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

performance of generalized estimating equations in practical situations moment methods for analyzing repeated binary responses have been proposed by liang and zeger 1986 biometrika 73 13 22 and extended by prentice 1988 biometrics 44 1033 1048 in their generalized estimating equations gee both liang and zeger 1986 and prentice 1988 estimate the parameters associated with the expected value of an individual s vector of binary responses as well as the correlations between pairs of binary responses in this paper we discuss one step estimators i e estimators obtained from one step of the generalized estimating equations and compare their performance to that of the fully iterated estimators in small samples in simulations we find the performance of the one step estimator to be qualitatively similar to that of the fully iterated estimator when the sample size is small and the association between binary responses is high we recommend using the one step estimator to circumvent convergence problems associated with the fully iterated gee algorithm furthermore we find the gee methods to be more efficient than ordinary logistic regression with variance correction for estimating the effect of a time varying covariate

comparing the areas under two or more correlated receiver operating characteristic curves a nonparametric approach methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed when a test is based on an observed variable that lies on a continuous or graded scale an assessment of the overall value of the test can be made through the use of a receiver operating characteristic roc curve the curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates when two or more empirical curves are constructed based on tests performed on the same individuals statistical analysis on differences between curves must take into account the correlated nature of the data this paper presents a nonparametric approach to the analysis of areas under correlated roc curves by using the theory on generalized u statistics to generate an estimated covariance matrix

nonparametric statistical inference method for partial areas under receiver operating characteristic curves with application to genomic studies recently roc50 index the area under the lower portion of the receiver operating characteristic roc curve up to the first 50 false positives has been increasingly widely used in genomic research unfortunately statistical inferences on the roc50 index are not commonly drawn due to a lack of handy statistical inference methods and or software tools in this paper we reviewed developments in statistical methods for the partial areas under roc curves and using nonparametric methods we derived a simple and direct variance calculation formula for the partial areas different from existing methods in the literature we have also verified our method through simulation studies and compared our method with existing bi normal approaches we have shown that the partial area has an asymptotic normal distribution using trimmed u statistics theory on the basis of this asymptotic normality we have given formulas for the confidence interval and the test statistic and we reported on their application to a genomic study of sample size approximately 10 000 copyright 2008 john wiley sons ltd

model checking for roc regression analysis the receiver operating characteristic roc curve is a prominent tool for characterizing the accuracy of a continuous diagnostic test to account for factors that might influence the test accuracy various roc regression methods have been proposed however as in any regression analysis when the assumed models do not fit the data well these methods may render invalid and misleading results to date practical model checking techniques suitable for validating existing roc regression models are not yet available in this article we develop cumulative residual based procedures to graphically and numerically assess the goodness of fit for some commonly used roc regression models and show how specific components of these models can be examined within this framework we derive asymptotic null distributions for the residual processes and discuss resampling procedures to approximate these distributions in practice we illustrate our methods with a dataset from the cystic fibrosis registry

partial auc estimation and regression accurate diagnosis of disease is a critical part of health care new diagnostic and screening tests must be evaluated based on their abilities to discriminate diseased from nondiseased states the partial area under the receiver operating characteristic roc curve is a measure of diagnostic test accuracy we present an interpretation of the partial area under the curve auc which gives rise to a nonparametric estimator this estimator is more robust than existing estimators which make parametric assumptions we show that the robustness is gained with only a moderate loss in efficiency we describe a regression modeling framework for making inference about covariate effects on the partial auc such models can refine knowledge about test accuracy model parameters can be estimated using binary regression methods we use the regression framework to compare two prostate specific antigen biomarkers and to evaluate the dependence of biomarker accuracy on the time prior to clinical diagnosis of prostate cancer

fuzzy localized subrings we characterize local rings in terms of certain fuzzy ideals we also characterize rings of fractions at a prime ideal in terms of fuzzy ideals we apply our results to fuzzy intersection equations in particular we show that the fuzzy ideal which represents a system of fuzzy intersection equations in a polynomial ring is such that its extension in a ring of fractions represents the same system of fuzzy intersection equations

higher order correlations of tchebyscheff maps we determine the higher order correlation functions of iterates of tchebyscheff polynomials of arbitrary order n the non vanishing correlations can be described by suitable sets of graphs these graphs allow for a perturbative evaluation of invariant densities for langevin like systems that are driven by deterministic chaotic forces generated by tchebyscheff maps rather than gaussian white noise

