estimating the proportion of voters preferring truman in a 1948 preelection poll mosteller hyman mccarthy marks and truman 1949 results may depend strongly on the model assumed for nonresponse goodness of fit tests are available for comparing alternative models

assessing rater agreement using marginal association models new models that are useful in the assessment of rater agreement particularly when the rating scale is ordered or partially ordered are presented the models are parameterized to address two important aspects of rater agreement i agreement in terms of the overall frequency in which raters assign categories and ii the extent to which raters agree on the category assigned to individual subjects or items we present methodology for the simultaneous modelling of univariate marginal responses and bivariate marginal associations in the k way contingency table representing the joint distribution of k rater responses the univariate marginal responses provide information for evaluating agreement in terms of the overall frequency of responses and the bivariate marginal associations provide information on category wise agreement among pairs of raters in addition estimated scores within a generalized log non linear model for bivariate associations facilitate the assessment of category distinguishability

a latent trait finite mixture model for the analysis of rating agreement this article presents a latent distribution model for the analysis of agreement on dichotomous or ordered category ratings the model includes parameters that characterize bias category definitions and measurement error for each rater or test parameter estimates can be used to evaluate rater performance and to improve classification or measurement with use of multiple ratings a simple maximum likelihood estimation procedure is described two examples illustrate the approach although considered in the context of analyzing rater agreement the model provides a general approach for mixture analysis using two or more ordered caregory measures

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

interval estimation under two study designs for kappa with binary classifications cornfield s test based method of setting a confidence interval on a parameter associated with a two by two contingency table is adapted for use with the measure of agreement kappa one sided confidence intervals derived in this way are compared to other intervals proposed for kappa under two study designs both designs involve two ratings per subject on a dichotomous scale in one design the same two raters make all evaluations in the other possibly different pairs of raters evaluate different subjects or the same rater carries out a pair of independent assessments for each subject it is shown through simulation that lower bounds based on cornfield s test based method attain the nominal coverage probability more often than other intervals proposed in the literature

estimating kappa from binocular data and comparing marginal probabilities suppose that two graders classify all eyes in a sample of patients for the presence or absence of a specified abnormality in the statistical analysis of the data possible correlation between the observations in the right and left eyes should be taken into account recently general methods have been developed to analyse grader agreement in rating paired body structures in the present paper an alternative procedure is proposed that utilizes existing formulae for the computation of weighted kappa and its standard error in addition the mcnemar test is generalized to compare the probabilities of a positive judgement by the two graders

confidence intervals for the mean of a variable taking the values 0 1 and 2 the problem of setting a confidence interval for the mean of a variable on 0 1 2 arises in three contexts paired organ procedure rates estimation of gene frequency and three point likert scales an effective approach to its estimation may be based on a good interval for a paired difference of proportions four good methods due to newcombe and tango are shown to have much better performance than more naive methods over regions of the parameter space corresponding to the above applications

comparing two diagnostic tests against the same gold standard in the same sample methods are proposed for comparing two diagnostic tests for the same data where a threshold for positive for each test is specified one method contrasts the diagnostic tests estimated risks a second method compares the two tests kappa coefficients when thresholds for positive test results are specified a priori maximum likelihood estimators and their asymptotic variances are derived and test statistics are presented for both case control and naturalistic methods of sampling the bootstrap is proposed as a method to assess differences in risk estimators when thresholds for positive test results are chosen by scanning the data examples are given to illustrate the methods

how reliable are change corrected measures of agreement chance corrected measures of agreement are prone to exhibit paradoxical and counter intuitive results when used as measures of reliability it is demonstrated that these problems arise with cohen s kappa as well as with aickin s alpha they are the consequence of an analogy to simpson s paradox in mixed populations it is further shown that chance corrected measures of agreement may yield misleading values for binary ratings it is concluded that improvements in the design and the analysis of reliability studies are a pre requisite for valid and pertinent results

testing the equality of two dependent kappa statistics procedures are developed and compared for testing the equality of two dependent kappa statistics in the case of two raters and a dichotomous outcome variable such problems may arise when each of a sample of subjects are rated under two distinct settings and it is of interest to compare the observed levels of inter observer and intra observer agreement the procedures compared are extensions of previously developed procedures for comparing kappa statistics computed from independent samples the results of a monte carlo simulation show that adjusting for the dependency between samples tends to be worthwhile only if the between setting correlation is comparable in magnitude to the within setting correlations in this case a goodness of fit procedure that takes into account the dependency between samples is recommended

modeling and evaluating repeatability and reproducibility of ordinal classifications this paper argues that currently available methods for the assessment of the repeatability and reproducibility of ordinal classifications are not satisfactory the paper aims to study whether we can modify a class of models from item response theory well established for the study of the reliability of categorical measurements in psychometrics and education for use in business and industry and whether the resulting approaches offer a satisfactory solution the fitted models can be presented graphically but also allow the calculation of probabilities of correct ordering and consistent classification in addition the model based approach allows refined diagnostics giving the user insight into the workings of a classification procedure which is vital information for a user willing to improve a poor classification procedure the approach is illustrated from a real life example and the proposed analysis is contrasted to two popular alternative analyses based on goodman and kruskal s gamma and kendall s

