GOF Indices

Chi-Square Test

• Reasonable indicator of fit for n=75-200
• In models where n \(\le 75\), probability of making a Type I error increases.

fitmeasures(fit, "chisq")

##  chisq 
## 30.723

• Some sources also report \(\frac{\chi^2}{df}\) ratio.
• Recommended: \(\le 3.0\)

fitmeasures(fit, "chisq")/10 

## chisq 
## 3.072

Incremental Fit Index

• Analogous to \(R^2\)
• Recommended: \(\ge 0.95\)

fitmeasures(fit,"ifi")

##   ifi 
## 0.946

Comparative Fit Index

• Only meaningful when comparing several different models.
• Recommended: \(\ge 0.95\)

fitmeasures(fit,"cfi")

##   cfi 
## 0.937

Bentler-Bonett Index or Normed Fit Index

• Compares hypothesized model to a null model where the null model is defined as:

The null model when there are causal paths would be to have all exogenous variables are correlated but the endogenous variables are uncorrelated with each other and the exogenous variables. The degrees of freedom for this null model are \(\frac{\left[k \cdot (k-1) - p \cdot (p-1) \right]}{2}\) where \(k\) is the number of variables in the model and \(p\) the number of exogenous variables or indicators in the model.

• Recommended: \(\ge 0.95\)

fitmeasures(fit,"nfi")

##   nfi 
## 0.922

Tucker Lewis Index or Non-normed Fit Index (NNFI)

• Similar to Bentler-Bonett Index; however, penalizes models with too many pathways.
• Recommended: \(\ge 0.95\)

fitmeasures(fit,"tli")

##   tli 
## 0.584

Root Mean Square Error of Approximation

• Small df and small n models are penalized.
• Some authors argue that small models should not even compute this value.
• Recommended: \(\le 0.10\) (especially for small models).

fitmeasures(fit,"rmsea")

## rmsea 
## 0.234

Standardized Root Mean Square Residual

• Does not penalize for complex models.
• Biased towards small df and small n models.
• Recommended: \(\le 0.10\) (especially for small models).

fitmeasures(fit,"srmr")

## srmr 
## 0.06

Akaike/Bayesia Information Criterion

• Similar to CFI, these measures are meaningful when comparing two models; however, do not amount to much when looked in isolation.
• When comparing models, the model with the lower AIC/BIC value indicates better fit.

fitmeasures(fit,"aic")

##     aic 
## 2362.83

fitmeasures(fit,"bic")

##      bic 
## 2472.549

Resources to be Consulted

• Barrett, P. (2007). Structural equation modelling: adjudging model fit. Personality and Individual Differences, 42, 815–824.

• Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness-of-fit in the analysis of covariance structures. Psychological Bulletin, 88, 588-600.

• Bentler, P. M., & Chou, C. P. (1987) Practical issues in structural modeling. Sociological Methods & Research, 16, 78-117.

• Bollen, K. A., & Long, J. S., Eds. (1993). Testing structural equation models. Newbury Park, CA: Sage.

• Enders, C.K., & Tofighi, D. (2008). The impact of misspecifying class-specific residual variances in growth mixture models. Structural Equation Modeling: A Multidisciplinary Journal, 15, 75-95.

• Hayduk, L., Cummings, G. G., Boadu, K., Pazderka-Robinson, H., & Boulianne, S. (2007). Testing! Testing! One, two three – Testing the theory in structural equation models! Personality and Individual Differences, 42, 841-50.

• Hu, L., & Bentler, P. M. (1998). Fit indices in covariance structure modeling: Sensitivity to underparameterized model misspecification. Psychological Methods, 3, 424–453.

• Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1–55.

• Kenny, D. A., Kaniskan, B., & McCoach, D. B. (2015). The performance of RMSEA in models with small degrees of freedom. Sociological Methods & Research, 44, 486-507.

• Kenny, D. A., & McCoach, D. B. (2003). Effect of the number of variables on measures of fit in structural equation modeling. Structural Equation Modeling, 10, 333-3511.

• MacCallum, R. C., Browne, M. W., & Sugawara, H. M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological Methods, 1, 130-149.

• O’Boyle, E. H., Jr., & Williams, L. J. (2011). Decomposing model fit: Measurement vs. theory in organizational research using latent variables. Journal of Applied Psychology, 96, 1-12.

• Satorra, A., & Saris,W. E. (1985). The power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50, 83–90.

• Sharma, S., Mukherjee, S., Kumar, A., & Dillon, W.R. (2005). A simulation study to investigate the use of cutoff values for assessing model fit in covariance structure models. Journal of Business Research, 58, 935-43.

• Tanaka, J.S. (1987). “How big is big enough?”: Sample size and goodness of fit in structural equation models with latent variables. Child Development, 58, 134-146.

• Tofghi, D., & Enders, C. K. (2007). Identifying the correct number of classes in mixture models. In G. R. Hancock & K. M. Samulelsen (Eds.), Advances in latent variable mixture models (pp. 317-341). Greenwich, CT: Information Age.

• Wui, W., West, S. B., & Taylor, A. B. (2009). Evaluating model fit for growth curve models: Integration of fit indices from SEM and MLM frameworks. Psychological Methods, 14,183-201.

• Carmines, E. G., & McIver, J. P. (1981). Analyzing Models with Unobserved Variables. In Steffel R. V., & Ellis, R. S. (Eds.). Structural and Social Bond of Commitment in Inter-Firm Relationsships. Journal of Applied Business and Economics, 10(1), 1-18.

• Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In Byrne, B. M. (Ed.). Structural equation modeling with AMOS: Basic Concepts, Applications, and Programming. (2th ed.)

• Jaccard, J., & Wan, C. K. (1996). Lisrel Approaches to Interaction Effect in Multiple Regression. California, USA: Sage Publications Inc.

• Hu, L.-T., & Bentler, P. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1–55.