optimal linear combination of biomarkers for multi category diagnosis the receiver operating characteristic roc curve and the area under the roc curve have been popularly employed in evaluating the diagnosis accuracy for diseases with binary outcome categories and have been naturally used as the utility measures for finding the optimal linear combination of multiple biomarkers in the hope to improve the diagnostic accuracy based on each single biomarker for diseases with more than two outcome categories the roc manifold and the hypervolume under the roc manifold hum have been analogously proposed as diagnostic accuracy measures however finding optimal combinations of biomarkers based on the hum criterion is less easily feasible in computation especially when the number of disease categories is more than three and the number of biomarkers is large in this study we propose two new indices for evaluating the diagnostic accuracy for multi category diagnosis which are related to the lower and upper bounds of hum and involve only diagnostic accuracies for comparing adjacent pairs of outcome categories we then propose finding the optimal linear combinations of biomarkers for multi category diagnosis using the new indices as the criterion functions simulations and real data examples show that the optimal linear combinations identified by the new proposal perform quite well in diagnostic accuracy and can be much more efficient in computation than the hum based method

a caveat concerning independence estimating equations with multivariate binary data clustered binary data occur commonly in both the biomedical and health sciences in this paper we consider logistic regression models for multivariate binary responses where the association between the responses is largely regarded as a nuisance characteristic of the data in particular we consider the estimator based on independence estimating equations iee which assumes that the responses are independent this estimator has been shown to be nearly efficient when compared with maximum likelihood ml and generalized estimating equations gee in a variety of settings the purpose of this paper is to highlight a circumstance where assuming independence can lead to quite substantial losses of efficiency in particular when the covariate design includes within cluster covariates assuming independence can lead to a considerable loss of efficiency in estimating the regression parameters associated with those covariates

assessing accuracy of mammography in the presence of verification bias and intrareader correlation la performance des tests diagnostiques medicaux est souvent evaluee en comparant le resultat du test au vrai etat de la maladie du patient l analyse des courbes de caracteristiques de fonctionnement roc peut etre utilisee pour resumer la precision du test cependant une telle analyse peut rencontrer plusieurs difficultes dans la pratique reelle une difficulte est la verification du biais c est a dire que l evaluation de l etat de la maladie peut etre partiellement disponible et la probabilite de determination de la maladie peut dependre aussi bien du resultat du test que des caracteristiques du sujet une deuxieme difficulte est que les tests interpretes par le meme lecteur peuvent ne pas etre independants a l aide des equations d estimation nous generalisons des methodes proposees pour prendre en compte ces problemes nous comparons la performance de differents estimateurs de la precision en utilisant les estimateurs robustes de la variance sandwich pour permettre une inference asymptotique valide nous montrons que dans le contexte d une cohorte d observation ou de bonnes informations sur les covariables sont disponibles une approche par des equations d estimations ponderees wee peut etre preferable pour sa robustesse a un modele mal specifie nous appliquons la methodologie a la mammographie comme realisee dans la communaute des radiologues

three methods for analysing correlated roc curves a comparison in real data sets from multi reader multi case studies with a factorial design this paper compares three published methods for analysing multiple correlated roc curves a method using generalized estimating equations with marginal non proportional ordinal regression models a method using jackknifed pseudovalues of summary statistics a method using a corrected f test from analysis of variance of summary statistics use of these methods is illustrated through six real data examples from studies with the common factorial design that is multiple readers interpreting images obtained with each test modality on each study subject the issue of the difference between typical summary statistics and summary statistics from typical roc curves is explored the examples also address similarities and differences among the analytical methods in particular while point estimates of differences between test modalities are similar the standard errors of these differences do not agree for all three methods a simulation study supports the standard errors provided by the generalized estimating equations with marginal non proportional ordinal regression models copyright 2003 john wiley sons ltd

