How To Calculator Chi Square Satorra Bentler

Calculate the adjusted chi-square statistic for non-normal data using the Satorra-Bentler method

Comprehensive Guide to Calculating Satorra-Bentler Scaled Chi-Square

The Satorra-Bentler scaled chi-square statistic is a corrected test statistic used in structural equation modeling (SEM) when the assumption of multivariate normality is violated. This guide explains the theoretical foundation, calculation process, and practical interpretation of this important statistical adjustment.

Why the Satorra-Bentler Correction is Needed

Traditional chi-square statistics in SEM assume:

• Multivariate normality of observed variables
• Large sample sizes
• Correct model specification

When these assumptions are violated – particularly the normality assumption – the standard chi-square test becomes:

• Inflated (too large) with non-normal data
• More likely to reject true models (Type I error)
• Less reliable for model comparison

Key Advantages

• Robust to non-normality
• Maintains proper Type I error rates
• Works with ML, GLS, and WLS estimation
• Allows valid model comparisons

When to Use

• Non-normal continuous data
• Ordinal data with ≥5 categories
• Small to moderate sample sizes
• Models with complex structures

The Mathematical Foundation

The Satorra-Bentler scaled chi-square (T_SB) is calculated as:

T_SB = T_ML/c

where:
T_ML = maximum likelihood chi-square statistic
c = scaling factor derived from model’s information matrix

The scaling factor c is computed as:

c = (d_M + d_V)/(d_M * tr(UΓ)^-1)

Where:

• d_M = degrees of freedom for the model
• d_V = degrees of freedom for the variance-covariance matrix
• U = weight matrix
• Γ = asymptotic covariance matrix of sample moments

Step-by-Step Calculation Process

1. Estimate your SEM model

   Run your structural equation model using maximum likelihood estimation. Most SEM software (LISREL, Mplus, lavaan in R) will provide the standard chi-square statistic (T_ML) and degrees of freedom.

2. Obtain the scaling factor

   The scaling factor (c) can be:

   • Directly provided by your SEM software
   • Calculated from the model’s information matrix
   • Estimated using the formula above for advanced users
3. Compute the scaled chi-square

   Divide the original chi-square by the scaling factor: T_SB = T_ML/c

4. Adjust the degrees of freedom

   The adjusted degrees of freedom become: df_SB = df_M/c

5. Calculate the p-value

   Use the chi-square distribution with adjusted df to find the p-value for your scaled statistic.

6. Make model fit decision

   Compare p-value to your significance level (typically 0.05):

   • p > 0.05: Fail to reject null (model fits well)
   • p ≤ 0.05: Reject null (model doesn’t fit well)

Interpreting Your Results

Satorra-Bentler χ²	Adjusted df	p-value	Interpretation	Recommended Action
12.45	8.2	0.132	Good fit	Accept model as plausible
24.78	12.5	0.016	Marginal fit	Consider modifications with theoretical justification
45.32	18.7	0.0004	Poor fit	Significant misspecification likely; reconsider model
8.92	6.1	0.258	Excellent fit	Model fits data very well; proceed with interpretation

Note that interpretation should always consider:

• The substantive meaning of model parameters
• Other fit indices (CFI, RMSEA, SRMR)
• Sample size and model complexity
• Theoretical justification for the model

Comparison with Other Robust Methods

Method	Normality Assumption	Sample Size Requirements	Implementation Complexity	Best Use Case
Satorra-Bentler	Robust to non-normality	Moderate (n > 100)	Moderate	Continuous non-normal data
Bollen-Stine Bootstrap	No distributional assumptions	Large (n > 200)	High	Small samples, complex models
Yuan-Bentler T2*	Robust to non-normality	Moderate to large	High	Ordinal data with ≥4 categories
ADF (WLS)	No normality assumption	Very large (n > 1000)	Very high	Severely non-normal data
ML with Robust SE	Robust to mild non-normality	Moderate (n > 150)	Low	Slightly non-normal continuous data

Practical Example with Real Data

Consider a confirmatory factor analysis model with:

• Original χ² = 185.42, df = 84
• Scaling factor c = 1.28
• Significance level α = 0.05

Calculation steps:

1. Scaled χ² = 185.42 / 1.28 = 144.86
2. Adjusted df = 84 / 1.28 ≈ 65.63
3. p-value = P(χ²(65.63) > 144.86) ≈ 0.0001

Interpretation: The very small p-value (0.0001) indicates poor model fit. The researcher should:

• Examine modification indices for theoretically justified changes
• Check for misspecified factor loadings
• Consider alternative model specifications
• Verify the appropriateness of the scaling factor

Common Mistakes to Avoid

Incorrect Scaling Factor

Using the wrong scaling factor can lead to:

• Overcorrection (if c > actual)
• Undercorrection (if c < actual)
• Invalid p-values

Solution: Always verify the scaling factor from your SEM output or calculate it properly.

