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

use of the mann whitney u test for clustered data the mann whitney u test is ubiquitous in statistical practice for the comparison of measures of location for two samples where the assumption of normality is questionable frequently one has replicate data for each individual in a group and would like to compare measures of central tendency between groups without assuming normality for this purpose we present a generalization of the mann whitney u test for clustered data the test is performed by computing zc wc mu c sigma c approximately n 0 1 under h0 where wc mu c are the observed and expected mann whitney u statistic based on a comparison of all pairs of replicates in the two groups and sigma c is the standard deviation of wc that is modified to account for clustering effects within a cluster we obtain an explicit variance formula that is a function of four clustering parameters we validate the properties of the test procedure in a simulation study we illustrate the methods with an example comparing the baseline humphrey visual field between two treatment groups in a randomized clinical trial of patients with retinitis pigmentosa rp

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

on root loci in several variables continuity in the high gain limit in this paper we discuss the continuity of the closed loop poles of a linear multivariable system with respect to a multidimensional polynomial family of direct output gains k 1 r this is based on and contains an exposition of the geometric formulation for including infinite gains which was developed in the lectures 2 and extended and applied in 1 to the study of output feedback systems this has been a basic tool in recent work on the classical problem of poleplacement by output feedback and in 1 the lack of continuity of the root loci in certain situations was discussed with special emphasis on the complex case here after presenting two somewhat surprising counterexamples to this continuity we give in theorem 1 and the ensuing discussion necessary and sufficient conditions for continuity of the root loci at a real infinite gain this should have significant impact on the problem of constructing graphical tests for the stability of systems subject to 2 dimensional variations in the gain parameter

an application of maximum likelihood and generalized estimating equations to the analysis of ordinal data from a longitudinal study with cases missing at random data are analysed from a longitudinal psychiatric study in which there are no dropouts that do not occur completely at random a marginal proportional odds model is fitted that relates the response severity of side effects to various covariates two methods of estimation are used generalized estimating equations gee and maximum likelihood ml both the complete set of data and the data from only those subjects completing the study are analysed for the completers only data the gee and ml analyses produce very similar results these results differ considerably from those obtained from the analyses of the full data set there are also marked differences between the results obtained from the gee and ml analysis of the full data set the occurrence of such differences is consistent with the presence of a non completely random dropout process and it can be concluded in this example that both the analyses of the completers only and the gee analysis of the full data set produce misleading conclusions about the relationships between the response and covariates

the order restricted rc model for ordered contingency tables estimation and testing for fit the rc model has been proposed as a model for ordered contingency tables it involves parametric scores that are assigned to the rows and columns of the table so that these scores reflect the ordinality of the row and column categories efficient estimation of these parameters subject to order constraints remained an open problem mainly due to severe difficulties in computing these estimates and difficulties in deriving an appropriate asymptotic goodness of fit test a nonstandard yet very simple algorithm is derived which produces the desired order restricted maximum likelihood estimates with probability converging to 1 testing the order restricted rc model for fit is also discussed

weighted rank regression for clustered data analysis we consider ranked based regression models for clustered data analysis a weighted wilcoxon rank method is proposed to take account of within cluster correlations and varying cluster sizes the asymptotic normality of the resulting estimators is established a method to estimate covariance of the estimators is also given which can bypass estimation of the density function simulation studies are carried out to compare different estimators for a number of scenarios on the correlation structure presence absence of outliers and different correlation values the proposed methods appear to perform well in particular the one incorporating the correlation in the weighting achieves the highest efficiency and robustness against misspecification of correlation structure and outliers a real example is provided for illustration

a weighted multivariate sign test for cluster correlated data we consider the multivariate location problem with cluster correlated data a family of multivariate weighted sign tests is introduced for which observations from different clusters can receive different weights under weak assumptions the test statistic is asymptotically distributed as a chi squared random variable as the number of clusters goes to infinity the asymptotic distribution of the test statistic is also given for a local alternative model under multivariate normality optimal weights maximizing pitman asymptotic efficiency are provided these weights depend on the cluster sizes and on the intracluster correlation several approaches for estimating these weights are presented using pitman asymptotic efficiency we show that appropriate weighting can increase substantially the efficiency compared to a test that gives the same weight to each cluster a multivariate weighted t test is also introduced the finite sample performance of the weighted sign test is explored through a simulation study which shows that the proposed approach is very competitive a real data example illustrates the practical application of the methodology

the invariance principle and income wealth conservation laws application of lie groups and related transformations abstract the purpose of this paper is to uncover hidden conservation laws in the models with heterogeneous capital goods the first conservation law uncovered is a pseudo net productivity relation which implies that the rate of change in national income is equal to the discount rate multiplied by the utility value of investment secondly the modified income conservation law is confirmed when the economy possesses taste change and or technical change from an empirical perspective knowledge of conservation laws can assist in the econometric testing of optimal growth models in addition to the capital labor ratio and saving ratio the empirical validity of the models can be tested by the appropriate income wealth ratios

