Nonparametric estimation of copula-based measures of multivariate association from contingency tables

Nonparametric estimation of copula-based measures of multivariate association in a continuous random vector X=(X₁, …, X_d) is usually based on complete continuous data. In many practical applications, however, these types of data are not readily available; instead aggregated ordinal observations are given, for example, ordinal ratings based on a latent continuous scale. This article introduces a purely nonparametric and data-driven estimator of the unknown copula density and the corresponding copula based on multivariate contingency tables. Estimators for multivariate Spearman's rho and Kendall's tau are based thereon. The properties of these estimators in samples of medium and large size are evaluated in a simulation study. An increasing bias can be observed along with an increasing degree of association between the components. As it is to be expected, the bias is severely influenced by the amount of information available. Additionally, the influence of sample size is only marginal. We further give an empirical illustration based on daily returns of five German stocks.     

On Modelling Agreement and Category Distinguishability on an Ordinal Scale

It is quite common that raters may need to classify a sample of subjects on a categorical scale. Perfect agreement can rarely be observed partly because of different perceptions about the meanings of the category labels between raters and partly because of factors such as intrarater variability. Usually, category indistinguishability occurs between adjacent categories. In this article, we propose a simple log-linear model combining ordinal scale information and category distinguishability between ordinal categories for modelling agreement between two raters. For the proposed model, no score assignment is required to the ordinal categories. An algorithm and statistical properties will be provided.     

Concordance coefficients to measure the agreement among several sets of ranks

In this paper, two measures of agreement among several sets of ranks, Kendall's concordance coefficient and top-down concordance coefficient, are reviewed. In order to illustrate the utility of these measures, two examples, in the fields of health and sports, are presented. A Monte Carlo simulation study was carried out to compare the performance of Kendall's and top-down concordance coefficients in detecting several types and magnitudes of agreements. The data generation scheme was developed in order to induce an agreement with different intensities among m (m>2) sets of ranks in non-directional and directional rank agreement scenarios. The performance of each coefficient was estimated by the proportion of rejected null hypotheses, assessed at 5% significance level, when testing whether the underlying population concordance coefficient is sufficiently greater than zero. For the directional rank agreement scenario, the top-down concordance coefficient allowed to achieve a percentage of significant concordances that was higher than the one achieved by Kendall's concordance coefficient. Mainly, when the degree of agreement was small, the results of the simulation study pointed to the advantage of using a weighted rank concordance, namely the top-down concordance coefficient, simultaneously with Kendall's concordance coefficient, enabling the detection of agreement (in a top-down sense) in situations not detected by Kendall's concordance coefficient.     

A bootstrap method for comparing correlated kappa coefficients

Cohen's kappa coefficient is traditionally used to quantify the degree of agreement between two raters on a nominal scale. Correlated kappas occur in many settings (e.g., repeated agreement by raters on the same individuals, concordance between diagnostic tests and a gold standard) and often need to be compared. While different techniques are now available to model correlated κ coefficients, they are generally not easy to implement in practice. The present paper describes a simple alternative method based on the bootstrap for comparing correlated kappa coefficients. The method is illustrated by examples and its type I error studied using simulations. The method is also compared with the generalized estimating equations of the second order and the weighted least-squares methods.     

A “Reverse Order” Elementary Statistics Course

A family of coefficients for measuring monotone association is presented. These include measures of association of ordinal or interval variables such as gamma of Goodman and Kruskal, Somers's d_yx , Kendall's tau, or Spearman's rho as special cases. The article shows how a large number of measures of association can be put into a single general form. These coefficients are used as a basis for defining a variety of data analysis techniques.     

Regression modelling of weighted κ by using generalized estimating equations

In many clinical studies more than one observer may be rating a characteristic measured on an ordinal scale. For example, a study may involve a group of physicians rating a feature seen on a pathology specimen or a computer tomography scan. In clinical studies of this kind, the weighted κ coefficient is a popular measure of agreement for ordinally scaled ratings. Our research stems from a study in which the severity of inflammatory skin disease was rated. The investigators wished to determine and evaluate the strength of agreement between a variable number of observers taking into account patient-specific (age and gender) as well as rater-specific (whether board certified in dermatology) characteristics. This suggested modelling κ as a function of these covariates. We propose the use of generalized estimating equations to estimate the weighted κ coefficient. This approach also accommodates unbalanced data which arise when some subjects are not judged by the same set of observers. Currently an estimate of overall κ for a simple unbalanced data set without covariates involving more than two observers is unavailable. In the inflammatory skin disease study none of the covariates were significantly associated with κ, thus enabling the calculation of an overall weighted κ for this unbalanced data set. In the second motivating example (multiple sclerosis), geographic location was significantly associated with κ. In addition we also compared the results of our method with current methods of testing for heterogeneity of weighted κ coefficients across strata (geographic location) that are available for balanced data sets.     

