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Satorra–Bentler adjustments

Satorra–Bentler adjustments


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.

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