Ignoring Model Complexity

Complex models with many parameters:

• May require larger samples
• Can have unstable scaling factors
• Might benefit from parceling

Solution: Consider model simplification or use of latent variable scores.

Overinterpreting p-values

Common pitfalls:

• Treating p = 0.051 as “good” and p = 0.049 as “bad”
• Ignoring effect sizes
• Disregarding practical significance

Solution: Always consider multiple fit indices and theoretical meaning.

Software Implementation

Most major SEM software packages support Satorra-Bentler scaling:

R (lavaan package)

library(lavaan)
model <- 'visual  =~ x1 + x2 + x3
         textual =~ x4 + x5 + x6'
fit <- cfa(model, data = mydata, estimator = "ML")
fit.scaled <- lavTestLRT(fit, test = "satorra.bentler")
summary(fit.scaled)
        

Mplus

ANALYSIS:
  TYPE = MEANSTRUCTURE;
  ESTIMATOR = ML;
  INFORMATION = EXPECTED;
MODEL:
  visual BY x1-x3;
  textual BY x4-x6;
OUTPUT:
  SAMPSTAT;
        

LISREL

Satorra-Bentler Scaled Chi-Square Test Statistic: 124.32
Degrees of Freedom: 8
P-Value: 0.00045
Scaling Factor: 1.18
        

Advanced Considerations

For researchers working with particularly challenging data, consider these advanced topics:

Nested Model Comparisons

The Satorra-Bentler scaled difference test allows comparison of nested models:

ΔT_SB = (T_ML1 - T_ML2)/(c₁ - c₂)

Where c₁ and c₂ are scaling factors for the two models being compared.

Categorical Data Applications

For ordinal data with fewer than 5 categories:

• Consider robust weighted least squares (WLSMV)
• Use polychoric correlations
• Apply the Yuan-Bentler T2* statistic

Small Sample Adjustments

With n < 100:

• Use Bollen-Stine bootstrap
• Consider Bayesian SEM approaches
• Report confidence intervals for fit indices

Frequently Asked Questions

Q: When should I not use the Satorra-Bentler correction?

A: Avoid using it when:

• Your data are multivariate normal
• You have very small samples (n < 100)
• Your model is very simple (few parameters)
• You're using WLS estimation (use robust WLS instead)

Q: How does the scaling factor relate to kurtosis?

A: The scaling factor is directly influenced by:

• Multivariate kurtosis of the data
• Model complexity (number of parameters)
• Sample size

Higher kurtosis typically leads to larger scaling factors (more correction needed).

Q: Can I use this with multi-group models?

A: Yes, but consider:

• Calculating separate scaling factors for each group
• Using the scaled difference test for group comparisons
• Checking measurement invariance before comparing groups

Authoritative Resources

For deeper understanding, consult these authoritative sources:

• Kelly (2016) - Comprehensive review of Satorra-Bentler scaling in Journal of Counseling Psychology
• UCLA Statistical Consulting Group - SEM resources including robust estimation methods
• Bentler & Dudgeon (1996) - Original paper on scaled test statistics (see Chapter 6)
• NCSSM - Normality testing resources to determine when robust methods are needed

Conclusion

The Satorra-Bentler scaled chi-square statistic provides researchers with a powerful tool for evaluating structural equation models when the assumption of multivariate normality is violated. By properly applying this correction, researchers can:

• Make more accurate inferences about model fit
• Reduce Type I error rates with non-normal data
• Compare nested models appropriately
• Maintain valid statistical conclusions

Remember that while the Satorra-Bentler correction addresses normality violations, it doesn't solve all potential issues in SEM. Always consider:

• Multiple fit indices (CFI, RMSEA, SRMR)
• Substantive meaning of parameter estimates
• Replication of findings
• Theoretical justification for your model

For complex applications or when working with small samples, consider consulting with a statistical expert to ensure proper implementation of robust SEM methods.