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

interval estimation for cohen s kappa as a measure of agreement cohen s kappa statistic is a very well known measure of agreement between two raters with respect to a dichotomous outcome several expressions for its asymptotic variance have been derived and the normal approximation to its distribution has been used to construct confidence intervals however information on the accuracy of these normal approximation confidence intervals is not comprehensive under the common correlation model for dichotomous data we evaluate 95 per cent lower confidence bounds constructed using four asymptotic variance expressions exact computation rather than simulation is employed specific conditions under which the use of asymptotic variance formulae is reasonable are determined

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

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

homogeneity score test for the intraclass version of the kappa statistics and sample size determination in multiple or stratified studies quand le coefficient de correlation intra classe ou sa version equivalente du coefficient d agrement kappa ont ete estimes a partir de plusieurs etudes independantes ou d une etude avec stratification nous nous trouvons devant le probleme de la comparaison de statistiques kappa et de la combinaison d information sur ces statistiques en un kappa commun quand leur supposition d homogeneite est satisfaite dans cet article en utilisant la theorie de vraisemblance du score etendue aux parametres de nuisance tarone 1988 nous presentons un test d homogeneite efficace pour comparer plusieurs statistiques kappa independantes et de surcroit nous donnons une methode du score d homogeneite modifiee en employant comme alternative un estimateur non iteratif et consistant nous fournissons la taille d echantillon en employant la methode du score d homogeneite modifiee et nous la comparons a celle de la qualite d ajustement gof donner eliasziw et klar 1996 une etude de simulation pour des tailles d echantillon faibles et moderes a montre que le niveau actuel du test du score d homogeneite employant les estimateurs du maximum de vraisemblance mles des parametres est de maniere satisfaisante proche du nominal et qu il est plus faible que ceux du score d homogeneite mortifiee et des tests de la qualite d ajustement nous etudions les proprietes statistiques de plusieurs estimateurs non iteratifs d un kappa commun l estimateur donner eliasziw et klar 1996 est essentiellement efficace et peut etre employe comme une alternative au mle iteratif nous presentons une estimation par intervalle efficace d un kappa commun en employant la methode de vraisemblance du score

measures of agreement between many raters for ordinal classifications screening and diagnostic procedures often require a physician s subjective interpretation of a patient s test result using an ordered categorical scale to define the patient s disease severity because of wide variability observed between physicians ratings many large scale studies have been conducted to quantify agreement between multiple experts ordinal classifications in common diagnostic procedures such as mammography however very few statistical approaches are available to assess agreement in these large scale settings many existing summary measures of agreement rely on extensions of cohen s kappa these are prone to prevalence and marginal distribution issues become increasingly complex for more than three experts or are not easily implemented here we propose a model based approach to assess agreement in large scale studies based upon a framework of ordinal generalized linear mixed models a summary measure of agreement is proposed for multiple experts assessing the same sample of patients test results according to an ordered categorical scale this measure avoids some of the key flaws associated with cohen s kappa and its extensions simulation studies are conducted to demonstrate the validity of the approach with comparison with commonly used agreement measures the proposed methods are easily implemented using the software package r and are applied to two large scale cancer agreement studies copyright 2015 john wiley sons ltd

simultaneous estimation of intrarater and interrater agreement for multiple raters under order restrictions for a binary trait it is valuable in many studies to assess both intrarater and interrater agreement most measures of intrarater agreement do not adjust for unequal estimates of prevalence between the separate rating occasions for a given rater and measures of interrater agreement typically ignore data from the second set of assessments when raters make duplicate assessments in the event when both measures are assessed there are instances where interrater agreement is larger than at least one of the corresponding intrarater agreements this implies that a rater agrees less with him herself and more with another rater in the situation of multiple raters making duplicate assessments on all subjects the authors propose properties for an agreement measure based on the odds ratio for a dichotomous trait i estimate a single prevalence across two reading occasions for each rater ii estimate pairwise interrater agreement from all available data iii bound the pairwise interrater agreement above by the corresponding intrarater agreements estimation of odds ratios under these properties is done by maximizing the multinomial likelihood with constraints using generalized log linear models in combination with a generalization of the lemke dykstra iterative incremental algorithm an example from a mammography examination reliability study is used to demonstrate the new method