on the partitioning of goodness of fit statistics for multivariate categorical response models abstract numerical and asymptotic stochastic partitioning of goodness of fit statistics are considered for a broad class of simultaneous multivariate categorical response models these simultaneous models impose constraints on the joint and marginal distributions of categorical response variables under certain conditions the tenability of the corresponding simultaneous hypothesis can be assessed by separately testing the two subhypotheses one regarding the joint distributions and the other regarding the marginal distributions specifically easily verifiable sufficient conditions are introduced that allow us to partition the overall goodness of fit statistic into two interesting goodness of fit statistics one for testing whether the joint distribution model holds and the other for testing whether the marginal distribution model holds moreover it is proven that when the sufficient conditions hold and the simultaneous hypothesis is true the two component goodness of fit statistics are asymptotically

the invariance principle and income wealth conservation laws in the early part of the nineteenth century william rowan hamilton discovered a principle which can be generalized to encompass many areas of physics engineering and applied mathematics hamilton s principle roughly states that the evolution in time of a dynamic system takes place in such a manner that integral of the difference between the kinetic and potential energies for the system is stationary if the action integral is free of the time variable the sum of the kinetic and potential energies the hamiltonian is constant the conservation law of the total energy

the wilcoxon signed rank test for paired comparisons of clustered data the wilcoxon signed rank test is a frequently used nonparametric test for paired data e g consisting of pre and posttreatment measurements based on independent units of analysis this test cannot be used for paired comparisons arising from clustered data e g if paired comparisons are available for each of two eyes of an individual to incorporate clustering a generalization of the randomization test formulation for the signed rank test is proposed where the unit of randomization is at the cluster level e g person while the individual paired units of analysis are at the subunit within cluster level e g eye within person an adjusted variance estimate of the signed rank test statistic is then derived which can be used for either balanced same number of subunits per cluster or unbalanced different number of subunits per cluster data with an exchangeable correlation structure with or without tied values the resulting test statistic is shown to be asymptotically normal as the number of clusters becomes large if the cluster size is bounded simulation studies are performed based on simulating correlated ranked data from a signed log normal distribution these studies indicate appropriate type i error for data sets with or 20 clusters and a superior power profile compared with either the ordinary signed rank test based on the average cluster difference score or the multivariate signed rank test of puri and sen finally the methods are illustrated with two data sets i an ophthalmologic data set involving a comparison of electroretinogram erg data in retinitis pigmentosa rp patients before and after undergoing an experimental surgical procedure and ii a nutritional data set based on a randomized prospective study of nutritional supplements in rp patients where vitamin e intake outside of study capsules is compared before and after randomization to monitor compliance with nutritional protocols

analysis of repeated categorical data using generalized estimating equations moment methods for analysing repeated binary responses have been proposed by liang and zeger and extended by prentice and zhao and prentice in these estimating equations models are proposed for the correlation between the repeated binary responses we extend liang and zeger s method to models for the correlation between repeated nominal or ordinal categorical responses in particular when the repeated responses are binary our methods reduce to liang and zeger s method our method is illustrated with two datasets one dataset contains repeated observations of self assessment of arthritis an ordered variable with three categories collected during a randomized comparative study of alternative treatments of patients with rheumatoid arthritis the second dataset is a longitudinal study of the health effects of air pollution in which the repeated ordered multinomial response is the wheezing status no wheeze wheeze with cold wheeze apart from cold of a child at ages 9 10 11 and 12 years

power and sample size estimation for the clustered wilcoxon test the wilcoxon rank sum test is widely used for two group comparisons of nonnormal data an assumption of this test is independence of sampling units both within and between groups which will be violated in the clustered data setting such as in ophthalmological clinical trials where the unit of randomization is the subject but the unit of analysis is the individual eye for this purpose we have proposed the clustered wilcoxon test to account for clustering among multiple subunits within the same cluster rosner glynn and lee 2003 biometrics 59 1089 1098 2006 biometrics 62 1251 1259 however power estimation is needed to plan studies that use this analytic approach we have recently published methods for estimating power and sample size for the ordinary wilcoxon rank sum test rosner and glynn 2009 biometrics 65 188 197 in this article we present extensions of this approach to estimate power for the clustered wilcoxon test simulation studies show a good agreement between estimated and empirical power these methods are illustrated with examples from randomized trials in ophthalmology enhanced power is achieved with use of the subunit as the unit of analysis instead of the cluster using the ordinary wilcoxon rank sum test

analysis of sets of two way contingency tables using association models abstract a class of models is introduced for the analysis of group differences in the association between two discrete variables the rc m association model for two way tables is reviewed and alternative weighting systems for identifying interaction parameters are presented this model is generalized for the setting where a two way contingency table is available for two or more groups various restricted models can be used to examine possible sources of intergroup heterogeneity in the association these sources pertain to heterogeneity in the intrinsic association and or in the scores for the row and column variables the importance of weights used to identify the row and column scores is emphasized a classical set of data previously analyzed by many authors is used to illustrate the advantages of the models and methods developed here

robustness of the two independent samples t test when applied to ordinal scaled data one may encounter the application of the two independent samples t test to ordinal scaled data for example data that assume only the values 0 1 2 3 from small samples this situation clearly violates the underlying normality assumption for the t test and one cannot appeal to large sample theory for validity in this paper we report the results of an investigation of the t test s robustness when applied to data of this form for samples of sizes 5 to 20 our approach consists of complete enumeration of the sampling distributions and comparison of actual levels of significance with the significance level expected if the data followed a normal distribution we demonstrate under general conditions the robustness of the t test in that the maximum actual level of significance is close to the declared level