equivalence of binormal likelihood ratio and bi chi squared roc curve models a basic assumption for a meaningful diagnostic decision variable is that there is a monotone relationship between it and its likelihood ratio this relationship however generally does not hold for a decision variable that results in a binormal receiver operating characteristic roc curve as a result roc curve estimation based on the assumption of a binormal roc curve model produces improper roc curves which have hooks are not concave over the entire domain and cross the chance line although in practice this improperness is usually not noticeable sometimes it is evident and problematic to avoid this problem metz and pan proposed basing roc curve estimation on the assumption of a binormal likelihood ratio binormal lr model which states that the decision variable is an increasing transformation of the likelihood ratio function of a random variable having normal conditional diseased and nondiseased distributions however their development is not easy to follow i show that the binormal lr model is equivalent to a bi chi squared model in the sense that the families of corresponding roc curves are the same the bi chi squared formulation provides an easier to follow development of the binormal lr roc curve and its properties in terms of well known distributions copyright 2015 john wiley sons ltd

fuzzy direct sums of fuzzy rings abstract in this paper we introduce the concepts of the fuzzy weak direct sum and the fuzzy complete direct sum of fuzzy subrings of commutative rings we prove some basic properties analogous to those of ordinary rings

generalized estimating equations for correlated binary data using the odds ratio as a measure of association summary moment methods for analyzing repeated binary responses have been proposed by liang zeger 1986 and extended by prentice 1988 in their generalized estimating equations both liang zeger 1986 and prentice 1988 estimate the parameters associated with the expected value of an individual s vector of binary responses as well as the correlations between pairs of binary responses because the odds ratio has many desirable properties and some investigators may find the odds ratio is easier to interpret we discuss modelling the association between binary responses at pairs of times with the odds ratio we then modify the estimating equations of prentice to estimate the odds ratios in simulations the parameter estimates for the logistic regression model for the marginal probabilities appear slightly more efficient when using the odds ratio parameterization

the partial area under the summary roc curve the area under the curve auc is commonly used as a summary measure of the receiver operating characteristic roc curve it indicates the overall performance of a diagnostic test in terms of its accuracy at various diagnostic thresholds used to discriminate cases and non cases of disease the auc measure is also used in meta analyses where each component study provides an estimate of the test sensitivity and specificity these estimates are then combined to calculate a summary roc sroc curve which describes the relationship between test sensitivity and specificity across studies the partial auc has been proposed as an alternative measure to the full auc when using the partial auc one considers only those regions of the roc space where data have been observed or which correspond to clinically relevant values of test sensitivity or specificity in this paper we extend the idea of using the partial auc to sroc curves in meta analysis theoretical and numerical results describe the variation in the partial auc and its standard error as a function of the degree of inter study heterogeneity and of the extent of truncation applied to the roc space a scaled partial area measure is also proposed to restore the property that the summary measure should range from 0 to 1 the results suggest several disadvantages of the partial auc measures in contrast to earlier findings with the full auc the partial auc is rather sensitive to heterogeneity comparisons between tests are more difficult especially if an empirical truncation process is used finally the partial area lacks a useful symmetry property enjoyed by the full auc although the partial auc may sometimes have clinical appeal on balance the use of the full auc is preferred

partial auc maximization in a linear combination of dichotomizers classifier combination is a useful and common methodology to design an effective classification system a large number of combination rules has been proposed hitherto mostly aimed at minimizing the error rate recently some methods have been presented that are devoted to maximize the area under the roc curve auc a more suitable performance measure when dealing with two class problems with imprecise environment and or imbalanced class priors however there are several applications that do not operate in the complete range of the roc curve but only in particular regions of it in these cases it is better to analyze the performance only in a part of the curve and to use the partial auc pauc this paper presents a new method that aims at maximizing the pauc by means of linear combination of classifiers the effectiveness of the proposed method has been proved on two biometric databases

the analysis of longitudinal polytomous data generalized estimating equations and connections with weighted least squares in recent years methods have been developed for modelling repeated observations of a categorical response obtained over time on the same individual although situations in which the repeated response is binary or poisson have been studied extensively relatively little attention has been given to polytomous categorical response variable in this paper we extend the estimating equations initially developed for clustered discrete data by liang and zeger 1986 biometrika 73 13 22 and subsequently extended by prentice 1988 biometrics 44 1033 1048 to polytomous response variables under certain assumptions we illustrate that these estimating equations simplify to the weighted least squares wls equations formalized by koch et al 1977 biometrics 33 133 158 this connection provides a formal framework for obtaining iterated weighted least squares model parameter estimates cumulative logit models are developed and applied to a representative longitudinal data set simulation results comparing wls an iterative form of wls and independence estimating equations using a robust estimate of the variance are presented

