Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

AbstractA scaled difference test statistic $\tilde{T}{}_{d}$ that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (Psychometrika 66:507–514, 2001). The statistic $\tilde{T}_{d}$ is asymptotically equivalent to the scaled difference test statistic $\bar{T}_{d}$ introduced in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neudecker, pp. 233–247, 2000), which requires more involved computations beyond standard output of SEM software. The test statistic $\tilde{T}_{d}$ has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic $\bar{T}_{d}$ that avoids negative chi-square values.

[1]  Kenneth A. Bollen,et al.  Latent curve models: A structural equation perspective , 2005 .

[2]  Sik-Yum Lee.  Handbook of latent variable and related models , 2007 .

[3]  K. Bollen.  Latent variables in psychology and the social sciences. , 2002, Annual review of psychology.

[4]  J. Fraumeni,et al.  Cigarette smoking and cancers of the urinary tract: geographic variation in the United States. , 1968, Journal of the National Cancer Institute.

[5]  Albert Satorra,et al.  Scaled and Adjusted Restricted Tests in Multi Sample Analysis of Moment Structures , 1999 .

[6]  Albert Satorra,et al.  A scaled difference chi-square test statistic for moment structure analysis , 1999 .

[7]  A. Satorra.  Alternative test criteria in covariance structure analysis: A unified approach , 1989 .

[8]  John Fox,et al.  TEACHER'S CORNER: Structural Equation Modeling With the sem Package in R , 2006 .

[9]  A. Satorra,et al.  Smoking and Cancers: Case-Robust Analysis of a Classic Data Set , 2009, Structural equation modeling : a multidisciplinary journal.

[10]  Karl G. Jöreskog,et al.  Lisrel 8: User's Reference Guide , 1997 .

[11]  C. Stein,et al.  Structural equation modeling. , 2012, Methods in molecular biology.

[12]  Clifford C. Clogg,et al.  Latent Variables Analysis: Applications for Developmental Research. , 1995 .

[13]  Douglas G. Bonett,et al.  Estimating p-values for Mardia's coefficients of multivariate skewness and kurtosis , 2002, Comput. Stat..

[14]  M. Browne.  Asymptotically distribution-free methods for the analysis of covariance structures. , 1984, The British journal of mathematical and statistical psychology.

[15]  A. Satorra,et al.  Corrections to test statistics and standard errors in covariance structure analysis. , 1994 .

[16]  James B. Grace,et al.  Structural Equation Modeling and Natural Systems , 2006 .

[17]  A. Satorra,et al.  Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic - eScholarship , 2008 .

[18]  Peter M. Bentler,et al.  EQS : structural equations program manual , 1989 .

[19]  Barbara M. Byrne,et al.  Cross-Cultural Comparisons and the Presumption of Equivalent Measurement and Theoretical Structure , 1999 .

[20]  Albert Satorra,et al.  Innovations in multivariate statistical analysis : a festschrift for Heinz Neudecker , 2000 .