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

comparison of alternative regression models for paired binary data we used simulated data derived from real ophthalmologic examples to evaluate the performance of alternative logistic regression approaches for paired binary data approaches considered were standard logistic regression ignoring the correlation between fellow eyes treating individuals classified on the basis of their more impaired eye as the unit of analysis or considering only right eyes marginal logistic regression models fitted by the maximum likelihood approach of lipsitz laird and harrington or the estimating equation approach of liang and zeger and conditional logistic regression models fitted by the maximum likelihood approach of rosner or the estimating equation approach of connolly and liang taylor series approximations were used to compare conditional and marginal parameter estimates consideration of type i and ii error rates found application of standard logistic regression to be inferior to methods that treated the eye as the unit of analysis and accounted for the correlation between fellow eyes among these latter approaches none was uniformly superior to the others across the range of condition considered

assessing interrater agreement from dependent data estimation of interrater agreement for ordered categorical data is examined when the same sample is being assessed by various raters with different methods we investigate the use of a latent model proposed by qu piedmonte and medendorp 1995 biomerics 51 268 275 to estimate the correlation between raters for each method and test for their equality for each of the assessment methods these correlations can be interpreted as the variance components of random effects representing subject and rater this method is applied to an hiv study in which the amount of ectopy on a woman s cervix is measured by both direct visual assessment and a computer planimetry method

incorporation of clustering effects for the wilcoxon rank sum test a large sample approach the wilcoxon rank sum test is frequently used in statistical practice for the comparison of measures of location when the underlying distributions are far from normal or not known in advance an assumption of the ordinary rank sum test is that individual sampling units are independent in many ophthalmologic clinical trials the early treatment for diabetic retinopathy scale etdrs is a principal endpoint used for measuring the level of diabetic retinopathy this is an ordinal scale and it is natural to consider the wilcoxon rank sum test for the comparison of the level of diabetic retinopathy between treatment groups however under this design unlike the usual wilcoxon rank sum test the subject is the unit of randomization but the eye is the unit of analysis furthermore a person will tend to have different but correlated etdrs scores for fellow eyes thus we propose a correction to the variance of the wilcoxon rank sum statistic that accounts for clustering effects and that can be used for both balanced same number of subunits per cluster or unbalanced different number of subunits per cluster data both in the presence or absence of ties with p value adjusted accordingly in this article we present large sample theory and simulation results for this test procedure and apply it to diabetic retinopathy data from type i diabetics in the sorbinil retinopathy trial

rank sum tests for clustered data the wilcoxon rank sum test is widely used to test the equality of two populations because it makes fewer distributional assumptions than parametric procedures such as the t test however the wilcoxon rank sum test can be used only if data are independent when data are clustered tests based on generalized estimating equations gees that generalize the t test have been proposed here we develop a rank sum test that can be used when data are clustered as an application we use our rank sum test to develop a nonparametric test of association between a genetic marker and a quantitative trait locus we also give a rank sum test for equivalence of three or more populations that generalizes the kruskal wallis test to situations with clustered data unlike previous rank tests for clustered data our proposal is valid when members of the same cluster belong to different groups or when the correlation between cluster members differs across groups

improved tests for comparing treatments against a control and other one sided problems abstract three multivariate one sided testing problems are studied all problems concern a independent normal populations with unknown means and common unknown variance for the first problem the null hypothesis is that all means are zero while the alternative is that each mean is nonnegative and at least one is positive for the second problem the null hypothesis is that all means are equal and the alternative is that all of the first k 1 means are greater than or equal to the mean of the last population with strict inequality for at least one the alternative for the second problem is called the tree top alternative for the third problem the null hypothesis is that all means are equal while the alternative is that the means are ordered say nonincreasing this latter alternative is called the simple order alternative all three of these problems are discussed extensively in the book by robertson wright and dykstra we will assume that random samples of size n are drawn from each population

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

multivariate methods for clustered ordinal data with applications to survival analysis clustered data are the rule in many clinical specialties such as ophthalmology methods have been developed for the treatment of clustered continuous or binary outcome data less attention has been given to ordinal outcomes which occur frequently in ophthalmology for example grading systems of cataract and diabetic retinopathy are commonly used where a photograph is graded by comparison with a series of reference photographs of increasing severity some commonly used methods for the analysis of ordered categorical data include the proportional odds and continuation ratio models it is difficult however to incorporate clustering effects into these models instead for clusters of size two we propose a generalization of the adjacent category model given by log pr i 1 j pr i j ui j 1 x where pr i j denotes the probability that the right left eye has grade i j x is a vector of person or eye specific covariates for the right eye u and are vectors of location and covariate parameters and is a clustering parameter based on this model and a similar model interchanging the role of i and j we derive a closed form expression for pr i j as a function of u and and use newton raphson methods to maximize the likelihood an extension of the method allows for extra agreement along the diagonal and is then a generalization of the agreement plus linear by linear association model proposed by agresti in the setting of no covariates we apply these methods to a data set of 43 diabetic subjects from the harvard clinical cataract research center where cortical cataract grade was the outcome we also extend this methodology to a survival setting where both censored and uncensored outcomes are available for individual cluster members and one wishes to take clustering into account we apply the survival analysis model to a data set of 1807 children two ears per child in the greater boston area who were followed for the development of otitis media over the first year of life 1997 by john wiley sons ltd

