Generating Nonnormal Multivariate Data Using Copulas: Applications to SEM

REFERENCES

• Bentler , P. M. 2006 . EQS 6 structural equations program manual Encino , CA : Multivariate Software. .
   Google Scholar
• Braeken , J. 2011 . A boundary mixture approach to violations of conditional independence . Psychometrika , 76 : 57 – 76 .
   Web of Science ®Google Scholar
• Braeken , J. , Tuerlinckx , T. and De Boeck , P. 2007 . Copula functions for residual dependency . Psychometrika , 72 : 393 – 411 .
   Web of Science ®Google Scholar
• Browne , M. W. 1984 . Asymptotically distribution-free methods for the analysis of covariance structures . British Journal of Mathematical and Statistical Psychology , 37 : 62 – 83 .
   PubMed Web of Science ®Google Scholar
• Chen , L. A. and Portnoy , S. 1996 . Two-stage regression quantiles and two-stage trimmed least squares estimators for structural equation models . Communications in Statistics: Theory and Methods , 25 : 1005 – 1032 .
   Web of Science ®Google Scholar
• Clayton , D. G. 1978 . A model for association in bivariate life tables and its applications in epidemiological studies of familial tendency in chronic disease incidence . Biometrika , 65 : 141 – 151 .
   Web of Science ®Google Scholar
• Clemen , R. T. and Reilly , T. 1999 . Correlations and copulas for decision and risk analysis . Management Science , 45 : 208 – 224 .
   Web of Science ®Google Scholar
• Curran , P. J. , West , S. G. and Finch , J. F. 1996 . The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis . Psychological Methods , 1 : 16 – 29 .
   Web of Science ®Google Scholar
• Farlie , D. J. G. 1960 . The performance of some correlation coefficients for a general bivariate distribution . Biometrika , 47 : 307 – 323 .
   Web of Science ®Google Scholar
• Finch , J. F. , West , S. G. and MacKinnon , D. P. 1997 . Effects of sample size and nonnormality on the estimation of mediated effects in latent variable models . Structural Equation Modeling: A Multidisciplinary Journal , 4 : 87 – 107 .
   Web of Science ®Google Scholar
• Fleishman , A. I. 1978 . A method for simulation nonnormal distributions . Psychometrika , 43 : 521 – 532 .
   Web of Science ®Google Scholar
• Frank , M. J. 1979 . On the simultaneous associativity of f(x,y) and x+y-f(x,y) . Aequationes Mathematicae , 19 : 194 – 226 .
   Google Scholar
• Gumbel , E. J. 1958 . Statistics of extremes. New York , NY : Columbia University Press. .
   Google Scholar
• Gumbel , E. J. 1960 . Bivariate exponential distributions . Journal of the American Statistical Association , 55 : 698 – 707 .
   Web of Science ®Google Scholar
• Headrick , T. C. 2002 . Fast fifth-order polynomial transforms for generating univariate and multivariate nonnormal distributions . Computational Statistics & Data Analysis , 40 : 685 – 711 .
   Web of Science ®Google Scholar
• Headrick , T. C. 2010 . Statistical simulation: Power methods polynomials and other transformations. Boca Raton , FL : Chapman & Hall/CRC. .
   Google Scholar
• Headrick , T. C. and Sawilowsky , S. S. 1999 . Simulating correlated multivariate nonnormal distributions: Extending the Fleishman power method . Psychometrika , 64 : 25 – 35 .
   Web of Science ®Google Scholar
• Headrick , T. C. and Sawilowsky , S. S. 2000 . Weighted simplex procedures for determining boundary points and constants for the univariate and multivariate power methods . Journal of Educational and Behavioral Statistics , 25 : 417 – 436 .
   Web of Science ®Google Scholar
• Hörmann , W. , Leydold , J. and Derflinger , G. 2004 . Automatic nonuniform random variate generation. Berlin , , Germany : Springer. .
   Google Scholar
• Hu , L. , Bentler , P. M. and Kano , Y. 1992 . Can test statistics in covariance structure analysis be trusted? . Psychological Bulletin , 112 : 351 – 362 .
   PubMed Web of Science ®Google Scholar
• Joe , H. 1997 . Multivariate models and dependence concepts. London , , UK : Chapman & Hall. .
