IDEAS home Printed from https://ideas.repec.org/a/sae/jedbes/v44y2019i2p127-143.html
   My bibliography  Save this article

A Note on the Solution Multiplicity of the Vale–Maurelli Intermediate Correlation Equation

Author

Listed:
  • Oscar L. Olvera Astivia
  • Bruno D. Zumbo

    (The University of British Columbia)

Abstract

The Vale and Maurelli algorithm is a widely used method that allows researchers to generate multivariate, nonnormal data with user-specified levels of skewness, excess kurtosis, and a correlation structure. Before obtaining the desired correlation structure, a transitional step requires the user to calculate the roots of a cubic polynomial referred to as the intermediate correlation equation. The Cardano method and a corollary of Rouché’s theorem were used to derive closed-form solutions to this equation. These solutions highlight the fact that three real-valued roots are possible, and solutions can either contain multiple roots within the allowable correlation range or can all be greater than 1 for many combinations of skewness and excess kurtosis. It was also found that large values of excess kurtosis exacerbate the issue of multiple solutions or solutions greater than 1, bringing further restrictions into the combinations of higher order moments that researchers can simulate from. A small computer study on the power of the t test for the correlation coefficient also uncovered that different values of the intermediate correlation can influence the results from Monte Carlos simulations. This note is intended to inform both researchers who conduct simulations with nonnormal data and users who inform their data analysis practice from simulation studies.

Suggested Citation

  • Oscar L. Olvera Astivia & Bruno D. Zumbo, 2019. "A Note on the Solution Multiplicity of the Vale–Maurelli Intermediate Correlation Equation," Journal of Educational and Behavioral Statistics, , vol. 44(2), pages 127-143, April.
    • Handle: RePEc:sae:jedbes:v:44:y:2019:i:2:p:127-143
    DOI: 10.3102/1076998618803381
    as

    Download full text from publisher

    File URL: https://journals.sagepub.com/doi/10.3102/1076998618803381
    Download Restriction: no
    File URL: https://libkey.io/10.3102/1076998618803381?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. C. Vale & Vincent Maurelli, 1983. "Simulating multivariate nonnormal distributions," Psychometrika, Springer;The Psychometric Society, vol. 48(3), pages 465-471, September.
    2. Allen Fleishman, 1978. "A method for simulating non-normal distributions," Psychometrika, Springer;The Psychometric Society, vol. 43(4), pages 521-532, December.
    3. Henry Kaiser & Kern Dickman, 1962. "Sample and population score matrices and sample correlation matrices from an arbitrary population correlation matrix," Psychometrika, Springer;The Psychometric Society, vol. 27(2), pages 179-182, June.
    4. Headrick, Todd C., 2002. "Fast fifth-order polynomial transforms for generating univariate and multivariate nonnormal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 40(4), pages 685-711, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Max Auerswald & Morten Moshagen, 2015. "Generating Correlated, Non-normally Distributed Data Using a Non-linear Structural Model," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 920-937, December.
    2. Mohan D. Pant & Todd C. Headrick, 2017. "Simulating Uniform- and Triangular- Based Double Power Method Distributions," Journal of Statistical and Econometric Methods, SCIENPRESS Ltd, vol. 6(1), pages 1-1.
    3. Al-Subaihi, Ali A., 2004. "Simulating Correlated Multivariate Pseudorandom Numbers," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 9(i04).
    4. repec:jss:jstsof:09:i04 is not listed on IDEAS
    5. Hakan Demirtas, 2016. "A Note on the Relationship Between the Phi Coefficient and the Tetrachoric Correlation Under Nonnormal Underlying Distributions," The American Statistician, Taylor & Francis Journals, vol. 70(2), pages 143-148, May.
    6. repec:jss:jstsof:19:i03 is not listed on IDEAS
    7. Hakan Demirtas & Robab Ahmadian & Sema Atis & Fatma Ezgi Can & Ilker Ercan, 2016. "A nonnormal look at polychoric correlations: modeling the change in correlations before and after discretization," Computational Statistics, Springer, vol. 31(4), pages 1385-1401, December.
    8. Gary van Vuuren & Riaan de Jongh, 2017. "A comparison of risk aggregation estimates using copulas and Fleishman distributions," Applied Economics, Taylor & Francis Journals, vol. 49(17), pages 1715-1731, April.
    9. Nagahara, Yuichi, 2004. "A method of simulating multivariate nonnormal distributions by the Pearson distribution system and estimation," Computational Statistics & Data Analysis, Elsevier, vol. 47(1), pages 1-29, August.
    10. Emanuela Raffinetti & Pier Alda Ferrari, 2021. "A dependence measure flow tree through Monte Carlo simulations," Quality & Quantity: International Journal of Methodology, Springer, vol. 55(2), pages 467-496, April.
    11. M Hashem Pesaran & Takashi Yamagata, 2012. "Testing CAPM with a Large Number of Assets," Discussion Papers 12/05, Department of Economics, University of York.
    12. Beasley, T. Mark & Zumbo, Bruno D., 2003. "Comparison of aligned Friedman rank and parametric methods for testing interactions in split-plot designs," Computational Statistics & Data Analysis, Elsevier, vol. 42(4), pages 569-593, April.
    13. Pasquale Dolce & Cristina Davino & Domenico Vistocco, 2022. "Quantile composite-based path modeling: algorithms, properties and applications," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 16(4), pages 909-949, December.
    14. Headrick, Todd C. & Mugdadi, Abdel, 2006. "On simulating multivariate non-normal distributions from the generalized lambda distribution," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3343-3353, July.
    15. Ke-Hai Yuan & Peter Bentler, 2002. "On robusiness of the normal-theory based asymptotic distributions of three reliability coefficient estimates," Psychometrika, Springer;The Psychometric Society, vol. 67(2), pages 251-259, June.
    16. Jeff Jones & Niels Waller, 2015. "The Normal-Theory and Asymptotic Distribution-Free (ADF) Covariance Matrix of Standardized Regression Coefficients: Theoretical Extensions and Finite Sample Behavior," Psychometrika, Springer;The Psychometric Society, vol. 80(2), pages 365-378, June.
    17. Njål Foldnes & Steffen Grønneberg, 2015. "How General is the Vale–Maurelli Simulation Approach?," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 1066-1083, December.
    18. Fosgerau, Mogens & Mabit, Stefan L., 2013. "Easy and flexible mixture distributions," Economics Letters, Elsevier, vol. 120(2), pages 206-210.
    19. Reinartz, Werner & Haenlein, Michael & Henseler, Jörg, 2009. "An empirical comparison of the efficacy of covariance-based and variance-based SEM," International Journal of Research in Marketing, Elsevier, vol. 26(4), pages 332-344.
    20. Pesaran, M. H. & Yamagata, T., 2012. "Testing CAPM with a Large Number of Assets (Updated 28th March 2012)," Cambridge Working Papers in Economics 1210, Faculty of Economics, University of Cambridge.
    21. Headrick, Todd C. & Sheng, Yanyan & Hodis, Flaviu-Adrian, 2007. "Numerical Computing and Graphics for the Power Method Transformation Using Mathematica," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 19(i03).
    22. Mahul, Olivier, 2002. "Hedging Price Risk in the Presence of Crop Yield and Revenue Insurance," 2002 International Congress, August 28-31, 2002, Zaragoza, Spain 24881, European Association of Agricultural Economists.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:sae:jedbes:v:44:y:2019:i:2:p:127-143. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: SAGE Publications (email available below). General contact details of provider: .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.