methods for estimating the parameters of a linear model for ordered categorical data in many empirical analyses the response of interest is categorical with an ordinal scale attached many investigators prefer to formulate a linear model assigning scores to each category of the ordinal response and treating it as continuous when the covariates are categorical haber 1985 computational statistics and data analysis 3 1 10 has developed a method to obtain maximum likelihood ml estimates of the parameters of the linear model using lagrange multipliers however when the covariates are continuous the only method we found in the literature is ordinary least squares ols performed under the assumption of homogeneous variance the ols estimates are unbiased and consistent but since variance homogeneity is violated the ols estimates of variance can be biased and may not be consistent we discuss a variance estimate white 1980 econometrica 48 817 838 that is consistent for the true variance of the ols parameter estimates the possible bias encountered by using the naive ols variance estimate is discussed an estimated generalized least squares egls estimator is proposed and its efficiency relative to ols is discussed finally an empirical comparison of ols egls and ml estimators is made

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

non parametric two sample tests for repeated ordinal responses consider data on two groups of clusters where each cluster consists of many units that respond on an ordinal scale we develop a mann whitney type test to determine whether a typical response from the first group is larger or smaller than a typical response from the second group

the score test for independence in r x c contingency tables with missing data in this paper the score test statistic for testing independence in r x c contingency tables with missing data is proposed under the null hypothesis of independence the statistic has an approximate chi squared distribution with r 1 c 1 degrees of freedom the proposed test statistic is quite similar to the pearson chi squared statistic with complete data and unlike the likelihood ratio statistic for testing independence its computation is simple and noniterative in addition a score test statistic is proposed for testing independence when the rows and columns of the r x c table are ordinal finally extensions of the score statistics to test for conditional independence in a set of r x c contingency tables with missing data are described this yields score test statistics that are natural extensions of the mantel haenszel statistic an example using a subset of data from the six cities study is presented to illustrate the methods

generalized lyapunov equation and factorization of matrix polynomials abstract this paper presents an algorithm for the construction of a solution of the generalized lyapunov equation it is proved that the polynomial matrix factorization relative to the imaginary axis may be reduced to the successive solution of lyapunov equations i e the factorization is reduced to the solution of a sequence of generalized lyapunov equations not to the solution of generalized riccati equation

a class of regression models for multivariate categorical responses summary when data composed of several categorical responses together with categorical or continuous predictors are observed it is often useful to relate the response probabilities to the predictors via a generalised linear model with a composite link function this paper discusses a class of link functions that lie between the two extremes of the multivariate logistic transform of mccullagh nelder 1989 and the log linear decomposition of contingency table analysis the models derived from these link functions are shown to inherit various desirable properties of both the multivariate logistic regression models and the log linear regression models a computational scheme for implementing these models is derived and they are demonstrated to be computationally more tractable than the multivariate logistic regression models their application is illustrated in a numerical example

two sample rank tests under complex sampling rank tests are widely used for exploratory and formal inference in the health and social sciences with the widespread use of data from complex survey samples in medical and social research there is increasing demand for versions of rank tests that account for the sampling design we propose a general approach to constructing design based rank tests when comparing groups within a complex sample and when using a national survey as a reference distribution and illustrate both scenarios with examples we show that the tests have asymptotically correct levels and that the relative power of different rank tests is not greatly affected by complex sampling copyright 2013 oxford university press

marginal regression models for clustered ordinal measurements abstract this article constructs statistical models for clustered ordinal measurements we specify two regression models one for the marginal means and one for the marginal pairwise global odds ratios of particular interest are problems in which the odds ratio regression is a focus simple assumptions about higher order conditional moments give a quadratic exponential likelihood function with second order estimating equations gee2 as score equations but computational difficulty can arise for large clusters when both the mean response and the association between measures is of interest first we present gee1 as an alternative estimation strategy second we extend to repeated ordinal measurements the method developed by carey et al for binary observations that is based on alternating logistic regressions alr for the marginal mean parameters and the pairwise log odds ratio parameters we study the efficiency of gee1 and alr relative to full maximum likelihood we demonstrate the utility of our regr

power and sample size calculations for exact conditional tests with ordered categorical data we develop an algorithm for computing sample sizes equal or unequal for categorical data we illustrate its use in the two sample setting using the wilcoxon rank sum statistic but the algorithm accommodates the entire class of linear rank statistics and can be extended to include nonlinear rank statistics as well the sample size determinations can be based on either exact power or on a very precise monte carlo estimate of it to reduce the computations further power can be computed as a function of asymptotic critical values when the number of categories is not too small for the wilcoxon statistic we show that this approximation works well if there are more than five response categories

multiple gain parameter multivariable root locus abstract the multivariable root locus concept is generalized here by allowing for independently variable gain parameters in the different feedback paths sufficient conditions are found under which the matrix of markov parameters of the open loop transfer matrix g s will guarantee stability for every gain matrix in the gain space this generalizes the recent work of byrnes 1981 and cook and hall 1982 and is useful in the design of proportional plus integral controllers for multivariable feedback control systems

sample size formula for clustered survival data using weighted log rank statistics summary we present a simple sample size formula for weighted log rank statistics applied to clustered survival data with variable cluster sizes and arbitrary treatment assignments within clusters this formula is based on the asymptotic normality of weighted log rank statistics under certain local alternatives in the clustered data context we also provide consistent variance estimators the derived sample size formula reduces to schoenfeld s 1983 formula for cases of no clustering or independence within clusters simulation results verify control of the type i error and accuracy of the sample size formula use of the sample size formula in an event driven clinical trial design is illustrated using data from the early treatment diabetic retinopathy study

