Bivariate Distributions Underlying Responses to Ordinal Variables

The association between two ordinal variables can be expressed with a polychoric correlation coefficient. This coefficient is conventionally based on the assumption that responses to ordinal variables are generated by two underlying continuous latent variables with a bivariate normal distribution. When the underlying bivariate normality assumption is violated, the estimated polychoric correlation coefficient may be biased. In such a case, we may consider other distributions. In this paper, we aimed to provide an illustration of fitting various bivariate distributions to empirical ordinal data and examining how estimates of the polychoric correlation may vary under different distributional assumptions. Results suggested that the bivariate normal and skew-normal distributions rarely hold in the empirical datasets. In contrast, mixtures of bivariate normal distributions were often not rejected.

Angular similarity in test parameters

Through N-dimensional person space, the article gives measures of test parameters and item statistics, including difficulty/discriminating value of test, correlations between a pair of items, and item-total correlations with binary items using angular similarity between two vectors. Relationships between difficulty value and discriminating value of items and test were derived, including relationship between test reliability and test discriminating value. Reliability of a test as per theoretical definition in terms of length of score vectors of two parallel subtests and angle between such vectors was derived. The method was extended to find reliability of a battery of tests. Reliability and discriminating value of a Likert-type item and scale was found in terms of angular similarity without involving assumptions of continuous nature or linearity or normality for the observed variables, or the underlying variable being measured. The proposed methods also avoid test of unidimensionality or assumption of normality or bivariate normality associated with the polychoric correlations. Thus, the proposed methods are in fact nonparametric and considered as improvement over the existing ones. Reliability as a measure of association of two vectors and discrimination as a measure of distance between the vectors are likely to show a negative relationship.

Logistics Performance Index: Methodological Issues

This article addresses limitations of Logistics Performance Index (LPI) and suggests remedies. Reliability of the instrument used in LPI may be better found by Angular Association method or Bhattacharyya’s measure, using only the frequencies or probabilities of item–response categories without involving assumptions of continuous nature or linearity or normality for the observed variables or the underlying variable being measured. The suggested methods also avoid test of uni-dimensionality, assumption of normality, bivariate normality. The problems of outlying observations and linear assumptions in principal component analysis for finding reliability theta are also avoided in each proposed method. Geometric mean approach provides a better alternative to compute LPI scores avoiding scaling and calculation of weights satisfies many desired properties and reduces level of substitutability between components, facilitates statistical test of equality of two geometric means and identifies critical areas for corrective measures. Such identifications are important from a policy point of view. The graph of LPI for a country over a long period of time reflects pattern of growth of LPI for the country. The method helps to rank and benchmark the countries, if the target vector is taken as LPI score of the best performing country. JEL Codes: C43, C54

Testing Categorized Bivariate Normality With Two-Stage Polychoric Correlation Estimates

Structural equation modeling (SEM) with ordinal indicators rely on an assumption of categorized normality. This assumption may be tested for pairs of variables using the likelihood ratio G2 or Pearson’s X2 statistics. For increased computational efficiency, SEM programs usually estimate polychoric correlations in two stages. However, two-stage polychoric estimates are not asymptotically efficient and G2 and X2 need not be asymptotically chi-square when the estimator is not efficient. Recently, Maydeu-Olivares and Joe (2005) have introduced a new statistic, Mn , that is asymptotically chi-square even for estimators that are not efficient. We investigate the behavior of G2, X2, and Mn when testing underlying bivariate normality with polychoric correlations estimated in two stages.