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Original Articles

Noncentral Chi-Square Versus Normal Distributions in Describing the Likelihood Ratio Statistic: The Univariate Case and Its Multivariate Implication

Pages 109-136 | Published online: 19 Mar 2008

References

  • Bentler , P. M. 1990 . Comparative fit indexes in structural models . Psychological Bulletin. , 107 : 238 – 246 .
  • Bentler , P. M. and Dijkstra , T. K. 1985 . “ Efficient estimation via linearization in structural models ” . In Multivariate analysis VI , Edited by: Krishnaiah , P. R. 9 – 42 . Amsterdam : North-Holland .
  • Bollen , K. A. 1989 . Structural equations with latent variables , New York : Wiley .
  • Bollen , K. A. 2002 . Latent variables in psychology and the social sciences . Annual Review of Psychology. , 53 : 605 – 634 .
  • Boomsma , A. 2000 . Reporting on structural equation analyses . Structural Equation Modeling. , 7 : 461 – 483 .
  • Browne , M. W. 1984 . Asymptotic distribution-free methods for the analysis of covariance structures . British Journal of Mathematical and Statistical Psychology. , 37 : 62 – 83 .
  • Browne , M. W. and Arminger , G. 1995 . “ Specification and estimation of mean and covariance structure models ” . In Handbook of statistical modeling for the social and behavioral sciences , Edited by: Arminger , G. , Clogg , C. C. and Sobel , M. E. 185 – 249 . New York : Plenum .
  • Browne , M. W. and Cudeck , R. 1993 . “ Alternative ways of assessing model fit ” . In Testing structural equation models , Edited by: Bollen , K. A. and Long , J. S. 136 – 162 . Newbury Park , CA : Sage .
  • Cohen , J. J. 1977 . Statistical power analysis for the behavioral sciences, , 2nd ed. , Hillsdale , NJ : Lawrence Erlbaum Associates, Inc .
  • Ferguson , T. S. 1996 . A course in large sample theory , London : Chapman & Hall .
  • Hancock , G. R. 2001 . Effect size, power, and sample size determination for structured means modeling and MIMIC approaches to between-groups hypothesis testing of means on a single latent construct . Psychometrika , 66 : 373 – 388 .
  • Holzinger , K. J. and Swineford , F. 1939 . A study in factor analysis: The stability of a bi-factor solution , Chicago : University of Chicago Supplementary Educational Monographs . (Monograph No. 48)
  • Howell , D. C. 2007 . Statistical methods for psychology, , 6th ed. , Belmont , CA : Thomson Wadsworth .
  • Kano , Y. 2001 . “ Structural equation modeling for experimental data ” . In Structural equation modeling: Present and future , Edited by: Cudeck , R. , du Toit , S. H.C. and Sörbom , D. 381 – 402 . Lincolnwood , IL : Scientific Software International .
  • Kim , K. 2005 . The relationship among fit indexes, power, and sample size in structural equation modeling . Structural Equation Modeling. , 12 : 368 – 390 .
  • Li , L. and Bentler , P. M. 2006 . Robust statistical tests for evaluating the hypothesis of close fit of misspecified mean and covariance structural models , UCLA Statistics Electronic Publications . http://preprints.stat.ucla.edu/
  • MacCallum , R. C. and Austin , J. T. 2000 . Applications of structural equation modeling in psychological research . Annual Review of Psychology. , 51 : 201 – 226 .
  • MacCallum , R. C. , Browne , M. W. and Sugawara , H. M. 1996 . Power analysis and determination of sample size for covariance structure modeling . Psychological Methods , 1 : 130 – 149 .
  • Olkin , I. and Siotani , M. 1976 . “ Asymptotic distribution of functions of a correlation matrix ” . In Essays in probability and statistics , 235 – 251 . Tokyo : Shinko Tsusho .
  • Olsson , U. H. , Foss , T. and Breivik , E . 2004 . Two equivalent discrepancy functions for maximum likelihood estimation: Do their test statistics follow a non-central chi-square distribution under model misspecification? . Sociological Methods & Research , 32 : 453 – 500 .