min and max scorings for two sample ordinal data abstract to analyze two sample ordinal data one must often assign some increasing numerical scores to the ordinal categories the choice of appropriate scores in these types of analyses is often problematic this article presents a new approach for reporting the results of such analyses using techniques of order restricted inference we obtain the minimum and maximum of standard two sample test statistics over all possible assignments of increasing scores if the range of the min and max values does not include the critical value for the test statistics then we can immediately conclude that the result of the analysis remains the same no matter what choice of increasing scores is used on the other hand if the range includes a critical value the choice of scores used in the analysis must be carefully justified numerous examples are given to clarify our approach

improving tests for superior treatment in contingency tables abstract when comparing two treatments on the basis of ordinal data a natural alternative is stochastic order as it implies that more favorable outcomes receive greater probability results of eaton can be used to provide a complete class of tests we find that many tests in current use are not in this class and hence are inadmissible more important we present methods of improving such tests often the increase in power is substantial although the size remains the same

a reduced rank multivariate regression approach to aquatic joint toxicity experiments classical multivariate regression techniques can lead to an asymptotically efficient restricted estimator of the regression matrix if there is evidence that the regression matrix is of reduced rank a sublethal joint toxicity experiment involving two agents is used to illustrate this approach

distribution free fitting of logit models with random effects for repeated categorical responses this article discusses random effects models for within subject comparisons of repeated responses on the same categorical scale the models account for the correlation that normally occurs between repeated responses the standard way of fitting such models maximizes the marginal likelihood after integrating with respect to a distribution for the random effect an alternative non parametric approach does not assume a distributional form for the random effects recent literature shows that for certain simple logit models this approach yields essentially the same model parameter estimates as conditional maximum likelihood moreover these estimates also result from fitting corresponding quasi symmetric log linear models for simple data sets in which primary interest relates to subject specific comparisons of the repeated responses one can easily obtain the estimates with standard software for log linear models examples include data from crossover designs and from comparisons of treatment and control groups regarding the change between baseline and follow up observations

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

sample size calculations for clustered binary data in this paper we propose a sample size calculation method for testing on a binomial proportion when binary observations are dependent within clusters in estimating the binomial proportion in clustered binary data two weighting systems have been popular equal weights to clusters and equal weights to units within clusters when the number of units varies cluster by cluster performance of these two weighting systems depends on the extent of correlation among units within each cluster in addition to them we will also use an optimal weighting method that minimizes the variance of the estimator a sample size formula is derived for each of the estimators with different weighting schemes we apply these methods to the sample size calculation for the sensitivity of a periodontal diagnostic test simulation studies are conducted to evaluate a finite sample performance of the three estimators we also assess the influence of misspecified input parameter values on the calculated sample size the optimal estimator requires equal or smaller sample sizes and is more robust to the misspecification of an input parameter than those assigning equal weights to units or clusters

analyzing bivariate ordinal data using a global odds ratio abstract a moment method is proposed for regression analysis of bivariate ordered categorical data using the global odds ratio as the measure of association for modeling the margins this method utilizes the stochastic ordering implicit in the data this method allows for covariate effects in modeling the association between responses an application of the proposed model is illustrated using the ophthalmological data from the wisconsin epidemiologic study of diabetic retinopathy for identifying risk factors among younger onset diabetics this model is also extended to a longitudinal data setting with more than two repeated measures

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

a likelihood based method for analysing longitudinal binary responses summary in this paper we discuss a likelihood based method for analysing correlated binary responses based on a multivariate model it is related to the pseudo maximum likelihood approach suggested recently by zhao prentice 1990 their parameterization results in a simple pairwise model in which the association between responses is modelled in terms of correlations while the present paper uses conditional log odds ratios with this approach higher order associations can be incorporated in a natural way one important advantage of this parameterization is that the maximum likelihood estimates of the marginal mean parameters are robust to misspecification of the time dependence we describe an iterative two stage procedure for obtaining the maximum likelihood estimates two examples are presented to illustrate this methodology

sample size calculations for ordered categorical data many clinical trials yield data on an ordered categorical scale such as very good good moderate poor under the assumption of proportional odds such data can be analysed using techniques of logistic regression in simple comparisons of two treatments this approach becomes equivalent to the mann whitney test in this paper sample size formulae consistent with an eventual logistic regression analysis are derived the influence on efficiency of the number and breadth of categories will be examined effects of misclassification and of stratification are discussed and examples of the calculations are given

a proportional odds model with subject specific effects for repeated ordered categorical responses summary suppose subjects make repeated responses on the same ordered categorical scale we propose a generalization of the rasch model that expresses the cumulative logit of the response distribution using subject parameters and a proportional odds structure for item effects parameters in the model describe subject specific rather than populationaveraged effects consistent estimation of the effects requires eliminating the subject parameters we accomplish this by simultaneous fitting of rasch models conditional on sufficient statistics for those parameters for the possible binary collapsings of the response the fitting process uses an improved newton raphson algorithm for fitting generalized loglinear models by maximum likelihood estimation subject to constraints for the case of two items we give simple expressions for an effect estimate and its standard error and suggest a test of marginal homogeneity for ordinal matched pairs