fuzzy chain subrings and their fuzzy direct sum abstract we characterize chain rings in terms of certain fuzzy ideals and by using the concept of minimial generating set we also employ the concept of fuzzy direct sum to characterize artinian principal ideal rings and to find the necessary and sufficient fuzzy conditions under which an artinian ring is a direct sum of fields

a non inferiority test for diagnostic accuracy based on the paired partial areas under roc curves non inferiority is a reasonable approach to assessing the diagnostic accuracy of a new diagnostic test if it provides an easier administration or reduces the cost the area under the receiver operating characteristic roc curve is one of the common measures for the overall diagnostic accuracy however it may not differentiate the various shapes of the roc curves with different diagnostic significances the partial area under the roc curve pauroc may present an alternative that can provide additional and complimentary information for some diagnostic tests which require false positive rate that does not exceed a certain level non parametric and maximum likelihood methods can be used for the non inferiority tests based on the difference in paired paurocs however their performance has not been investigated in finite samples we propose to use the concept of generalized p value to construct a non inferiority test for diagnostic accuracy based on the difference in paired paurocs simulation results show that the proposed non inferiority test not only adequately controls the size at the nominal level but also is uniformly more powerful than the non parametric methods the proposed method is illustrated with a numerical example using published data

fuzzy maximal radical and primary ideals of a ring we introduce the concepts of fuzzy maximal ideal the fuzzy radical of a fuzzy ideal and fuzzy primary ideal of a ring we show that a fuzzy left right ideal a of a ring r is a fuzzy maximal ideal if and only a 0 1 and a x r a x a 0 is a maximal left right ideal of r we also show that a fuzzy ideal a of a commutative ring r with unity is a fuzzy primary ideal of r if and only a 0 1 a is two valued and a is a primary ideal of r

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

measures for evaluation of prognostic improvement under multivariate normality for nested and nonnested models when comparing performances of two risk prediction models several metrics exist to quantify prognostic improvement including the change in the area under the receiver operating characteristic curve the integrated discrimination improvement the net reclassification index at event rate the change in standardized net benefit the change in brier score and the change in scaled brier score we explore the behavior and interrelationships between these metrics under multivariate normality in nested and nonnested model comparisons we demonstrate that within the framework of linear discriminant analysis all six statistics are functions of squared mahalanobis distance a robust metric that properly measures discrimination by quantifying the separation between the risk scores of events and nonevents these relationships are important for overall interpretability and clinical usefulness through simulation we demonstrate that the performance of the theoretical estimators under normality is comparable or superior to empirical estimation methods typically used by investigators in particular the theoretical estimators for the net reclassification index and the change in standardized net benefit exhibit less variability in their estimates as compared to their empirically estimated counterparts finally we explore how these metrics behave with potentially nonnormal data by applying these methods in a practical example based on the sex specific cardiovascular disease risk models from the framingham heart study our findings aim to give greater insight into the behavior of these measures and the connections existing among them and to provide additional estimation methods with less variability for the net reclassification index and the change in standardized net benefit

from the perron frobenius equation to the fokker planck equation we show that for certain classes of deterministic dynamical systems the perron frobenius equation reduces to the fokker planck equation in an appropriate scaling limit by perturbative expansion in a small time scale parameter we also derive the equations that are obeyed by the first and second order correction terms to the fokker planck limit case in general these equations describe non gaussian corrections to a langevin dynamics due to an underlying deterministic chaotic dynamics for double symmetric maps the first order correction term turns out to satisfy a kind of inhomogeneous fokker planck equation with a source term for a special example we are able solve the first and second order equations explicitly

combining multiple biomarkers linearly to maximize the partial area under the roc curve it is now common in clinical practice to make clinical decisions based on combinations of multiple biomarkers in this paper we propose new approaches for combining multiple biomarkers linearly to maximize the partial area under the receiver operating characteristic curve pauc the parametric and nonparametric methods that have been developed for this purpose have limitations when the biomarker values for populations with and without a given disease follow a multivariate normal distribution it is easy to implement our proposed parametric approach which adopts an alternative analytic expression of the pauc when normality assumptions are violated a kernel based approach is presented which handles multiple biomarkers simultaneously we evaluated the proposed as well as existing methods through simulations and discovered that when the covariance matrices for the disease and nondisease samples are disproportional traditional methods such as the logistic regression are more likely to fail to maximize the pauc while the proposed methods are more robust the proposed approaches are illustrated through application to a prostate cancer data set and a rank based leave one out cross validation procedure is proposed to obtain a realistic estimate of the pauc when there is no independent validation set available