  • Satorra , A. 1989 . Alternative test criteria in covariance structure analysis: A unified approach . Psychometrika. , 54 : 131 – 151 .
  • Satorra , A. and Saris , W. 1985 . Power of the likelihood ratio test in covariance structure analysis . Psychometrika. , 50 : 83 – 90 .
  • Satorra , A. , Saris , W. E. and de Pijper , W. M. 1991 . A comparison of several approximations to the power function of the likelihood ratio test in covariance structure analysis . Statistica Neerlandica. , 45 : 173 – 185 .
  • Shapiro , A. 1983 . Asymptotic distribution theory in the analysis of covariance structures (a unified approach) . South African Statistical Journal. , 17 : 33 – 81 .
  • Shapiro , A. and Browne , M. 1987 . Analysis of covariance structures under elliptical distributions . Journal of the American Statistical Association. , 82 : 1092 – 1097 .
  • Snedecor , G. W. and Cochran , W. G. 1989 . Statistical methods, , 8th, ed. , Ames : Iowa State University Press .
  • Steiger , J. H. and Lind , J. M. Statistically based tests for the number of common factors . Paper presented at the annual meeting of the Psychometric Society . Iowa City , IA . June .
  • Steiger , J. H. , Shapiro , A. and Browne , M. W. 1985 . On the multivariate asymptotic distribution of sequential chi-square statistics . Psychometrika , 50 : 253 – 264 .
  • Sugiura , N. 1969 . Asymptotic expansions of the distributions of the likelihood ratio criteria for covariance matrix . Annals of Mathematical Statistics. , 40 : 2051 – 2063 .
  • Sugiura , N. and Fujikoshi , Y. 1969 . Asymptotic expansions of the non-null distributions of the likelihood ratio criteria for multivariate linear hypothesis and independence . Annals of Mathematical Statistics. , 40 : 942 – 952 .
  • van der Vaart , A. W. 1998 . Asymptotic statistics , Cambridge , UK : Cambridge University Press .
  • Vuong , Q. H. 1989 . Likelihood ratio tests for model selection and nonnested hypotheses . Econometrica , 57 : 307 – 333 .
  • Wald , A. 1943 . Tests of statistical hypotheses concerning several parameters when the number of observations is large . Transactions of the American Mathematical Society. , 54 : 426 – 482 .
  • Yanagihara , H. , Tonda , T. and Matsumoto , C. 2005 . The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption . Journal of Multivariate Analysis. , 96 : 237 – 264 .
  • Yuan , K.-H. and Bentler , P. M. 2006 . Mean comparison: Manifest variable versus latent variable . Psychometrika. , 71 : 139 – 159 .
  • Yuan , K.-H. and Bentler , P. M. 2007 . “ Structural equation modeling ” . In Handbook of statistics 26: Psychometrics , Edited by: Rao , C. R. and Sinharay , S. 297 – 358 . Amsterdam : North-Holland .
  • Yuan , K.-H. , Bentler , P. M. and Chan , W. 2004 . Structural equation modeling with heavy tailed distributions . Psychometrika. , 69 : 421 – 436 .
  • Yuan , K.-H. , Bentler , P. M. and Zhang , W. 2005 . The effect of skewness and kurtosis on mean and covariance structure analysis: The univariate case and its multivariate implication . Sociological Methods & Research. , 34 : 249 – 258 .
  • Yuan , K.-H. and Chan , W. 2005 . On nonequivalence of several procedures of structural equation modeling . Psychometrika. , 70 : 791 – 798 .
  • Yuan , K.-H. and Hayashi , K. 2003 . Bootstrap approach to inference and power analysis based on three statistics for covariance structure models . British Journal of Mathematical and Statistical Psychology. , 56 : 93 – 110 .
  • Yuan , K.-H. , Hayashi , K. and Bentler , P. M. 2007 . Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses . Journal of Multivariate Analysis. , 98 : 1262 – 1282 .
  • Yuan , K.-H. and Marshall , L. L. 2004 . A new measure of misfit for covariance structure models . Behaviormetrika , 31 : 67 – 90 .

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