# Generating Multivariate Ordinal Data via Entropy Principles

- 195 Downloads

## 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 \(\chi ^2\) and robust fit indices, especially when the sample size is small, the data are severely non-normal, and the fitted model is complex.

## Keywords

Non-normal data generation Entropy Discrete data## Notes

### Acknowledgements

We acknowledge Professors Stephen Wright and Michael Ferris for their valuable suggestions on solving the optimization problems. Professor Chunming Zhang suggested ways for proving the uniqueness when \(k_{j}=3\). The discussion between the first author and Professor Chun-Ping Cheng have inspired this paper.

## Supplementary material

## References

- Asparouhov, T., Muthén, B. (2010).
*Simple second order chi-square correction*. Retrieved from Mplus website: http://www.statjnodel.com/dowrJoad/WLSMV_new_chi21. - Babakus, E., Ferguson, E. J., & Joreskog, K. G. (1987). The sensitivity of confirmatory maximum likelihood factor analysis to violations of measurement scale and distributional assumptions.
*Journal of Marketing Research*,*24*, 222–228.CrossRefGoogle Scholar - Blair, R. C. (1981). A reaction to consequence of failure to meet assumptions underlying fixed effects analysis of variance and covariance.
*Review of Educational Research*,*51*, 499–507.CrossRefGoogle Scholar - Bollen, K. A. (1989).
*Structural equations with latent variables*. New York: John Wiley and Sons Inc.CrossRefGoogle Scholar - Bradley, J. V. (1982). The insidious L-shaped distribution.
*Bulletin of the Psychonomic Society*,*20*, 85–88.CrossRefGoogle Scholar - Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.),
*Testing Structural Equation Models*(pp. 136–162). Thousand Oaks, CA: Sage Publications.Google Scholar - DiStefano, C., & Morgan, G. B. (2014). A comparison of diagonal weighted least squares robust estimation techniques for ordinal data.
*Structural Equation Modeling: A Multidisciplinary Journal*,*213*, 425–438.CrossRefGoogle Scholar - Ethington, C. A. (1987). The robustness of LISREL estimates in structural equation models with categorical variables.
*The Journal of Experimental Education*,*55*, 80–88.CrossRefGoogle Scholar - Fleishman, A. I. (1978). A method for simulating non-normal distributions.
*Psychometrika*,*43*, 521–532.CrossRefGoogle Scholar - Flora, D. B., & Curran, P. J. (2004). An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data.
*Psychological Methods*,*9*, 466–491.CrossRefPubMedPubMedCentralGoogle Scholar - Genz, A. (1992). Numerical computation of multivariate normal probabilities.
*Journal of computational and graphical statistics*,*1*(2), 141–149.Google Scholar - Genz, A., & Bretz, F. (2002). Comparison of methods for the computation of multivariate t probabilities.
*Journal of Computational and Graphical Statistics*,*11*(4), 950–971.CrossRefGoogle Scholar - Genz, A., Bretz, F., Miwa, T., Mi, X , Leisch, F., Scheipl, F., Hothorn, T. (2015).
*mvtnorm: Multivariate normal and t distributions.*http://CRAN.R-project.org/package=mvtnorm (R package version 1.0-3) - Golan, A., Judge, G., & Miller, D. (1997).
*Maximum entropy econometrics: Robust estimation with limited data*. Chichester: Wiley.Google Scholar - Headrick, T. C. (2010).
*Statistical simulation: Power method polynomials and other transformations*. Boca Raton, FL: Chapman and Hall.Google Scholar - Headrick, T. C., & Sawilosky, S. S. (1999). Simulating correlated multivariate nonnormal distributions: Extending the Fleishman power method.
*Psychometrika*,*64*, 25–35.CrossRefGoogle Scholar - Hipp, J. R., & Bollen, K. A. (2003). Model fit in structural equation models with censored, ordinal, and dichotomous variables: Testing vanishing tetrads.
*Sociological Methodology*,*33*, 267–305.CrossRefGoogle Scholar - Hooper, D., Coughlan, J., & Mullen, M. R. (2008). Structural equation modelling: Guidelines for determining model fit.
*Electronic Journal of Business Research Methods*,*6*, 53–60.Google Scholar - Jaynes, E. T. (1957). Information theory and statistical mechanics.
*Physics Review*,*106*, 620–630.CrossRefGoogle Scholar - Jaynes, E. T. (1982). On the rationale of maximum-entropy methods.
*IEEE*,*70*, 939–952.CrossRefGoogle Scholar - Jorgensen, T. (2016).