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

from dynamical systems to the langevin equation abstract it is shown that some dynamical systems which in microscopic time are fully deterministic exhibit a long time stochastic behavior governed by the langevin equation the limit in which this becomes a precise statement is obtained by an appropriate scaling of time and force the results are exemplified by a computer experiment

ordinal regression methodology for roc curves derived from correlated data we present an approach for the analysis of correlated roc data using ordinal regression models in conjunction with generalized estimating equations the approach applies to the analysis of degree of suspicion data derived from multiple interpretations of the same diagnostic study and from the examination of the same patients with multiple diagnostic modalities the regression models make it possible to incorporate patient and reader characteristics into the analysis without having to resort to stratification we illustrate the potential of the approach with analysis of data from two studies in diagnostic oncology

on the statistical analysis of roc curves we introduce a new accuracy index for receiver operating characteristic roc curves namely the partial area under the binormal roc graph over any specified region of interest we propose a simple but general procedure based on a conventional analysis of variance for comparing accuracy indices derived from two or more different modalities the proposed method is related to and compared with existing methodology and is illustrated by results from an experiment on optimization of density and contrast yielded by multiform photographic images used for scintigraphy

semiparametric receiver operating characteristic analysis to evaluate biomarkers for disease the receiver operating characteristic roc curve is a popular method for characterizing the accuracy of diagnostic tests when test results are not binary various methodologies for estimating and comparing roc curves have been developed one approach due to pepe uses a parametric regression model rocx u g h0 u 0x with the baseline function h0 u specified up to a finite dimensional parameter in this article we extend the regression models by allowing arbitrary nonparametric baseline functions we also provide asymptotic distribution theory and procedures for making statistical inference we illustrate our approach with dataset from a prostate cancer biomarker study simulation studies suggest that the extra flexibility inherent in the semiparametric method is gained with little loss in statistical efficiency

semiparametric regression for the area under the receiver operating characteristic curve medical advances continue to provide new and potentially better means for detecting disease such is true in cancer for example where biomarkers are sought for early detection and where improvements in imaging methods may pick up the initial functional and molecular changes associated with cancer development in other binary classification tasks computational algorithms such as neural networks support vector machines and evolutionary algorithms have been applied to areas as diverse as credit scoring object recognition and peptide binding prediction before a classifier becomes an accepted technology it must undergo rigorous evaluation to determine its ability to discriminate between states characterization of factors influencing classifier performance is an important step in this process analysis of covariates may reveal subpopulations in which classifier performance is greatest or identify features of the classifier that improve accuracy we develop regression methods for the nonparametric area

doubly robust estimation of the area under the receiver operating characteristic curve in the presence of verification bias the area under the receiver operating characteristic curve auc is a popular summary measure of the efficacy of a medical diagnostic test to discriminate between healthy and diseased subjects a frequently encountered problem in studies that evaluate a new diagnostic test is that not all patients undergo disease verification because the verification test is expensive invasive or both furthermore the decision to send patients to verification often depends on the new test and on other predictors of true disease status in such cases usual estimators of the auc based on verified patients only are biased in this article we develop estimators of the auc of markers measured on any scale that adjust for selection to verification these estimators adjust for measured patient covariates and diagnostic test results and also for an assumed degree of residual selection bias they can then be used in a sensitivity analysis to examine how the auc estimates change when different plausible degrees of residual asso

fuzzy prime ideals of rings abstract the notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers a procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established all fuzzy prime maximal ideals of a given ring are determined by establishing a one to one correspondence between fuzzy prime maximal ideals and the pairs p where p is a prime maximal ideal of the ring and is a prime element dual atom in the complete lattice

a regression modelling framework for receiver operating characteristic curves in medical diagnostic testing summary receiver operating characteristic curves roc s are used to evaluate diagnostic tests when test results are not binary they describe the inherent capacity of the test for distinguishing between truly diseased and nondiseased subjects although methodology for estimating and for comparing roc s is well developed to date no general framework exists for evaluating covariate effects on roc s we formulate a general regression model which allows the effects of covariates on test accuracy to be succinctly summarised such covariates might include for example characteristics of the patient or test environment test type or severity of disease the regression models are shown to arise naturally from some classic models for continuous or ordinal test data regression parameters are fitted using an estimating equation approach the method is illustrated on data from a study of multiformat photographic images used for scintigraphy