inference procedures for assessing interobserver agreement among multiple raters summary we propose a new procedure for constructing inferences about a measure of interobserver agreement in studies involving a binary outcome and multiple raters the proposed procedure based on a chi square goodness of fit test as applied to the correlated binomial model bahadur 1961 in studies in item analysis and prediction 158 176 is an extension of the goodness of fit procedure developed by donner and eliasziw 1992 statistics in medicine11 1511 1519 for the case of two raters the new procedure is shown to provide confidence interval coverage levels that are close to nominal over a wide range of parameter combinations the procedure also provides a sample size formula that may be used to determine the required number of subjects and raters for such studies

testing the homogeneity of kappa statistics procedures are compared for testing the homogeneity of k or 2 independent kappa statistics in the case of two raters and a dichotomous outcome one of the procedures is based on the estimated large sample variance derived under a model frequently adopted for inferences concerning interobserver agreement the other is based on a goodness of fit approach to this model the results of a monte carlo simulation show that the two approaches have similar properties if the number of subjects in each sample is large 100 and the prevalence of the underlying trait of interest is not extreme while the goodness of fit approach is recommended for comparisons involving smaller numbers of subjects or in which the prevalence of the underlying trait is small 0 3

dispersion weighted kappa an integrative framework for metric and nominal scale agreement coefficients the rater agreement literature is complicated by the fact that it must accommodate at east two different properties of rating data the number of raters two versus more than two and the rating scale level nominal versus metric while kappa statistics are most widely used for nominal scales intraclass correlation coefficients have been preferred for metric scales in this paper we suggest a dispersion weighted kappa framework for multiple raters that integrates some important agreement statistics by using familiar dispersion indices as weights for expressing disagreement these weights are applied to ratings identifying cells in the traditional inter judge contingency table novel agreement statistics can be obtained by applying less familiar indices of dispersion in the same way

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

some paradoxical results for the quadratically weighted kappa the quadratically weighted kappa is the most commonly used weighted kappa statistic for summarizing interrater agreement on an ordinal scale the paper presents several properties of the quadratically weighted kappa that are paradoxical for agreement tables with an odd number of categories n it is shown that if one of the raters uses the same base rates for categories 1 and n categories 2 and n 1 and so on then the value of quadratically weighted kappa does not depend on the value of the center cell of the agreement table since the center cell reflects the exact agreement of the two raters on the middle category this result questions the applicability of the quadratically weighted kappa to agreement studies if one wants to report a single index of agreement for an ordinal scale it is recommended that the linearly weighted kappa instead of the quadratically weighted kappa is used

a kraemer type rescaling that transforms the odds ratio into the weighted kappa coefficient this paper presents a simple rescaling of the odds ratio that transforms the association measure into the weighted kappa statistic for a 2 2 table

separation of systematic and random differences in ordinal rating scales we introduce a new statistical method which separates and measures different types of variability between paired ordered categorical measurements the key to the separation is a two way augmented ranking approach of observations in a contingency table it means that cases classified in a specific category by one rater will be internally ranked according to the classifications from the other this enables us to extract the component of interobserver variation which is not systematic the variance of the rank differences between judgements is a suitable measure of this interrater variability which we characterize as random the empirical measure of random interjudge disagreement which lies between 0 and 1 is called the relative rank variance and is an estimate of a parameter defined on the multinomial probability distribution in the contingency table the systematic differences are determined by the marginals and described by two empirical measures relative position and relative concentration both measures lie between 1 and 1 our method is applied to data sets from a reliability study of two clinical rating scales for assessing hydrocephalus and subarachnoid haemorrhage

maximum likelihood estimation and model selection in contingency tables with missing data abstract in many studies the values of one or more variables are missing for subsets of the original sample this article focuses on the problem of obtaining maximum likelihood estimates mle for the parameters of log linear models under this type of incomplete data the appropriate systems of equations are presented and the expectation maximization em algorithm dempster laird and rubin 1977 is suggested as one of the possible methods for solving them the algorithm has certain advantages but other alternatives may be computationally more effective tests of fit for log linear models in the presence of incomplete data are considered the data from the protective services project for older persons blenkner bloom and nielsen 1971 blenkner bloom and weber 1974 are used to illustrate the procedures discussed in the article