population averaged and cluster specific models for clustered ordinal response data we compare population averaged and cluster specific models for clustered ordinal data we consider generalized estimating equations and constrained equations maximum likelihood estimation of population averaged cumulative logit regression models and mixed effects estimation of cluster specific cumulative logit regression models a previously reported relationship between population averaged and cluster specific parameters for the binary logistic link appears to hold for analogous parameters under the cumulative logit link we address these issues in the context of data from two cross over clinical trials

rank based inference for covariate and group effects in clustered data in presence of informative intra cluster group size there have been numerous attempts to extend the wilcoxon rank sum test to clustered data recently one such rank sum test dutta datta 2016 biometrics 72 432 440 was developed to compare the group specific marginal distributions of outcomes in clustered data where the conditional distributions of outcomes depend on the number of observations from that group in a given cluster a phenomenon referred to as informative intra cluster group icg size however comparison of group specific marginal distributions may not be sufficient in presence of some potentially useful covariables that are observed in the study in addition not accounting for the effect of these covariates can lead to biased and misleading inference for the group comparisons thus the purpose of this article is twofold first we develop a method to estimate the covariate effects using rank based weighted estimating equations that are appropriate when the icg size is informative second we construct an aligned rank sum test based on the covariate adjusted outcomes asymptotic distributions of the r estimators and the test statistic are provided through simulation studies we show the importance of selecting proper weights in constructing the estimating equations when informativeness is present through the cluster or icg sizes we also demonstrate the superiority and the robustness of our method in comparison to regular parametric linear mixed models in clustered data we apply our method to analyze different real life data sets including a data on birthweights of rat pups in different litters and a dental data on tooth attachment loss

modeling within subject dependencies in ordinal paired comparison data this paper presents two probabilistic models based on the logistic and the normal distribution for the analysis of dependencies in individual paired comparison judgments it is argued that a core assumption of latent class choice models independence of individual decisions may not be well suited for the analysis of paired comparison data instead the analysis and interpretation of paired comparison data may be much simplified by allowing for within person dependencies that result from repeated evaluations of the same options in different pairs moreover by relating dependencies among the individual level responses to in consistencies in the judgmental process we show that the proposed graded paired comparison models reduce to ranking models under certain conditions three applications are presented to illustrate the approach

extension of the rank sum test for clustered data two group comparisons with group membership defined at the subunit level the wilcoxon rank sum test is widely used for two group comparisons for nonnormal data an assumption of this test is independence of sampling units both between and within groups in ophthalmology data are often collected on two eyes of an individual which are highly correlated in ophthalmological clinical trials randomization is usually performed at the subject level but the unit of analysis is the eye if the eye is used as the unit of analysis then a modification to the usual wilcoxon rank sum variance formula must be made to account for the within cluster dependence for some clustered data designs where the unit of analysis is the subunit group membership may be defined at the subunit level for example in some randomized ophthalmologic clinical trials different treatments may be applied to fellow eyes of some patients while the same treatment may be applied to fellow eyes of other patients in general binary eye specific covariates may be present scored as exposed or unexposed and one wishes to compare nonnormally distributed outcomes between exposed and unexposed eyes using the wilcoxon rank sum test while accounting for the clustering in this article we present a corrected variance formula for the wilcoxon rank sum statistic in the setting of eye subunit specific covariates we apply it to compare ocular itching scores in ocular allergy patients between eyes treated with active versus placebo eye drops where some patients receive the same eye drop in both eyes while other patients receive different eye drops in fellow eyes we also present comparisons between the clustered wilcoxon test and each of the signed rank tests and mixed model approaches and show dramatic differences in power in favor of the clustered wilcoxon test for some designs

two sample test for correlated data under outcome dependent sampling with an application to self reported weight loss data standard methods for two sample tests such as the t test and wilcoxon rank sum test may lead to incorrect type i errors when applied to longitudinal or clustered data recent alternatives of two sample tests for clustered data often require certain assumptions on the correlation structure and or noninformative cluster size in this paper based on a novel pseudolikelihood for correlated data we propose a score test without knowledge of the correlation structure or assuming data missingness at random the proposed score test can capture differences in the mean and variance between two groups simultaneously we use projection theory to derive the limiting distribution of the test statistic in which the covariance matrix can be empirically estimated we conduct simulation studies to evaluate the proposed test and compare it with existing methods to illustrate the usefulness proposed test we use it to compare self reported weight loss data in a friends referral group with the data from the internet self joining group

maximum likelihood methods for a generalized class of log linear models we discuss maximum likelihood methods for fitting a broad class of multivariate categorical response data models in particular we derive the large sample distributions for maximum likelihood estimators of parameters of product multinomial generalized log linear models the large sample behavior of other relevant likelihood based statistics such as goodness of fit statistics and adjusted residuals is also described the asymptotic results are derived within the framework of the constraint specification rather than the more common freedom specification of the model we also outline an improved fitting algorithm for computing parameter maximum likelihood estimates and other relevant statistics the broad class of multivariate categorical response data models which are referred to as generalized log linear models can imply structure on several response configuration distributions e g joint and marginal distributions these models which include as special cases log linear logit and cumulative logit models enjoy a wide breadth of application including longitudinal rater agreement and crossover data analyses