from ruelle s transfer operator to the schro dinger operator abstract we show that ruelle s transfer operator defined for an appropriate class of strongly chaotic mappings reduces to the schrodinger operator with imaginary time in an appropriate scaling limit we also derive the equations for the next order corrections to the schrodinger behavior if the scaling limit is not completely performed in the simplest case the first order corrections turns out to obey a schrodinger equation with a source term this means a possible chaotic dynamics underlying the feynman kac formula on a very small time scale e g the planc scale leads generalized quantum mechanics with an additional source term

the analysis of placement values for evaluating discriminatory measures summary the idea of using measurements such as biomarkers clinical data or molecular biology assays for classification and prediction is popular in modern medicine the scientific evaluation of such measures includes assessing the accuracy with which they predict the outcome of interest receiver operating characteristic curves are commonly used for evaluating the accuracy of diagnostic tests they can be applied more broadly indeed to any problem involving classification to two states or populations d 0 or 1 we show that the roc curve can be interpreted as a cumulative distribution function for the discriminatory measure y in the affected population d 1 after y has been standardized to the distribution in the reference population d 0 the standardized values are called placement values if the placement values have a uniform 0 1 distribution then y is not discriminatory because its distribution in the affected population is the same as that in the reference population the degree to which the distribution of the standardized measure differs from uniform 0 1 is a natural way to characterize the discriminatory capacity of y and provides a nontraditional interpretation for the roc curve statistical methods for making inference about distribution functions therefore motivate new approaches to making inference about roc curves we demonstrate this by considering the roc glm regression model and observing that it is equivalent to a regression model for the distribution of placement values the likelihood of the placement values provides a new approach to roc parameter estimation that appears to be more efficient than previously proposed methods the method is applied to evaluate a pulmonary function measure in cystic fibrosis patients as a predictor of future occurrence of severe acute pulmonary infection requiring hospitalization finally we note the relationship between regression models for the mean placement value and recently proposed models for the area under the roc curve which is the classic summary index of discrimination

smooth semiparametric receiver operating characteristic curves for continuous diagnostic tests we propose a semiparametric kernel distribution function estimator based on which a new smooth semiparametric estimator of the receiver operating characteristic roc curve is constructed we derive the asymptotic bias and variance of the newly proposed distribution function estimator and show that it is more efficient than the traditional non parametric kernel distribution estimator we also derive the asymptotic bias and variance of our new roc curve estimator and show that it is more efficient than the smooth non parametric roc curve estimator proposed by zou et al stat med 1997 16 2143 2156 and lloyd j am stat assoc 1998 93 1356 1364 for our proposed estimators we derive data based methods for bandwidth selection in addition we present some results on the analysis of two real data sets finally a simulation study is presented to show that our estimators are better than the non parametric counterparts in terms of bias standard error and mean square error

probabilistic analysis of global performances of diagnostic tests interpreting the lorenz curve based summary measures several indices based on the receiver operating characteristic curve roc curve have previously been found to possess probabilistic interpretations however these interpretations are based on some unrealistic diagnostic scenarios in this paper the author presents a new approach using the lorenz curve the author found that the summary indices of the lorenz curve that is the pietra index and the gini index can be interpreted in several ways average change in post test probability per cent maximum prognostic information and probability of correct diagnosis these interpretations have a close tie with real world medical diagnosis suggesting that these indices are proper measures of test characteristics

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

an interpretation for the roc curve and inference using glm procedures the accuracy of a medical diagnostic test is often summarized in a receiver operating characteristic roc curve this paper puts forth an interpretation for each point on the roc curve as being a conditional probability of a test result from a random diseased subject exceeding that from a random nondiseased subject this interpretation gives rise to new methods for making inference about roc curves it is shown that inference can be achieved with binary regression techniques applied to indicator variables constructed from pairs of test results one component of the pair being from a diseased subject and the other from a nondiseased subject within the generalized linear model glm binary regression framework roc curves can be estimated and we highlight a new semiparametric estimator covariate effects can also be evaluated with the glm models the methodology is applied to a pancreatic cancer dataset where we use the regression framework to compare two different serum biomarkers asymptotic distribution theory is developed to facilitate inference and to provide insight into factors influencing variability of estimated model parameters