on the best constant in the moser onofri aubin inequality let s2 be the 2 dimensional unit sphere and let j denote the nonlinear functional on the sobolev space h1 s2 defined by j alpha u frac alpha 16 pi int s 2 nabla u 2 d mu 0 frac 1 4 pi int s 2 u d mu 0 rm ln int s 2 e u frac d mu 0 4 pi where d mu 0 sin theta d theta wedge d phi onofri had established that j is non negative on h1 s2 provided 1 in this note we show that if j is restricted to those u in h 1 s 2 that satisfies the aubin condition int s 2 e u x j d mu 0 0 quad rm for all 1 leq j leq 3 then the same inequality continues to hold i e j u 0 whenever alpha geq frac 2 3 epsilon 0 for some epsilon 0 0 the question of chang yang on whether this remains true for all alpha geq frac 1 2 remains open

estimating incidence and diagnostic error rates for bivariate progressive processes estimating the times until incidences of bivariate progressive processes that are categorical is a common problem in ophthalmology audiology pulmonary medicine and other fields of medical research we consider study designs in which diagnoses of subject s bivariate status are performed repeatedly across time and when diagnosis is subject to error in such situations error confounds the interpretation of the time until an event a composite model is proposed for parameterizing both the incidence and error distributions which allows for correlation between sites with respect to both incidence and diagnostic error an em algorithm is described for this model which allows categorical covariates for both incidence and error the methodology is applied to two examples the first represents a situation in which bivariate incidence and error can reasonably be assumed symmetric prospective data concerning the development of ocular lens opacities in a large pharmaceutical clinical trial the second example represents a situation in which bivariate incidence and error may not be symmetric clinical evaluations of sexual maturation status with respect to two different anatomical indices in the cooperative study of sickle cell disease the methodology described in this paper is used in each case to estimate incidence characterize error rates and assess bivariate correlations

2 x 2 kappa coefficients measures of agreement or association summary two general but different contexts in which kappa might be used are defined agreement and association two models one for agreement and one for utility of association are defined yielding different kappa coefficients and different sampling theory asymptotic results are derived for both models small sample evaluations are presented for the model for agreement

measuring agreement for multinomial data a kappa like statistic is proposed as a measure of agreement for a set of multinomial random variables arrayed in a two way layout this statistic is shown to arise either via the chancecorrection of a certain proportion of agreement or via an analysis of variance for a two way layout the asymptotic standard error is derived under the assumption of complete independence among the random variables

comparison of several goodness of fit tests for the kappa statistic based on exact power and coverage probability we present a general approach based on chi square goodness of fit tests for performing tests of significance about the kappa statistic we make recommendations for the particular goodness of fit test to use under different situations we also compare the test statistics in terms of the generated confidence intervals about the kappa statistic

quasi symmetric latent class models with application to rater agreement suppose we observe responses on several categorical variables having the same scale we consider latent class models for the joint classification that satisfy quasi symmetry the models apply when subject specific response distributions are such that i for a given subject responses on different variables are independent and ii odds ratios comparing marginal distributions of the variables are identical for each subject these assumptions are often reasonable in modeling multirater agreement when a sample of subjects is rated independently by different observers in this application the model parameters describe two components of agreement strength of association between classifications by pairs of observers and degree of heterogeneity among the observers marginal distributions we illustrate the models by analyzing a data set in which seven pathologists classified 118 subjects in terms of presence or absence of carcinoma yielding seven categorical classifications with the same binary scale a good fitting model has a latent classification that differentiates between subjects on whom there is agreement and subjects on whom there is disagreement

confidence interval estimation of the intraclass correlation coefficient for binary outcome data we obtain closed form asymptotic variance formulae for three point estimators of the intraclass correlation coefficient that may be applied to binary outcome data arising in clusters of variable size our results include as special cases those that have previously appeared in the literature fleiss and cuzick 1979 applied psychological measurement 3 537 542 bloch and kraemer 1989 biometrics 45 269 287 altaye donner and klar 2001 biometrics 57 584 588 simulation results indicate that confidence intervals based on the estimator proposed by fleiss and cuzick provide coverage levels close to nominal over a wide range of parameter combinations two examples are presented

using test retest reliability data to improve estimates of relative risk an application of latent class analysis the paper discusses the problem of interval estimation for the relative risk when some test retest data are available on random subsamples drawn from case control studies it is shown that although the upper portion of the interval may be very wide indeed the lower bound of the interval estimate may be improved considerably by the collection of even modest amounts of repeat measurements

