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## Highlights

• Satorra–Bentler scaled chi-squared statistic
• Adjustment for nonnormal data
• Robust standard errors
• Adjusted goodness-of-fit statistics and tests
• Adjusted posthoc tests and comparisons

## 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.