small sample adjustments for wald type tests using sandwich estimators the sandwich estimator of variance may be used to create robust wald type tests from estimating equations that are sums of k independent or approximately independent terms for example for repeated measures data on k individuals each term relates to a different individual these tests applied to a parameter may have greater than nominal size if k is small or more generally if the parameter to be tested is essentially estimated from a small number of terms in the estimating equation we offer some practical modifications to these robust wald type tests which asymptotically approach the usual robust wald type tests we show that one of these modifications provides exact coverage for a simple case and examine by simulation the modifications applied to the generalized estimating equations of liang and zeger 1986 conditional logistic regression and the cox proportional hazard model

fuzzy ideals of artinian rings abstract the object of this paper is to study fuzzy ideals over artinian rings among others we prove that a ring with unity is left right artinian if and only if every fuzzy left right ideal of r is finite valued this gives a characterization of artinian rings in terms of fuzzy ideals mukherjee and sen 1987 proved that if every fuzzy ideal of a ring with unity is finite valued then the ring is noetherian we give an example to show that the converse of this theorem is not true we also show that certain results of pan 1987 are not true

generalized estimating equations for ordinal categorical data arbitrary patterns of missing responses and missingness in a key covariate summary we propose methods for regression analysis of repeatedly measured ordinal categorical data when there is nonmonotone missingness in these responses and when a key covariate is missing depending on observables the methods use ordinal regression models in conjunction with generalized estimating equations gees we extend the gee methodology to accommodate arbitrary patterns of missingness in the responses when this missingness is independent of the unobserved responses we further extend the methodology to provide correction for possible bias when missingness in knowledge of a key covariate may depend on observables the approach is illustrated with the analysis of data from a study in diagnostic oncology in which multiple correlated receiver operating characteristic curves are estimated and corrected for possible verification bias when the true disease status is missing depending on observables

fuzzy principal ideals and fuzzy simple field extensions we examine minimal generating systems for certain fuzzy ideals we use the concept of a minimal generating set for a fuzzy ideal to characterize principal ideal rings and to characterize their equivalent fuzzy ideals we also use the concept of a minimal generating system to give a version of luroth s theorem for fuzzy intermediate fields and to characterize simple field extensions

three class roc analysis the equal error utility assumption and the optimality of three class roc surface using the ideal observer previously we have developed a decision model for three class receiver operating characteristic roc analysis based on decision theory the proposed decision model maximizes the expected decision utility under the assumption that incorrect decisions have equal utilities under the same hypothesis equal error utility assumption this assumption reduced the dimensionality of the general three class roc analysis and provided a practical figure of merit to evaluate the three class task performance however it also limits the generality of the resulting model because the equal error utility assumption will not apply for all clinical three class decision tasks the goal of this study was to investigate the optimality of the proposed three class decision model with respect to several other decision criteria in particular besides the maximum expected utility meu criterion used in the previous study we investigated the maximum correctness mc or minimum error maximum likelihood ml and nyman pearson n p criteria we found that by making assumptions for both meu and n p criteria all decision criteria lead to the previously proposed three class decision model as a result this model maximizes the expected utility under the equal error utility assumption maximizes the probability of making correct decisions satisfies the n p criterion in the sense that it maximizes the sensitivity of one class given the sensitivities of the other two classes and the resulting roc surface contains the maximum likelihood decision operating point while the proposed three class roc analysis model is not optimal in the general sense due to the use of the equal error utility assumption the range of criteria for which it is optimal increases its applicability for evaluating and comparing a range of diagnostic systems

a proposed design and analysis for comparing digital and analog mammography because randomized trials have shown a reduction in breast cancer mortality analog mammography for the early detection of breast cancer has gained widespread use recently several manufacturers have developed digital mammography which promises great advantages in the storage and transmission of images we were asked to design a study to compare the two types of mammography in terms of their performance for the early detection of breast cancer a standard measure of mammography performance is the receiver operating characteristic roc curve which is a plot of false and true positive rates for each ordered classification of the mammography images methods for study design and data analysis based on roc curves have been well developed for diagnostic tests particularly in radiology but for comparing the performance of mammography for the early detection of breast cancer among asymptomatic women special considerations motivate new designs and methodology first digital mammography may cost substantia

