Multivariate Methods > Structural Equation Models > Statistical Details for the Structural Equation Models Platform
Publication date: 03/23/2021

Image shown hereStatistical Details for the Structural Equation Models Platform

Image shown hereStatistical Details for Summary of Fit Measures

Image shown hereAICc and BIC

The formulas for AICc and BIC are defined as follows:

AICc = Equation shown here

BIC = Equation shown here

where:

-2logL is twice the negative log-likelihood.

n is the sample size.

k is the number of parameters.

For more information about the likelihood-based measures in the Model Comparisons report, see Likelihood, AICc, and BIC in Fitting Linear Models.

Image shown hereCFI

The comparative fit index (CFI) is calculated as follows:

CFI = Equation shown here

where:

Equation shown here is the chi-square statistic of the independence model.

df0 is the degrees of freedom of the independence model.

Equation shown here is the chi-square statistic of the fitted model.

dfmin is the degrees of freedom of the fitted model.

For more information about the CFI, see Bentler (1990).

Image shown hereTLI

The Tucker-Lewis index (TLI) is defined as follows:

TLI = Equation shown here

where:

Equation shown here is the chi-square statistic of the independence model.

df0 is the degrees of freedom of the independence model.

Equation shown here is the chi-square statistic of the fitted model.

dfmin is the degrees of freedom of the fitted model.

For more information, see West et al. (2012).

Image shown hereNFI

The Bentler-Bonett normed fit index (NFI) is defined as follows:

NFI = Equation shown here

where:

Equation shown here is the chi-square statistic of the independence model.

Equation shown here is the chi-square statistic of the fitted model.

For more information, see West et al. (2012).

Image shown hereRevised GFI and Revised AGFI

The revised goodness-of-fit index (Revised GFI) is defined as follows:

Revised GFI = Equation shown here

where:

Equation shown here is the chi-square statistic of the fitted model.

dfmin is the degrees of freedom of the fitted model.

p is number of observed variables in the fitted model.

n is the sample size.

The revised adjusted goodness-of-fit index (Revised AGFI) is defined as follows:

Revised AGFI = Equation shown here

where:

p* is the number of unique entries in the covariance matrix and the mean vector of the observed variables.

dfmin is the degrees of freedom of the fitted model.

For more information, see West et al. (2012).

Image shown hereRMSEA

The root mean square error of approximation (RMSEA) is calculated as follows:

RMSEA = Equation shown here

where:

n is the sample size.

dfmin is the degrees of freedom of the fitted model.

Equation shown here is the chi-square statistic of the fitted model.

The confidence limits for RMSEA are computed using the cumulative distribution function of the noncentral chi-square distribution Φ(x|λ, d). The 90% confidence limits are computed as follows:

Lower limit = Equation shown here

Upper limit = Equation shown here

where:

λL satisfies Φ(Equation shown here|λL, dfmin) = 0.95.

λU satisfies Φ(Equation shown here|λU, dfmin) = 0.05.

For more information, see Maydeu-Olivares et al. (2017).

Image shown hereRMR and SRMR

The formulas for RMR and SRMR are defined as follows:

RMR = Equation shown here

SRMR = Equation shown here

where:

p is the number of manifest variables.

b is the number of unique entries in the covariance matrix and the mean vector of the observed variables:

Equation shown here

sij is the (i, j)th element of the input covariance matrix.

Equation shown here is the (i, j)th element of the predicted covariance matrix.

Equation shown here is the ith element of the vector of sample means.

Equation shown here is the ith element of the vector predicted means.

For more information, see the CALIS Procedure chapter in SAS Institute Inc. (2020a).

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