inference for likelihood ratio ordering in the two sample problem abstract we obtain the maximum likelihood estimators of two multinomial probability vectors under the constraint that they are likelihood ratio ordered we extend this estimation approach to the case of two univariate distributions and show strong consistency of the estimators we also derive and study the asymptotic distribution of the likelihood ratio statistic for testing the equality of two discrete probability distributions against the alternative that one distribution is greater than the other in the likelihood ratio ordering sense finally we examine a data set pertaining to average daily insulin dose from the boston collaborative drug surveillance program and compare our testing procedure to testing procedures for other stochastic orderings

a random effects model for ordered categorical data abstract this article presents a random effects model for multivariate and grouped univariate ordered categorical data the assumed family of distributions for the random effects adopts a wide variety of forms and shapes the model s likelihood has a closed expression and can be computed without recourse to numerical integration or gaussian quadrature a bivariate and a grouped univariate example are used to illustrate the proposed model

evaluation of an adjusted chi square statistic as applied to observational studies involving clustered binary data a simple adjustment to the pearson chi square test has been proposed for comparing proportions estimated from clustered binary observations however the assumptions needed to assure the validity of this test have not yet been thoroughly addressed these assumptions will hold for experimental comparisons but could be violated for some observational comparisons in this paper we investigate the conditions under which the adjusted chi square statistic is valid and examine its performance when these assumptions are violated we also introduce some alternative test statistics that do not require these assumptions the test statistics considered are then compared through simulation and an example presented based on real data the simulation study shows that the adjusted chi square statistic generally produces empirical type i errors close to nominal under the assumption of a common intracluster correlation coefficient even if the intracluster correlations are different the adjusted chi square statistic performs well when the groups have equal numbers of clusters

nonparametric analysis of clustered multivariate data there is wide interest in extending univariate and multivariate nonparametric procedures to clustered and hierarchical data traditionally parametric mixed models have been used to account for the correlation structures among the dependent observational units in this work we extend multivariate nonparametric procedures for one sample and several sample location problems to clustered data settings the results are given for a general score function but with an emphasis on spatial sign and rank methods mixed models notation involving design matrices for fixed and random effects is used throughout the asymptotic variance formulas and limiting distributions of the test statistics under the null hypothesis and under a sequence of alternatives are derived as are the limiting distributions for the corresponding estimates the approach based on a general score function also shows for example how m estimates behave with clustered data efficiency studies demonstrate the practical advantages and disadvantag

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

multivariate methods in ophthalmology with application to other paired data situations methods are presented for performing multiple regression analyses and multiple logistic regression analyses on ophthalmologic data with normally and binomially distributed outcome variables while accounting for the intraclass correlation between eyes these methods are extended to more general nested data structures where a variable number of subunits are available for each primary unit of analysis as in familial data these methods can also be applied to other types of paired data as in matched studies with a variable matching ratio where one has a continuous outcome variable and wishes to control for other confounding variables while maintaining the matching examples are given of these methods with a group of over 400 patients with retinitis pigmentosa in which spherical refractive error and visual acuity are related to genetic type after the effects of age sex and the presence of cataract have been controlled

location scale cumulative odds models for ordinal data a generalized non linear model approach proportional odds regression models for multinomial probabilities based on ordered categories have been generalized in two somewhat different directions models having scale as well as location parameters for adjustment of boundaries on an unobservable underlying continuum between categories have been employed in the context of roc analysis partial proportional odds models having different regression adjustments for different multinomial categories have also been proposed this paper considers a synthesis and further generalization of these two families with use of a number of examples i discuss and illustrate properties of this extended family of models emphasis is on the computation of maximum likelihood estimates of parameters asymptotic standard deviations and goodness of fit statistics with use of non linear regression programs in standard statistical software such as sas

area under the free response roc curve froc and a related summary index free response assessment of diagnostic systems continues to gain acceptance in areas related to the detection localization and classification of one or more abnormalities within a subject a free response receiver operating characteristic froc curve is a tool for characterizing the performance of a free response system at all decision thresholds simultaneously although the importance of a single index summarizing the entire curve over all decision thresholds is well recognized in roc analysis e g area under the roc curve currently there is no widely accepted summary of a system being evaluated under the froc paradigm in this article we propose a new index of the free response performance at all decision thresholds simultaneously and develop a nonparametric method for its analysis algebraically the proposed summary index is the area under the empirical froc curve penalized for the number of erroneous marks rewarded for the fraction of detected abnormalities and adjusted for the effect of the target size or acceptance radius geometrically the proposed index can be interpreted as a measure of average performance superiority over an artificial guessing free response process and it represents an analogy to the area between the roc curve and the guessing or diagonal line we derive the ideal bootstrap estimator of the variance which can be used for a resampling free construction of asymptotic bootstrap confidence intervals and for sample size estimation using standard expressions the proposed procedure is free from any parametric assumptions and does not require an assumption of independence of observations within a subject we provide an example with a dataset sampled from a diagnostic imaging study and conduct simulations that demonstrate the appropriateness of the developed procedure for the considered sample sizes and ranges of parameters

symmetry and optimal control in economics this paper provides a new simple version of noether s theorem from symmetries of dynamic optimal programs this theorem gives invariant quantities along optimal paths it is suited to optimal control programs especially for economic models applications in growth economics are given

