Principles and Practice of Scaled Difference Chi-Square Testing

We highlight critical conceptual and statistical issues and how to resolve them in conducting Satorra–Bentler (SB) scaled difference chi-square tests. Concerning the original (Satorra & Bentler, 2001) and new (Satorra & Bentler, 2010) scaled difference tests, a fundamental difference exists in how to compute properly a model's scaling correction factor (c), depending on the particular structural equation modeling software used. Because of how LISREL 8 defines the SB scaled chi-square, LISREL users should compute c for each model by dividing the model's normal theory weighted least-squares (NTWLS) chi-square by its SB chi-square, to recover c accurately with both tests. EQS and Mplus users, in contrast, should divide the model's maximum likelihood (ML) chi-square by its SB chi-square to recover c. Because ML estimation does not minimize the NTWLS chi-square, however, it can produce a negative difference in nested NTWLS chi-square values. Thus, we recommend the standard practice of testing the scaled difference in ML chi-square values for models M 1 and M 0 (after properly recovering c for each model), to avoid an inadmissible test numerator. We illustrate the difference in computations across software programs for the original and new scaled tests and provide LISREL, EQS, and Mplus syntax in both single- and multiple-group form for specifying the model M 10 that is involved in the new test.

[1]  C. Carver,et al.  Optimism, coping, and health: assessment and implications of generalized outcome expectancies. , 2009, Health psychology : official journal of the Division of Health Psychology, American Psychological Association.

[2]  J. Henry,et al.  The positive and negative affect schedule (PANAS): construct validity, measurement properties and normative data in a large non-clinical sample. , 2004, The British journal of clinical psychology.

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

[4]  Albert Maydeu-Olivares,et al.  Assessing the dimensionality of optimism and pessimism using a multimeasure approach , 1994, Cognitive Therapy and Research.

[5]  S. Moreno,et al.  English and Spanish versions of the Body Shape Questionnaire: measurement equivalence across ethnicity and clinical status. , 2008, The International journal of eating disorders.

[6]  Y Kano,et al.  Can test statistics in covariance structure analysis be trusted? , 1992, Psychological bulletin.

[7]  K. Jöreskog,et al.  LISREL 8: New Statistical Features , 1999 .

[8]  Rick H. Hoyle,et al.  Confirmatory Factor Analysis , 1983 .

[9]  S. West,et al.  The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis. , 1996 .

[10]  Michael W. Bridges,et al.  Distinguishing optimism from neuroticism (and trait anxiety, self-mastery, and self-esteem): a reevaluation of the Life Orientation Test. , 1994, Journal of personality and social psychology.

[11]  Sara J. Finney,et al.  Measuring basic needs satisfaction: Evaluating previous research and conducting new psychometric evaluations of the Basic Needs Satisfaction in General Scale , 2010 .

[12]  R. P. McDonald,et al.  Structural Equations with Latent Variables , 1989 .

[13]  F. Bryant,et al.  Distinguishing Hope and Optimism: Two Sides of a Coin, or Two Separate Coins? , 2004 .

[14]  K. Yuan,et al.  On the Likelihood Ratio Test for the Number of Factors in Exploratory Factor Analysis , 2007 .

[15]  D. Vogel,et al.  The mediating roles of anticipated risks, anticipated benefits, and attitudes on the decision to seek professional help: An attachment perspective. , 2006 .

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

[17]  D. Kaplan.  Structural Equation Modeling: Foundations and Extensions , 2000 .

[18]  S. Vrana,et al.  Trait anxiety, disgust sensitivity, and the hierarchic structure of fears. , 2008, Journal of anxiety disorders.

[19]  A. Shapiro,et al.  On the multivariate asymptotic distribution of sequential Chi-square statistics , 1985 .

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

[21]  E. S. Pearson,et al.  ON THE USE AND INTERPRETATION OF CERTAIN TEST CRITERIA FOR PURPOSES OF STATISTICAL INFERENCE PART I , 1928 .

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

[23]  B. Muthén,et al.  Computing the Strictly Positive Satorra-Bentler Chi-Square Test in Mplus , 2013 .

[24]  A. Satorra,et al.  Scaled test statistics and robust standard errors for non-normal data in covariance structure analysis: a Monte Carlo study. , 1991, The British journal of mathematical and statistical psychology.

[25]  Peter M. Bentler,et al.  Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic , 2008, Psychometrika.

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