Publication date: 03/23/2021

The formulas for AICc and BIC are defined as follows:

AICc =

BIC =

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.

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

CFI =

where:

– is the chi-square statistic of the independence model.

– df0 is the degrees of freedom of the independence model.

– 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).

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

TLI =

where:

– is the chi-square statistic of the independence model.

– df0 is the degrees of freedom of the independence model.

– 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).

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

NFI =

where:

– is the chi-square statistic of the independence model.

– is the chi-square statistic of the fitted model.

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

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

Revised GFI =

where:

– 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 =

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).

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

RMSEA =

where:

– n is the sample size.

– dfmin is the degrees of freedom of the fitted model.

– 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 =

Upper limit =

where:

– λL satisfies Φ(|λL, dfmin) = 0.95.

– λU satisfies Φ(|λU, dfmin) = 0.05.

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

The formulas for RMR and SRMR are defined as follows:

RMR =

SRMR =

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:

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

– is the (i, j)th element of the predicted covariance matrix.

– is the ith element of the vector of sample means.

– 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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