bivariate modelling of clustered continuous and ordered categorical outcomes simultaneous observation of continuous and ordered categorical outcomes for each subject is common in biomedical research but multivariate analysis of the data is complicated by the multiple data types here we construct a model for the joint distribution of bivariate continuous and ordinal outcomes by applying the concept of latent variables to a multivariate normal distribution the approach is then extended to allow for clustering of the bivariate outcomes the model can be parameterized in a way that allows writing the joint distribution as a product of a standard random effects model for the continuous variable and a correlated cumulative probit model for the ordinal outcome this factorization suggests convenient parameter estimation using estimating equations foetal weight and malformation data from a developmental toxicity experiment illustrate the results

exact inference for contingency tables with ordered categories abstract this article proposes an efficient numerical algorithm for small sample exact inferences in contingency tables having ordinal classifications the inferences which apply conditional on the observed marginal totals also provide an exact analysis for the log linear model of linear by linear association for cell probabilities an exact test of independence has a one sided p value equal to the null probability that model based maximum likelihood estimates of odds ratios are at least as large as the observed estimates the conditional nonnull distribution yields confidence intervals for odds ratios having a linear by linear structure the computations utilize an extension of the network algorithm proposed by mehta and patel 1983

a mixed model procedure for analyzing ordered categorical data a mixed model procedure is developed for predicting the value of an ordered categorical response from knowledge of various predictor variables this procedure resembles the best linear unbiased procedure of henderson 1975 biometrics 31 423 447 for predicting the value of a quantitative response the approach is based on a mixed model version of the threshold model in which it is assumed that the observed category is determined by the value of an underlying unobservable continuous response that follows a mixed linear model the results are illustrated by an application to the problem of predicting the degree of difficulty that will be experienced by a dairy cow in the birth of her calf

discrete generalized algebraic riccati equations and polynomial matrix factorization abstract this paper presents an algorithm for solving discrete generalized algebraic riccati equations with the help of an orthogonal projector a generalization of the procedure of forming and correcting the orthogonal projector is considered and also that of correcting the proper solution by the newton raphson scheme the possibility to use the discrete generalized riccati equations for polynomial matrix factorization with respect to the unit circle is demonstrated a numerical example is given

a rank sum test for clustered data when the number of subjects in a group within a cluster is informative the wilcoxon rank sum test is a popular nonparametric test for comparing two independent populations groups in recent years there have been renewed attempts in extending the wilcoxon rank sum test for clustered data one of which datta and satten 2005 journal of the american statistical association 100 908 915 addresses the issue of informative cluster size i e when the outcomes and the cluster size are correlated we are faced with a situation where the group specific marginal distribution in a cluster depends on the number of observations in that group i e the intra cluster group size we develop a novel extension of the rank sum test for handling this situation we compare the performance of our test with the datta satten test as well as the naive wilcoxon rank sum test using a naturally occurring simulation model of informative intra cluster group size we show that only our test maintains the correct size we also compare our test with a classical signed rank test based on averages of the outcome values in each group paired by the cluster membership while this test maintains the size it has lower power than our test extensions to multiple group comparisons and the case of clusters not having samples from all groups are also discussed we apply our test to determine whether there are differences in the attachment loss between the upper and lower teeth and between mesial and buccal sites of periodontal patients

the analysis of longitudinal ordinal data with nonrandom drop out a model is proposed for longitudinal ordinal data with nonrandom drop out which combines the multivariate dale model for longitudinal ordinal data with a logistic regression model for drop out since response and drop out are modelled as conditionally independent given complete data the resulting likelihood can be maximised relatively simply using the em algorithm which with acceleration is acceptably fast and with appropriate additions can produce estimates of precision the approach is illustrated with an example such modelling of nonrandom drop out requires caution because the interpretation of the fitted models depends on assumptions that are unexaminable in a fundamental sense and the conclusions cannot be regarded as necessarily robust the main role of such modelling may be as a component of a sensitivity analysis

the analysis of multivariate contingency tables by restricted canonical and restricted association models abstract restricted canonical models and restricted association models are proposed and applied to multiway contingency tables these models have been previously applied to two way contingency tables however multivariate generalization has been impeded in the past since canonical and association models both depend on singular value decompositions that apply only to two way arrays in this article this restriction to two way arrays is overcome by division of the cross classified variables into explanatory and response variables the explanatory variables are treated as a single polytomous variable and the response variables are treated as a second single polytomous variable in this fashion the multiway table is reduced to a two way array to which traditional canonical and association models may be applied use of linear restrictions on parameters in canonical and association models is especially important in multiway tables if useful models are to be constructed the class of models considered in th

efficient estimation for rank based regression with clustered data rank based inference is widely used because of its robustness this article provides optimal rank based estimating functions in analysis of clustered data with random cluster effects the extensive simulation studies carried out to evaluate the performance of the proposed method demonstrate that it is robust to outliers and is highly efficient given the existence of strong cluster correlations the performance of the proposed method is satisfactory even when the correlation structure is misspecified or when heteroscedasticity in variance is present finally a real dataset is analyzed for illustration

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

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

a stability enhancing scaling procedure for schur riccati solvers abstract scaling riccati equations so that the solution has unit norm eliminates instabilities associated with the inversion step of invariant subspace based schur methods without changing the relative condition number of the underlying problem such optimal scaling makes it easy to estimate this condition number and to bound the relative error of the computed solution moreover the determination of whether the first step of newton s method will improve this solution is also simplified