using association models to analyse agreement data two examples two examples demonstrate how one can use association models to analyse agreement data the first example concerns intra rater variability in the classification of sputum cytology slides and the second deals with variability associated with the reporting of passive smoking histories the paper emphasizes models in which one estimates category scores from the data that is models that are not in the log linear family of models such models have use in assessment of category distinguishability and provide insights not easily obtained with log linear models

adams inequalities for bi laplacian and extremal functions in dimension four let be a smooth oriented bounded domain be the sobolev space and be the first eigenvalue of the bi laplacian operator 2 then for any a 0a o we have and the above supremum is infinity when a o this strengthens adams inequality in dimension 4 d adams a sharp inequality of j moser for high order derivatives ann of math 128 1988 365 398 where he proved the above inequality holds for a 0 moreover we prove that for sufficiently small a an extremal function for the above inequality exists as a special case of our results we thus show that there exists with such that this establishes the existence of an extremal function of the original adams inequality in dimension 4

log linear modelling of pairwise interobserver agreement on a categorical scale this article uses log linear models to describe pairwise agreement among several raters who classify a sample on a subjective categorical scale the models describe agreement structure simultaneously for second order marginal tables of a multidimensional cross classification of ratings practical difficulties arise in fitting the models because models refer to pairwise marginal tables of a very large and sparse table a standard analysis that treats the marginal tables as independent yields consistent estimates of model parameters but not of the covariance matrix of the estimates we estimate the covariance matrix using the jackknife we apply the models to describe agreement between evaluations made by seven pathologists of carcinoma in situ of the uterine cervix using a five level ordinal scale previous analyses showed differences among the pathologists in their pairwise levels of agreement but we observe near homogeneity in the dependence structure of their ratings

approximations for standard errors of estimators of fixed and random effects in mixed linear models abstract best linear unbiased estimators of the fixed and random effects of mixed linear models are available when the true values of the variance ratios are known if the true values are replaced by estimated values the mean squared errors of the estimators of the fixed and random effects increase in size the magnitude of this increase is investigated and a general approximation is proposed the performance of this approximation is investigated in the context of a the estimation of the effects of the balanced one way random model and b the estimation of treatment contrasts for balanced incomplete block designs

energy type estimates and global solvability in a two dimensional chemotaxis haptotaxis model with remodeling of non diffusible attractant this paper deals with the coupled chemotaxis haptotaxis model ut u u v u w u 1 u w x t 0 0 v u v x t 0 wt vw w 1 w u x t 0 which was initially proposed by chaplain and lolas 2006 10 to describe the interactions between cancer cells the matrix degrading enzyme and the host tissue in a process of cancer cell invasion of tissue extracellular matrix here r2 is a bounded domain with smooth boundary and and are positive parameters as compared to previous mathematical studies the novelty here consists of allowing for positive values of reflecting processes of self remodeling of the extracellular matrix under zero flux boundary conditions it is shown that for any sufficiently smooth initial data u0 w0 satisfying first order compatibility conditions the model admits a unique global smooth solution a crucial ingredient in the proof is an energy like inequality which given t 0 yields boundedness of u t in llogl this serves as a starting point for a bootstrap argument used to derive higher regularity estimates sufficient for global extensibility of solutions

boundedness in a keller segel system with external signal production abstract we study the neumann initial boundary problem for the chemotaxis system u t u u v x t 0 v t v v u f x t x t 0 u v 0 x t 0 u x 0 u 0 x v x 0 v 0 x x in a smooth bounded domain r n with n 2 and f l 0 l n 2 0 c 0 with some 0 and 0 0 1 first we prove local existence of classical solutions for reasonably regular initial values afterwards we show that in the case of n 2 and f being constant in time requiring the nonnegative initial data u 0 to fulfill the property u 0 d x 4 ensures that the solution is global and remains bounded uniformly in time thereby we extend the well known critical mass result by nagai senba and yoshida for the classical keller segel model coinciding with f 0 in the system above to the case f 0 under certain smallness conditions imposed on the initial data and f we furthermore show that for more general space dimension n 2 and f not necessarily constant in time the solutions are also global and remain bounded uniformly in time accordingly we extend a known result given by winkler for the classical keller segel system to the present situation

multilevel model prediction multilevel models are proven tools in social research for modeling complex hierarchical systems in multilevel modeling statistical inference is based largely on quantification of random variables this paper distinguishes among three types of random variables in multilevel modeling model disturbances random coefficients and future response outcomes and provides a unified procedure for predicting them these predictors are best linear unbiased and are commonly known via the acronym blup they are optimal in the sense of minimizing mean square error and are bayesian under a diffuse prior