# Generating Multivariate Ordinal Data via Entropy Principles

Generating Multivariate Ordinal Data via Entropy Principles 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Psychometrika Springer Journals

# Generating Multivariate Ordinal Data via Entropy Principles

, Volume 83 (1) – Jan 22, 2018
26 pages

/lp/springer_journal/generating-multivariate-ordinal-data-via-entropy-principles-cDAMOCzkIi
Publisher
Springer Journals
Subject
Psychology; Psychometrics; Assessment, Testing and Evaluation; Statistics for Social Science, Behavorial Science, Education, Public Policy, and Law; Statistical Theory and Methods
ISSN
0033-3123
eISSN
1860-0980
D.O.I.
10.1007/s11336-018-9603-3
Publisher site
See Article on Publisher Site

### 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.

### Journal

PsychometrikaSpringer Journals

Published: Jan 22, 2018

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