Bivariate Distributions Underlying Responses to Ordinal Variables

Laura Kolbe,Suzanne Jak,Frans Oort

doi:10.3390/psych3040037

Abstract

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.

Highlights

In the Type D personality dataset, the bivariate normal distribution was rejected for 83.52% of the variable pairs when the level of significance was 0.05
With a significance level of 0.05, the bivariate skew-normal distribution was rejected for 79.55% of the variable pairs
This study provides a first insight into the distributions underlying the responses to ordinal variables, but more data must be investigated in order to verify the results

Summary

Introduction

Data in the social and behavioral sciences commonly include observations from variables with ordinal response scales, such as Likert items. If ordinal variables do not possess metric properties, alternative techniques than those employed with continuous variables are required. The polychoric correlation coefficient proposed by Pearson [1] is a recommended measure of association between two ordinal variables. The polychoric correlation coefficient is based on the assumption that responses to ordinal variables are generated by two latent underlying continuous variables. The underlying variables are conventionally assumed to follow a bivariate normal distribution, referred to as underlying bivariate normality.

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Conclusion