*lavaanTabular Lavaan output .scaled —.robust*. https://groups.google.com/forum/#!topic/lavaan/rGitXu9h9zY (Online; accessed February 19, 2017). - Kapur, J. N., & Kesavan, H. K. (1992).
*Entropy optimization principles with applications*. Boston: Academic Press.CrossRefGoogle Scholar - Kullback, S., & Leibler, R. (1951). On information and sufficiency.
*Annals of Mathematical Statistics*,*22*, 79–86.CrossRefGoogle Scholar - Lee, Y. (2010). Generation of non-normal approximated discrete random variables. Master’s thesis, National Chengchi University, Taipei, Taiwan.Google Scholar
- Madsen, K., Nielsen, H. B., Tingleff, O. (2004).
*Optimization with constraints*. LyngbyIMM, Technical University of Denmark. http://orbit.dtu.dk/files/2721110/imm4213.pdf - Mair, P., Satorra, A., & Bentler, P. M. (2012). Generating nonnormal multivariate data using copulas: Applications to sem.
*Multivariate Behavioral Research*,*47*, 547–565.CrossRefPubMedGoogle Scholar - Mattson, S. (1997). How to generate non-normal data for simulation of structural equation models.
*Multivariate Behavioral Research*,*32*, 355–373.CrossRefPubMedGoogle Scholar - Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures.
*Psychological Bulletin*,*105*, 156–166.CrossRefGoogle Scholar - Muthén, B. (1983). Latent variable structural equation modeling with categorical data.
*Journal of Econometrics*,*22*, 486–5.CrossRefGoogle Scholar - Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators.
*Psychometrika*,*49*, 115–132.CrossRefGoogle Scholar - Muthén, B., & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables.
*British Journal of Mathematical and Statistical Psychology*,*38*, 171–189.CrossRefGoogle Scholar - Muthén, B., & Kaplan, D. (1992). A comparison of some methodologies for the factor analysis of non-normal likert variables: A note on the size of the model.
*British Journal of Mathematical and Statistical Psychology*,*45*, 19–30.CrossRefGoogle Scholar - Nocedal, J., & Wright, S. (2006).
*Numerical optimization*. Berlin: Springer.Google Scholar - Olsson, U. (1979). On the robustness of factor analysis against crude classification of the observations.
*Multivariate Behavioral Research*,*14*, 485–500.CrossRefPubMedGoogle Scholar - Pearson, E. S., & Please, N. W. (1975). Relation between the shape of population distribution and the robustness of four simple test statistics.
*Biometrika*,*62*, 223–241.CrossRefGoogle Scholar - R Core Team. (2014).
*R: A Language and Environment for Statistical Computing Vienna, Austria*. http://www.R-project.org/ - Ramachandran, K. M., & Tsokos, C. P. (2009).
*Mathematical statistics with applications*. Burlington, MA: Elsevier.Google Scholar - Rohatgi, V. K., & Székely, G. J. (1989). Sharp inequalities between skewness and kurtosis.
*Statistics & Probability Letters*,*84*, 297–299.CrossRefGoogle Scholar - Rosseel, Y. 2012. lavaan: An R package for structural equation modeling.
*Journal of Statistical Software4821-36*. http://jstatsoft.org/v48/i02/ - Ruscio, J., & Kaczetow, W. (2008). Simulating multivariate nonnormal data using an iterative technique.
*Multivariate Behavioral Research*,*48*, 355–381.CrossRefGoogle Scholar - Shannon, C. E. (1948). A mathematical theory of communication.
*The Bell System Technical Journal*,*27*(3), 379–423.CrossRefGoogle Scholar - Varadhan, R. (2015).
*alabama: Constrained nonlinear optimization*. http://CRAN.R-project.org/package=alabama (R package version 2015.3-1). - Weng, L., & Cheng, C. (2004). Effects of response order on Likert-type scale.
*Educational and Psychological Measurement*,*60*, 908–924.CrossRefGoogle Scholar - Wilkins, J. E. (1944). A note on skewness and kurtosis.
*The Annal of Mathematical Statistic*,*15*, 333–335.CrossRefGoogle Scholar - Wu, N. (1997).
*The maximum entropy method*. New York: Springer.CrossRefGoogle Scholar - Yang-Wallentin, F., Joreskog, K., & Luo, H. (2010). Confirmatory factor analysis of ordinal variables with misspecified models.
*Structural Equation Modeling*,*17*, 392–423.CrossRefGoogle Scholar - Zellner, A., & Highfield, R. A. (1988). Calculation of maximum entropy distributions and approximation of marginal posterior distributions.
*Journal of Econometrics*,*37*, 195–209.CrossRefGoogle Scholar