# moreFitIndices: Calculate more fit indices In semTools: Useful Tools for Structural Equation Modeling

## Description

Calculate more fit indices that are not already provided in lavaan.

## Usage

 1 moreFitIndices(object, fit.measures = "all", nPrior = 1) 

## Arguments

 object The lavaan model object provided after running the cfa, sem, growth, or lavaan functions. fit.measures Additional fit measures to be calculated. All additional fit measures are calculated by default nPrior The sample size on which prior is based. This argument is used to compute BIC*.

## Details

Gamma Hat (gammaHat; West, Taylor, & Wu, 2012) is a global fit index which can be computed (assuming equal number of indicators across groups) by

gammaHat =\frac{p}{p + 2 \times \frac{χ^{2}_{k} - df_{k}}{N}} ,

where p is the number of variables in the model, χ^{2}_{k} is the χ^2 test statistic value of the target model, df_{k} is the degree of freedom when fitting the target model, and N is the sample size (or sample size minus the number of groups if mimic is set to "EQS").

Adjusted Gamma Hat (adjGammaHat; West, Taylor, & Wu, 2012) is a global fit index which can be computed by

adjGammaHat = ≤ft(1 - \frac{K \times p \times (p + 1)}{2 \times df_{k}} \right) \times ≤ft( 1 - gammaHat \right) ,

where K is the number of groups (please refer to Dudgeon, 2004 for the multiple-group adjustment for agfi*).

Corrected Akaike Information Criterion (aic.smallN; Burnham & Anderson, 2003) is a corrected version of AIC for small sample size, often abbreviated AICc:

aic.smallN = AIC + \frac{2k(k + 1)}{N - k - 1},

where AIC is the original AIC: -2 \times LL + 2k (where k = the number of estimated parameters in the target model). Note that AICc is a small-sample correction derived for univariate regression models, so it is probably not appropriate for comparing SEMs.

Corrected Bayesian Information Criterion (bic.priorN; Kuha, 2004) is similar to BIC but explicitly specifying the sample size on which the prior is based (N_{prior}).

bic.priorN = f + k\log{(1 + N/N_{prior})},

Stochastic information criterion (SIC; Preacher, 2006) is similar to AIC or BIC. This index will account for model complexity in the model's function form, in addition to the number of free parameters. This index will be provided only when the χ^2 value is not scaled. The SIC can be computed by

sic = \frac{1}{2}≤ft(f - \log{\det{I(\hat{θ})}}\right),

where I(\hat{θ}) is the information matrix of the parameters.

Hannan-Quinn Information Criterion (hqc; Hannan & Quinn, 1979) is used for model selection similar to AIC or BIC.

hqc = f + 2k\log{(\log{N})},

Note that if Satorra–Bentler or Yuan–Bentler's method is used, the fit indices using the scaled χ^2 values are also provided.

See nullRMSEA for the further details of the computation of RMSEA of the null model.

## Value

1. gammaHat: Gamma Hat

2. adjGammaHat: Adjusted Gamma Hat

3. baseline.rmsea: RMSEA of the Baseline (Null) Model

4. aic.smallN: Corrected (for small sample size) Akaike Information Criterion

5. bic.priorN: Bayesian Information Criterion with specified prior sample size

6. sic: Stochastic Information Criterion

7. hqc: Hannan-Quinn Information Criterion

8. gammaHat.scaled: Gamma Hat using scaled χ^2

9. adjGammaHat.scaled: Adjusted Gamma Hat using scaled χ^2

10. baseline.rmsea.scaled: RMSEA of the Baseline (Null) Model using scaled χ^2

## Author(s)

Sunthud Pornprasertmanit (psunthud@gmail.com)

Terrence D. Jorgensen (University of Amsterdam; TJorgensen314@gmail.com)

Aaron Boulton (University of North Carolina, Chapel Hill; aboulton@email.unc.edu)

Ruben Arslan (Humboldt-University of Berlin, rubenarslan@gmail.com)

Yves Rosseel (Ghent University; Yves.Rosseel@UGent.be)

## References

Burnham, K., & Anderson, D. (2003). Model selection and multimodel inference: A practical–theoretic approach. New York, NY: Springer–Verlag.

Dudgeon, P. (2004). A note on extending Steiger's (1998) multiple sample RMSEA adjustment to other noncentrality parameter-based statistic. Structural Equation Modeling, 11(3), 305–319. doi: 10.1207/s15328007sem1103_1

Kuha, J. (2004). AIC and BIC: Comparisons of assumptions and performance. Sociological Methods Research, 33(2), 188–229. doi: 10.1177/0049124103262065

Preacher, K. J. (2006). Quantifying parsimony in structural equation modeling. Multivariate Behavioral Research, 43(3), 227-259. doi: 10.1207/s15327906mbr4103_1

West, S. G., Taylor, A. B., & Wu, W. (2012). Model fit and model selection in structural equation modeling. In R. H. Hoyle (Ed.), Handbook of Structural Equation Modeling (pp. 209–231). New York, NY: Guilford.

• miPowerFit For the modification indices and their power approach for model fit evaluation
• nullRMSEA For RMSEA of the null model
 1 2 3 4 5 6 7 8 9 HS.model <- ' visual =~ x1 + x2 + x3 textual =~ x4 + x5 + x6 speed =~ x7 + x8 + x9 ' fit <- cfa(HS.model, data = HolzingerSwineford1939) moreFitIndices(fit) fit2 <- cfa(HS.model, data = HolzingerSwineford1939, estimator = "mlr") moreFitIndices(fit2)