# Generating Multivariate Ordinal Data via Entropy Principles

Listed:
• Yen Lee

• David Kaplan

## Abstract

When conducting robustness research where the focus of attention is on the impact of non-normality, the marginal skewness and kurtosis are often used to set the degree of non-normality. Monte Carlo methods are commonly applied to conduct this type of research by simulating data from distributions with skewness and kurtosis constrained to pre-specified values. Although several procedures have been proposed to simulate data from distributions with these constraints, no corresponding procedures have been applied for discrete distributions. In this paper, we present two procedures based on the principles of maximum entropy and minimum cross-entropy to estimate the multivariate observed ordinal distributions with constraints on skewness and kurtosis. For these procedures, the correlation matrix of the observed variables is not specified but depends on the relationships between the latent response variables. With the estimated distributions, researchers can study robustness not only focusing on the levels of non-normality but also on the variations in the distribution shapes. A simulation study demonstrates that these procedures yield excellent agreement between specified parameters and those of estimated distributions. A robustness study concerning the effect of distribution shape in the context of confirmatory factor analysis shows that shape can affect the robust $$\chi ^2$$ χ 2 and robust fit indices, especially when the sample size is small, the data are severely non-normal, and the fitted model is complex.

## Suggested Citation

• Yen Lee & David Kaplan, 2018. "Generating Multivariate Ordinal Data via Entropy Principles," Psychometrika, Springer;The Psychometric Society, vol. 83(1), pages 156-181, March.
• Handle: RePEc:spr:psycho:v:83:y:2018:i:1:d:10.1007_s11336-018-9603-3
DOI: 10.1007/s11336-018-9603-3
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## References listed on IDEAS

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4. Todd Headrick & Shlomo Sawilowsky, 1999. "Simulating correlated multivariate nonnormal distributions: Extending the fleishman power method," Psychometrika, Springer;The Psychometric Society, vol. 64(1), pages 25-35, March.
5. Rosseel, Yves, 2012. "lavaan: An R Package for Structural Equation Modeling," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 48(i02).
6. Rohatgi, Vijay K. & Székely, Gábor J., 1989. "Sharp inequalities between skewness and kurtosis," Statistics & Probability Letters, Elsevier, vol. 8(4), pages 297-299, September.
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## Citations

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Cited by:

1. Alessandro Barbiero, 2021. "Inducing a desired value of correlation between two point-scale variables: a two-step procedure using copulas," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(2), pages 307-334, June.

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### Keywords

Non-normal data generation; Entropy; Discrete data;
All these keywords.

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