Estimation of symmetric disagreement using a uniform association model for ordinal agreement data

The Cohen kappa is probably the most widely used measure of agreement. Measuring the degree of agreement or disagreement in square contingency tables by two raters is mostly of interest. Modeling the agreement provides more information on the pattern of the agreement rather than summarizing the agreement by kappa coefficient. Additionally, the disagreement models in the literature they mentioned are proposed for the nominal scales. Disagreement and uniform association models are aggregated as a new model for the ordinal scale agreement data, thus in this paper, symmetric disagreement plus uniform association model that aims separating the association from the disagreement is proposed. Proposed model is applied to real uterine cancer data.     

The analysis of ordinal time-series data via a transition (Markov) model

While standard techniques are available for the analysis of time-series (longitudinal) data, and for ordinal (rating) data, not much is available for the combination of the two, at least in a readily-usable form. However, this data type is common place in the natural and health sciences where repeated ratings are recorded on the same subject. To analyse these data, this paper considers a transition (Markov) model where the rating of a subject at one time depends explicitly on the observed rating at the previous point of time by incorporating the previous rating as a predictor variable. Complications arise with adequate handling of data at the first observation (t=1), as there is no prior observation to use as a predictor. To overcome this, it is postulated the existence of a rating at time t=0; however it is treated as ‘missing data’ and the expectation–maximisation algorithm used to accommodate this. The particular benefits of this method are shown for shorter time series.     

Calculating power for the comparison of dependent κ-coefficients

Summary. In the psychosocial and medical sciences, some studies are designed to assess the agreement between different raters and/or different instruments. Often the same sample will be used to compare the agreement between two or more assessment methods for simplicity and to take advantage of the positive correlation of the ratings. Although sample size calculations have become an important element in the design of research projects, such methods for agreement studies are scarce. We adapt the generalized estimating equations approach for modelling dependent κ -statistics to estimate the sample size that is required for dependent agreement studies. We calculate the power based on a Wald test for the equality of two dependent κ -statistics. The Wald test statistic has a non-central χ ²-distribution with non-centrality parameter that can be estimated with minimal assumptions. The method proposed is useful for agreement studies with two raters and two instruments, and is easily extendable to multiple raters and multiple instruments. Furthermore, the method proposed allows for rater bias. Power calculations for binary ratings under various scenarios are presented. Analyses of two biomedical studies are used for illustration.     

A Note on Item Response Theory Modeling for Online Customer Ratings

ABSTRACT

Online consumer product ratings data are increasing rapidly. While most of the current graphical displays mainly represent the average ratings, Ho and Quinn proposed an easily interpretable graphical display based on an ordinal item response theory (IRT) model, which successfully accounts for systematic interrater differences. Conventionally, the discrimination parameters in IRT models are constrained to be positive, particularly in the modeling of scored data from educational tests. In this article, we use real-world ratings data to demonstrate that such a constraint can have a great impact on the parameter estimation. This impact on estimation was explained through rater behavior. We also discuss correlation among raters and assess the prediction accuracy for both the constrained and the unconstrained models. The results show that the unconstrained model performs better when a larger fraction of rater pairs exhibit negative correlations in ratings.     

The disagreeable behaviour of the kappa statistic

It is often of interest to measure the agreement between a number of raters when an outcome is nominal or ordinal. The kappa statistic is used as a measure of agreement. The statistic is highly sensitive to the distribution of the marginal totals and can produce unreliable results. Other statistics such as the proportion of concordance, maximum attainable kappa and prevalence and bias adjusted kappa should be considered to indicate how well the kappa statistic represents agreement in the data. Each kappa should be considered and interpreted based on the context of the data being analysed. Copyright © 2014 JohnWiley & Sons, Ltd.     

Bayesian variable selection in a class of mixture models for ordinal data: a comparative study

In this paper, we consider a special finite mixture model named Combination of Uniform and shifted Binomial (CUB), recently introduced in the statistical literature to analyse ordinal data expressing the preferences of raters with regards to items or services. Our aim is to develop a variable selection procedure for this model using a Bayesian approach. Bayesian methods for variable selection and model choice have become increasingly popular in recent years, due to advances in Markov chain Monte Carlo computational algorithms. Several methods have been proposed in the case of linear and generalized linear models (GLM). In this paper, we adapt to the CUB model some of these algorithms: the Kuo–Mallick method together with its ‘metropolized’ version and the Stochastic Search Variable Selection method. Several simulated examples are used to illustrate the algorithms and to compare their performance. Finally, an application to real data is introduced.     