   Google Scholar
• Kruskal , H. 1958 . Ordinal measures of association . Journal of the American Statistical Society , 53 : 814 – 861 .
   Web of Science ®Google Scholar
• Lei , M. and Lomax , R. G. 2005 . The effect of varying degrees of nonnormality in structural equation modeling . Structural Equation Modeling: A Multidisciplinary Journal , 12 : 1 – 27 .
   Web of Science ®Google Scholar
• Mair , P. , Wu , E. and Bentler , P. 2010 . EQS goes R: Simulations for SEM using the package REQS . Structural Equation Modeling: A Multidisciplinary Journal , 17 : 333 – 349 .
   Web of Science ®Google Scholar
• Mari , D. D. and Kotz , S. 2001 . Correlation and dependence. London , , UK : Imperial College Press. .
   Google Scholar
• Mattson , S. 1997 . How to generate non-normal data for simulation of structural equation models . Multivariate Behavioral Research , 32 : 355 – 373 .
   PubMed Web of Science ®Google Scholar
• Morgenstern , D. 1956 . Einfache Beispiele zweidimensionaler Verteilungen [Simple examples of two-dimensional distributions] . Mitteilungsblatt für Mathematische Statistik , 8 : 234 – 235 .
   Google Scholar
• Nelsen , R. B. 2005 . Dependence modeling with Archimedean copulas Lewis & Clark College, Department of Mathematical Sciences. . (Tech. Rep.)
   Google Scholar
• Nelsen , R. B. 2006 . An introduction to copulas, , 2nd ed. New York : Springer. .
   Google Scholar
• R Development Core Team . 2012 . R: A language and environment for statistical computing. Vienna , , Austria : R Foundation for Statistical Computing. . ISBN 3–900051-07–0
   Google Scholar
• Reinartz , W. J. , Echambadi , R. and Chin , W. W. 2002 . Generating nonnormal data for simulation of structural equation models using Mattson's method . Multivariate Behavioral Research , 37 : 227 – 244 .
   Web of Science ®Google Scholar
• Ruscio , J. and Kaczetow , W. 2008 . Simulating multivariate nonnormal data using an iterative algorithm . Multivariate Behavioral Research , 43 : 355 – 381 .
   PubMed Web of Science ®Google Scholar
• Satorra , A. 1989 . Alternative test criteria in covariance structure analysis: A unified approach . Psychometrika , 54 : 131 – 151 .
   Web of Science ®Google Scholar
• Satorra , A. and Bentler , P. M. 1994 . “ Corrections to test statistics and standard errors in covariance structure analysis. ” . In Latent variable analysis in developmental research , Edited by: von Eye , A. and Clogg , C. C. 285 – 305 . Thousand Oaks , CA : Sage. .
   Google Scholar
• Schölzel , C. and Friederichs , P. 2008 . Multivariate non-normally distributed random variables in climate research: Introduction to the copula approach . Nonlinear Processes in Geophysics , 15 : 761 – 772 .
   Web of Science ®Google Scholar
• Schweizer , B. and Wolff , E. F. 1981 . On nonparametric measures of dependence for random variables . The Annals of Statistics , 9 : 879 – 885 .
   Web of Science ®Google Scholar
• Sklar , A. 1959 . Fonctions de répartition à n dimensions et leurs marges [Distribution functions of n dimensions and their origins] . Publications de l'Institut de Statistique de l'Université de Paris , 8 : 229 – 231 .
   Google Scholar
• Vale , C. D. and Maurelli , V. A. 1983 . Simulating multivariate nonnormal distributions . Psychometrika , 48 : 465 – 471 .
   Web of Science ®Google Scholar
• Yan , J. 2007 . Enjoy the joy of copulas: With a copula package . Journal of Statistical Software , 21 : 1 – 21 .
   Web of Science ®Google Scholar
• Yuan , K.-H. and Bentler , P. M. 2007 . “ Robust procedures in structural equation modeling. ” . In Handbook of latent variable and related models , Edited by: Lee , S.-Y. 367 – 397 . Amsterdam , , The Netherlands : Elsevier. .
   Google Scholar
• Yule , G. U. 1912 . On the methods of measuring association between two attributes . Journal of the Royal Statistical Society , 75 : 579 – 652 .
   Google Scholar