Computer program for Satorra-Bentler scaled difference chi square test


This PC program is designed to test for a difference between the fit of two competing confirmatory factor analytic (CFA) models and implements a method described by Satorra and Bentler (2001). One of the models must be a more constrained version of the other (e.g., the more constrained model does not permit a correlation between two factors that is permitted in the less constrained model). Such models are termed nested models and can be compared using a test on the difference between chi square for the models. However, if the data depart markedly from multivariate normality, the Satorra-Bentler scaled chi square statistic (S-B c 2) should be used to provide an improved estimate of the fit of a model. The problem here, however, is that the difference between these scaled statistics is not itself distributed as chi square (Satorra & Bentler, 2001). To deal with this, Satorra & Bentler (2001) have provided a correction that, when applied, allows you to validly compare models. To run such a test requires entry of the following: the normal chi square values for the models being compared, the Satorra-Bentler scaled chi square values for these models, and the degrees of freedom for the models (the output from the structural equation modeling program EQS provides all of these statistics). The output consists of the scaled difference between the models and the significance of this difference (the scaled difference is evaluated against a chi square distribution with k degrees of freedom, where k = the df for the more constrained model minus the df for the less constrained model). The program has extensive error checking which will prevent inexperienced users going astray.

Essential background information and details of the correction can be found in:

Satorra, A., & Bentler, P. M. (2001). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66, 507-514.

A pre-print of this article (as a pdf) can be found here.

The program was written to accompany Crawford & Henry (2003). This paper, and the others listed below, contain examples of the use of the program:

Crawford, J. R., & Henry, J. D.  (2004). The Positive and Negative Affect Schedule (PANAS): Construct validity, measurement properties and normative data in a large non-clinical sample.  British Journal of Clinical Psychology, 43, 245-265.

reprint as pdf

Crawford, J. R., & Henry, J. D.  (2003).  The Depression Anxiety Stress Scales: Normative data and latent structure in a large non-clinical sample. British Journal of Clinical Psychology, 42, 111-131.

reprint as pdf

Henry, J. D., Crawford, J. R., Bedford, A., Crombie, C., & Taylor, E. P. (2002). The personal disturbance scale (sAD): normative data and latent structure in a large non-clinical sample. Personality and Individual Differences, 33, 1343-1360.

reprint as pdf

Click here to download the program. Your web browser is most probably configured to recognise that the file as an executable. If you encounter any problems (i.e. the browser treats it as a text file), try holding down the shift key when clicking. Click here to download a zipped version

Once downloaded, the program can be run by any of the normal Windows procedures i.e., by clicking on file in file manager; by using the Windows start menu; or by placing the program, or a shortcut to it, on the desktop etc.

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The author of this software (John R Crawford) and the University of Aberdeen make no representations about the suitability of the software or about any content or information made accessible by the software, for any purpose.

The software is provided 'as is' without express or implied warranties, including warranties of merchantability and fitness for a particular purpose or noninfringement.

The software is provided gratuitously and, accordingly, the author shall not be liable under any theory or any damages suffered by you or any user of the software.

If there are any problems please e-mail me at Further contact details are available in the footer of this page.