linear combinations of biomarkers to improve diagnostic accuracy with three ordinal diagnostic categories many researchers have addressed the problem of finding the optimal linear combination of biomarkers to maximize the area under receiver operating characteristic roc curves for scenarios with binary disease status in practice many disease processes such as alzheimer can be naturally classified into three diagnostic categories such as normal mild cognitive impairment and alzheimer s disease ad and for such diseases the volume under the roc surface vus is the most commonly used index of diagnostic accuracy in this article we propose a few parametric and nonparametric approaches to address the problem of finding the optimal linear combination to maximize the vus we carried out simulation studies to investigate the performance of the proposed methods we apply all of the investigated approaches to a real data set from a cohort study in early stage ad copyright 2012 john wiley sons ltd

entire fredholm determinants for evaluation of semiclassical and thermodynamical spectra proofs that fredholm determinants of transfer operators for hyperbolic flows are entire can be extended to a large new class of multiplicative evolution operators we construct such operators both for the gutzwiller semiclassical quantum mechanics and for classical thermodynamic formalism and introduce a new functional determinant which is expected to be entire for axiom a flows and whose zeros coincide with the zeros of the gutzwiller voros zeta function

regression models for convex roc curves the performance of a diagnostic test is summarized by its receiver operating characteristic roc curve under quite natural assumptions about the latent variable underlying the test the roc curve is convex empirical data on a test s performance often comes in the form of observed true positive and false positive relative frequencies under varying conditions this paper describes a family of regression models for analyzing such data the underlying roc curves are specified by a quality parameter delta and a shape parameter mu and are guaranteed to be convex provided delta 1 both the position along the roc curve and the quality parameter delta are modeled linearly with covariates at the level of the individual the shape parameter mu enters the model through the link functions log p mu log 1 p mu of a binomial regression and is estimated either by search or from an appropriate constructed variate one simple application is to the meta analysis of independent studies of the same diagnostic test illustrated on some data of moses shapiro and littenberg 1993 a second application to so called vigilance data is given where roc curves differ across subjects and modeling of the position along the roc curve is of primary interest

confidence intervals of the difference between areas under two roc curves in matched pair experiments in a matched pair study when outcomes of two diagnostic tests are ordinal continuous the difference between two correlated areas under roc curves aucs is usually used to compare the overall discriminatory ability of two diagnostic tests this article considers confidence interval ci construction problems of difference between two correlated aucs in a matched pair experiment and proposes 13 hybrid cis based on variance estimates recovery with the maximum likelihood estimation delong s statistic wilson score statistic ws and ws with continuity correction the modified wald statistic mw and mw with continuity correction and agresti coull statistic and three bootstrap resampling based cis for comparison we present traditional parametric and nonparametric cis simulation studies are conducted to assess the performance of the proposed cis in terms of empirical coverage probabilities empirical interval widths and ratios of the mesial noncoverage probabilities to the noncoverage probabilities two examples from clinical studies are illustrated by the proposed methodologies empirical results evidence that the hybrid agresti coull ci with the empirical estimation eac behaved most satisfactorily because its coverage probability was quite close to the prespecified confidence level with short interval width hence we recommend the usage of the eac ci in applications

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

properties of the summary receiver operating characteristic sroc curve for diagnostic test data the summary receiver operating characteristic sroc curve has been recommended to represent the performance of a diagnostic test based on data from a meta analysis however little is known about the basic properties of the sroc curve or its estimate in this paper the position of the sroc curve is characterized in terms of the overall diagnostic odds ratio and the magnitude of inter study heterogeneity in the odds ratio the area under the curve auc and an index q are discussed as potentially useful summaries of the curve it is shown that auc is maximized when the study odds ratios are homogeneous and that it is quite robust to heterogeneity an upper bound is derived for auc based on an exact analytic expression for the homogeneous situation and a lower bound based on the limit case q defined by the point where sensitivity equals specificity q is invariant to heterogeneity the standard error of auc is derived for homogeneous studies and shown to be a reasonable approximation with heterogeneous studies the expressions for auc and its standard error are easily computed in the homogeneous case and avoid the need for numerical integration in the more general case se auc and se q are found to be numerically close with se q being larger if the odds ratio is very large the methods are illustrated using data for the pap smear screening test for cervical cancer and for three tests for the diagnosis of metastases in cervical cancer patients