scholarly journals Bivariate Distributions Underlying Responses to Ordinal Variables

Psych ◽  
2021 ◽  
Vol 3 (4) ◽  
pp. 562-578
Author(s):  
Laura Kolbe ◽  
Frans Oort ◽  
Suzanne Jak
Keyword(s):  
Correlation Coefficient ◽  
Latent Variables ◽  
Ordinal Data ◽  
Bivariate Normal Distribution ◽  
Polychoric Correlation ◽  
Bivariate Distributions ◽  
Normal Distributions ◽  
Ordinal Variables ◽  
Bivariate Normal ◽  
Bivariate Normality

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.

Effects of Scaling on the Correlation Coefficient: Additional Considerations

1978 ◽  
Vol 15 (2) ◽  
pp. 304-308 ◽  
Author(s):  
Warren S. Martin
Keyword(s):  
Normal Distribution ◽  
Correlation Coefficient ◽  
Correlation Coefficients ◽  
Bivariate Normal Distribution ◽  
Information Loss ◽  
Simulation Data ◽  
Pearson Product Moment Correlation ◽  
Bivariate Normal ◽  
Amount Of Information ◽  
Restricted Number

Distortion in the Pearson product moment correlation due to a restricted number of scale points is evaluated in two ways. First, a simulation of the bivariate normal distribution is used to estimate the effects of varying the number of scale points on the product moment correlation. This procedure demonstrates a substantial amount of information loss. Second, other correlation coefficients and some methods to correct for this loss are discussed and related to the simulation data.

A Monte Carlo Investigation of the Fisher Z Transformation for Normal and Nonnormal Distributions

2000 ◽  
Vol 87 (3_suppl) ◽  
pp. 1101-1114 ◽  
Author(s):  
Kenneth J. Berry ◽  
Paul W. Mielke
Keyword(s):  
Monte Carlo ◽  
Correlation Coefficient ◽  
Bivariate Normal Distribution ◽  
Small Samples ◽  
Bivariate Distributions ◽  
Nonnormal Distributions ◽  
Monte Carlo Investigation ◽  
Sample Correlation ◽  
Heavy Tailed ◽  
Z Transformation

The Fisher transformation of the sample correlation coefficient r (1915, 1921) and two related techniques by Gayen (1951) and Jeyaratnam (1992) are examined for robustness to nonnormality. Monte Carlo analyses compare combinations of sample sizes and population parameters for seven bivariate distributions. The Fisher, Gayen, and Jeyaratnam approaches are shown to provide useful results for a bivariate normal distribution with any population correlation coefficient ρ and for nonnormal bivariate distributions when ρ = 0. In contrast, the techniques are virtually useless for nonnormal bivariate distributions when ρ#0.0. Surprisingly, small samples are found to provide better estimates than large samples for skewed and symmetric heavy-tailed bivariate distributions.

Measuring Correlation in Ordered Two-Way Contingency Tables

1980 ◽  
Vol 17 (3) ◽  
pp. 391-394 ◽  
Author(s):  
Ulf Olsson
Keyword(s):  
Correlation Coefficient ◽  
Ordinal Data ◽  
Contingency Tables ◽  
Tetrachoric Correlation ◽  
Polychoric Correlation

The product moment correlation coefficient is often used even for ordinal data with only a few scale steps. This procedure may lead to biased results, where the bias depends on the number of scale steps and on the skewnesses of the observed variables. The polychoric correlation coefficient, which is a generalization of the tetrachoric correlation to the general case, is discussed as a possible measure of correlation for this kind of data.

scholarly journals Goodman and Kruskal’s Gamma Coefficient for Ordinalized Bivariate Normal Distributions

Psychometrika ◽  
2020 ◽  
Author(s):  
Alessandro Barbiero ◽  
Asmerilda Hitaj
Keyword(s):  
Normal Distribution ◽  
Rank Correlation ◽  
Real Data ◽  
Random Variable ◽  
Bivariate Normal Distribution ◽  
Normal Distributions ◽  
Bivariate Normal ◽  
Ordinal Variable ◽  
Before And After ◽  
Random Components

Abstract We consider a bivariate normal distribution with linear correlation $$\rho $$ ρ whose random components are discretized according to two assigned sets of thresholds. On the resulting bivariate ordinal random variable, one can compute Goodman and Kruskal’s gamma coefficient, $$\gamma $$ γ , which is a common measure of ordinal association. Given the known analytical monotonic relationship between Pearson’s $$\rho $$ ρ and Kendall’s rank correlation $$\tau $$ τ for the bivariate normal distribution, and since in the continuous case, Kendall’s $$\tau $$ τ coincides with Goodman and Kruskal’s $$\gamma $$ γ , the change of this association measure before and after discretization is worth studying. We consider several experimental settings obtained by varying the two sets of thresholds, or, equivalently, the marginal distributions of the final ordinal variables. This study, confirming previous findings, shows how the gamma coefficient is always larger in absolute value than Kendall’s rank correlation; this discrepancy lessens when the number of categories increases or, given the same number of categories, when using equally probable categories. Based on these results, a proposal is suggested to build a bivariate ordinal variable with assigned margins and Goodman and Kruskal’s $$\gamma $$ γ by ordinalizing a bivariate normal distribution. Illustrative examples employing artificial and real data are provided.

scholarly journals A Modification of Linfoot's Informational Correlation Coefficient

2017 ◽  
Vol 46 (3-4) ◽  
pp. 99-105
Author(s):  
Georgy Shevlyakov ◽  
Nikita Vasilevskiy
Keyword(s):  
Normal Distribution ◽  
Correlation Coefficient ◽  
Strong Correlation ◽  
Bivariate Normal Distribution ◽  
Correlation Measure ◽  
Bivariate Normal ◽  
Large Samples ◽  
Wide Range ◽  
Symmetric Version

Performance of the Linfoot's informational correlation coefficient is experimentally studied at the bivariate normal distribution. It is satisfactory in the case of a strong correlation and on large samples. To reduce the bias of estimation, a symmetric version of this correlation measure is proposed. On small and large samples, this modified informational correlation coefficient outperforms Linfoot's correlation measure at the bivariate normal distribution in the wide range of the correlation coefficient.

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