What's this about?

Stata's linear sem now provides the Satorra–Bentler scaled chi-squared test for model goodness of fit versus the saturated model. Why do you care? The likelihood-ratio test comparing your estimated model to the saturated model is derived under the assumption that the observed variables in your model are normally distributed. If they are not, that test is not appropriate. The Satorra–Bentler scaled chi-squared test is robust to nonnormality. Because many other goodness-of-fit statistics are derived from the model using the chi-squared test, they too become robust to nonnormality.

What's more

The same adjustment that gives you the Satorra–Bentler scaled chi-squared test makes a host of other things robust to nonnormality: standard errors, p-values, and confidence intervals reported by sem and standard errors, p-values, and confidence intervals for most posthoc comparisons and tests, including joint tests, nonlinear tests, linear and nonlinear expressions of parameters, estimated marginal means and marginal effects, equation-level Wald tests, direct and indirect effects, and tests of standardized parameters.

What else might you like to know

Stata's sem already had an adjustment that makes everything in "What's more" true. It is often called the Huber or White method, or just called the linearized estimator. Whatever you call it, this estimator and the Satorra–Bentler adjustment are making your inferences robust to similar things. They are derived and computed differently, so they produce different estimates. As samples become very large, however, they converge to the same estimates.

Let's see it work

Suppose we have six variables, dep1–dep6, that are each intended to measure different aspects of depression. The measurements of depression are continuous with scores ranging from 0 to 100. They do not follow a multivariate normal distribution. We fit a one-factor CFA model for depression.

To fit this model and request the Satorra–Bentler scaled chi-squared statistic using sem, we type

. sem (Depression -> dep1-dep6), vce(sbentler)

Endogenous variables

Measurement:  dep1 dep2 dep3 dep4 dep5 dep6

Exogenous variables

Latent:       Depression

Fitting target model:

Iteration 0:   log pseudolikelihood = -8883.5322
Iteration 1:   log pseudolikelihood = -8882.5925
Iteration 2:   log pseudolikelihood =  -8882.576
Iteration 3:   log pseudolikelihood =  -8882.576

Structural equation model                      Number of obs     =        500
Estimation method  = ml
Log pseudolikelihood=  -8882.576

 ( 1)  [dep1]Depression = 1

             
           Satorra–Bentler
             
      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]

Measurement  

  dep1 <-    

  Depression 
          1  (constrained)
       _cons 
    31.1175   .3406304    91.35   0.000     30.44987    31.78512

  dep2 <-    

  Depression 
   .5134465   .0452706    11.34   0.000     .4247177    .6021753
       _cons 
   23.17739   .2970506    78.03   0.000     22.59518     23.7596

  dep3 <-    

  Depression 
   1.242734   .0485588    25.59   0.000     1.147561    1.337908
       _cons 
   34.95304   .3448572   101.36   0.000     34.27713    35.62895

  dep4 <-    

  Depression 
   1.317912   .0514062    25.64   0.000     1.217158    1.418666
       _cons 
   32.48394   .3546712    91.59   0.000      31.7888    33.17908

  dep5 <-    

  Depression 
   1.331852   .0552345    24.11   0.000     1.223595     1.44011
       _cons 
    34.6336   .3791664    91.34   0.000     33.89044    35.37675

  dep6 <-    

  Depression 
   1.297788   .0509321    25.48   0.000     1.197963    1.397613
       _cons 
    32.0842   .3508572    91.45   0.000     31.39654    32.77187

  var(e.dep1)
   24.97784   2.618343                      20.33885    30.67493
  var(e.dep2)
   35.35251   4.519579                      27.51692    45.41932
  var(e.dep3)
   8.502031   .5768576                      7.443363    9.711274
  var(e.dep4)
   5.590374   .5799466                      4.561808    6.850854
  var(e.dep5)
   13.34413   .9658573                      11.57923    15.37804
  var(e.dep6)
   5.980518   .5544732                      4.986791    7.172267
var(Depres~n)
   32.92067   3.276629                      27.08618    40.01195

LR test of model vs. saturated: chi2(9)   =     15.89, Prob > chi2 = 0.0693
Satorra–Bentler scaled test:    chi2(9)   =     15.08, Prob > chi2 = 0.0889

Here the Satorra–Bentler scaled statistic is 15.08 with a p-value of 0.0889, while the normal-theory statistic is 15.89 with a p-value of 0.0693. With a 0.05 significance level, we fail to reject the null hypothesis, indicating good fit using either the normal-theory statistic or the Satorra–Bentler scaled statistic. Clearly, for other models, the two statistics can lead us to draw different conclusions about the fit of our model.

By specifying the vce(sbentler) option, we also obtain standard errors that are robust to nonnormality. The robust standard errors are used in computing test statistics and confidence intervals for each of the parameters.

Many other goodness-of-fit statistics are a function of the model chi-squared statistic. estat gof automatically reports adjusted versions of each of these statistics as well.

. estat gof, stats(all)


Fit statistic        
      Value   Description

Likelihood ratio     
      Value   Description
          chi2_ms(9) 
     15.887   model vs. saturated
            p > chi2 
      0.069
         chi2_bs(15) 
   2993.029   baseline vs. saturated
            p > chi2 
      0.000
                     
      
  Satorra–Bentler    

        chi2sb_ms(9) 
     15.075
            p > chi2 
      0.089
       chi2sb_bs(15) 
   2920.693
            p > chi2 
      0.000

Population error     

               RMSEA 
      0.039   Root mean squared error of approximation
 90% CI, lower bound 
      0.000
         upper bound 
      0.070
              pclose 
      0.680   Probability RMSEA <= 0.05
                     

  Satorra–Bentler    

            RMSEA_SB 
      0.037   Root mean squared error of approximation

Information criteria 

                 AIC 
  17801.152   Akaike's information criterion
                 BIC 
  17877.015   Bayesian information criterion

Baseline comparison  

                 CFI 
      0.998   Comparative fit index
                 TLI 
      0.996   Tucker-Lewis index
                     

  Satorra–Bentler    

              CFI_SB 
      0.998   Comparative fit index
              TLI_SB 
      0.997   Tucker-Lewis index

Size of residuals    

                SRMR 
      0.011   Standardized root mean squared residual
                  CD 
      0.969   Coefficient of determination

In this case, the RMSEA, CFI, and TLI goodness-of-fit statistics that were computed using the Satorra–Bentler scaled chi-squared statistic are similar to those based on the normal-theory statistic. Based on the goodness-of-fit statistics reported here, our model appears to fit well.

Tell me more

Read more in Stata Structural Equation Modeling Reference Manual.

Read the overview from the Stata News.