Heine process as a q-analog of the Poisson process—waiting and interarrival times

In this study, we introduce the Heine process, {X_q(t), t > 0}, 0 < q < 1, where the random variable X_q(t), for every t > 0, represents the number of events (occurrences or arrivals) during a time interval (0, t]. The Heine process is introduced as a q-analog of the basic Poisson process. Also, in this study, we prove that the distribution of the waiting time W_{ν, q}, ν ? 1, up to the νth arrival, is a q-Erlang distribution and the interarrival times T_{k, q} = W_{k, q} ? W_{k ? 1, q},?k = 1, 2, …, ν with W_{0, q} = 0 are independent and equidistributed with a q-Exponential distribution.     

Some characterizations of discrete distributions based on weak records

Let X ₁, X ₂,... be iid random variables (rv's) with the support on nonnegative integers and let (W _n , n≥0) denote the corresponding sequence of weak record values. We obtain new characterization of geometric and some other discrete distributions based on different forms of partial independence of rv's W _n and W _n+r —W _n for some fixed n≥0 and r≥1. We also prove that rv's W ₀ and W _n+1 —W _n have identical distribution if and only if (iff) the underlying distribution is geometric.     

General location multivariate latent variable models for mixed correlated bounded continuous,ordinal, and nominal responses with non-ignorable missing data

Using a multivariate latent variable approach, this article proposes some new general models to analyze the correlated bounded continuous and categorical (nominal or/and ordinal) responses with and without non-ignorable missing values. First, we discuss regression methods for jointly analyzing continuous, nominal, and ordinal responses that we motivated by analyzing data from studies of toxicity development. Second, using the beta and Dirichlet distributions, we extend the models so that some bounded continuous responses are replaced for continuous responses. The joint distribution of the bounded continuous, nominal and ordinal variables is decomposed into a marginal multinomial distribution for the nominal variable and a conditional multivariate joint distribution for the bounded continuous and ordinal variables given the nominal variable. We estimate the regression parameters under the new general location models using the maximum-likelihood method. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms of the model on the maximal normal curvature. The proposed models are applied to two data sets: BMI, Steatosis and Osteoporosis data and Tehran household expenditure budgets.     

Comparison of alternative imputation methods for ordinal data

In this article, we compare alternative missing imputation methods in the presence of ordinal data, in the framework of CUB (Combination of Uniform and (shifted) Binomial random variable) models. Various imputation methods are considered, as are univariate and multivariate approaches. The first step consists of running a simulation study designed by varying the parameters of the CUB model, to consider and compare CUB models as well as other methods of missing imputation. We use real datasets on which to base the comparison between our approach and some general methods of missing imputation for various missing data mechanisms.     

A latent variable model for analyzing mixed longitudinal (k,l)-inflated count and ordinal responses

A random effects model for analyzing mixed longitudinal count and ordinal data is presented where the count response is inflated in two points (k and l) and an (k,l)-Inflated Power series distribution is used as its distribution. A full likelihood-based approach is used to obtain maximum likelihood estimates of parameters of the model. For data with non-ignorable missing values models with probit model for missing mechanism are used.The dependence between longitudinal sequences of responses and inflation parameters are investigated using a random effects approach. Also, to investigate the correlation between mixed ordinal and count responses of each individuals at each time, a shared random effect is used. In order to assess the performance of the model, a simulation study is performed for a case that the count response has (k,l)-Inflated Binomial distribution. Performance comparisons of count-ordinal random effect model, Zero-Inflated ordinal random effects model and (k,l)-Inflated ordinal random effects model are also given. The model is applied to a real social data set from the first two waves of the national longitudinal study of adolescent to adult health (Add Health study). In this data set, the joint responses are the number of days in a month that each individual smoked as the count response and the general health condition of each individual as the ordinal response. For the count response there is incidence of excess values of 0 and 30.     

Evaluation of open items using the many-facet Rasch model

The goal of this study is to analyze the quality of ratings assigned to two constructed response questions for evaluating the written ability of essays in Portuguese language from the perspective of the many-facet Rasch (MFR [15 J.M. Linacre, Many-facet Rasch Measurement, 2nd ed., MESA Press, Chicago, 1994. [Google Scholar]]) model. The analyzed data set comes from 350 written tests with two open-item tasks that were developed based on a rating process independently marked by two rater coordinators and a group of 42 raters. The MFR model analysis shows the measurement quality related to the examinees, raters, tasks and items, and classification scale that has been used for the task rating process. The findings indicate significant differences amongst the rater severities and show that the raters cannot be interchanged. The results also suggest that the comparison between the two task difficulties needs further investigation. An additional study has been done on the scale structure of the classification used by each rater for each item. The result suggests that there have been some similarities amongst the tasks and a need of revision for some criteria of the rating process. Overall, the scale of evaluation has shown to be efficient for a classification of the examinees.