Global permutation tests for multivariate ordinal data: alternatives, test statistics, and the null dilemma

We discuss two-sample global permutation tests for sets of multivariate ordinal data in possibly high-dimensional setups, motivated by the analysis of data collected by means of the World Health Organisation's International Classification of Functioning, Disability and Health. The tests do not require any modelling of the multivariate dependence structure. Specifically, we consider testing for marginal inhomogeneity and direction-independent marginal order. Max-T test statistics are known to lead to good power against alternatives with few strong individual effects. We propose test statistics that can be seen as their counterparts for alternatives with many weak individual effects. Permutation tests are valid only if the two multivariate distributions are identical under the null hypothesis. By means of simulations, we examine the practical impact of violations of this exchangeability condition. Our simulations suggest that theoretically invalid permutation tests can still be 'practically valid'. In particular, they suggest that the degree of the permutation procedure's failure may be considered as a function of the difference in group-specific covariance matrices, the proportion between group sizes, the number of variables in the set, the test statistic used, and the number of levels per variable

Robust Method for Testing the Significance of Bivariate Correlation of Ordinal Data

The Tau’s statistics were introduced to solve the problems of tied data but its effect on shape of table cannot be ascertained. This study compares the non parametric approaches for testing the significance of bivariate correlation for ordinal data based on table shape (square and rectangular table). The bootstrap method was used to compare the magnitude of the correlation values and the associated standard error of the values of  tau’s -b, c and Gamma over a square and rectangular table. The Gamma was found to be the most robust statistic for computing the correlation of ordinal data Keywords: Tau statistics, Gamma, ordinal correlation, Bivariate correlatio

Model-Free Descriptive Modeling for Multivariate Categorical Data with An Ordinal Dependent Variable

In the process of statistical modeling, the descriptive modeling plays an essential role in accelerating the formulation of plausible hypotheses in the subsequent explanatory modeling and facilitating the selection of potential variables in the subsequent predictive modeling. Especially, for multivariate categorical data analysis, it is desirable to use the descriptive modeling methods for uncovering and summarizing the potential association structure among multiple categorical variables in a compact manner. However, many classical methods in this case either rely on strong assumptions for parametric models or become infeasible when the data dimension is higher. To this end, we propose a model-free method for the descriptive modeling to delineate and quantify the association structure between an ordinal dependent variable and a set of categorical independent variables in a multi-dimensional contingency table. The proposed method consists of four components: subcopula score, subcopula regression, subcopula regression based association measure and its (sequential/non-sequenti-al) decompositions. The subcopula score is a data-dependent scoring method for an ordinal variable reflecting the ordered nature of its categories. The subcopula regression leverages the subcopula scores to identify the association structure between the ordinal dependent variable and a set of categorical independent variables. The subcopula regression based association measure exploits the subcopula regression to quantify the strength of the association structure in a model-free manner. The sequential and non-sequential decompositions of the proposed association measure evaluate the contribution of the subsets of independent variables to the overall association in various forms such as marginal, conditional, interactive and correlative association. We first study the theoretical properties of the subcopula score, subcopula regression, subcopula regression based association measure and its (sequential/non-sequential) decompositions. Next we develop the statistical inference for the proposed method including point estimation, (asymptotic/bootstrap) confidence intervals and permutation based hypothesis testing. Then we examine the finite-sample properties of the proposed overall, marginal and conditional association measures in multi-dimensional contingency tables. Finally, we demonstrate the potential use of the proposed method in real-world applications

An Overview of Methods in the Analysis of Dependent ordered catagorical Data: Assumptions and Implications

Subjective assessments of pain, quality of life, ability etc. measured by rating scales and questionnaires are common in clinical research. The resulting responses are categorical with an ordered structure and the statistical methods must take account of this type of data structure. In this paper we give an overview of methods for analysis of dependent ordered categorical data and a comparison of standard models and measures with nonparametric augmented rank measures proposed by Svensson. We focus on assumptions and issues behind model specifications and data as well as implications of the methods. First we summarise some fundamental models for categorical data and two main approaches for repeated ordinal data; marginal and cluster-specific models. We then describe models and measures for application in agreement studies and finally give a summary of the approach of Svensson. The paper concludes with a summary of important aspects.Dependent ordinal data; GEE; GLMM; Logit; modelling