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An index of fit is a catch-all term for a variety of methods to tell you how well observed data fits a particular probability distribution.
An index of fit is typically normalized (i.e. units of measurement are removed), and the values will usually be between 0 and 1. What these values mean depends on which index of fit you are using; but it tells you, numerically, how well your data matches a particular distribution. That said, the cut-offs for what indicates a “good enough” fit varies wildly, which can be confusing if you’re new to fit indices (Hooper et. al, 2008).
The following list of fit indices is not exhaustive, but does include the more popular indices.
Absolute Fit Indices
Absolute fit indexes have formulas which are based on the discrepancies in a data set, as well as sample size. Absolute fit indices do not compare the model with a particular distribution; they use the data to generate a model. Specifically, the obtained and implied covariance matrices and the ML (Maximum Likelihood) minimization function (Tanaka, 1993). In other words, what counts is how well the hypothesized covariances from the specified model’s fixed and free parameters match the observed covariances estimated from the free parameters in the model (Hoyle, 1995).
- Akaike’s Information Criterion (AIC): AIC is usually calculated with software. The basic formula is defined as: AIC = -2(log – likelihood) + 2K. Click here for the main AIC article.
- Bayesian Information Criterion (BIC): closely related to the AIC, the BIC is partly based on the likelihood function— a function of a model’s parameters given data.
- Chi-squared: the “original” measure of fit. Most other absolute fit indices (with the exception of SRMR) are simple transformations of the chi-squared (Newsom, J, 2017). See: Goodness of fit / chi-squared.
- The Goodness of Fit Index (GFI) and the Adjusted Goodness of Fit Index (AGFI): calculates the proportion of variance accounted for by the estimated population covariance (Tabachnick and Fidell, 2007). It doesn’t compare to a baseline or null hypothesis, instead, it can be generalized as 1-νresidual/νtotal. Here νresidual is the variance that isn’t explained by the model, and νtotal is the total variance in the covariance matrix.
- Hoelter’s Critical N: a seldom used statistic, indicating the largest sample size for accepting the hypothesis that a model is correct.
- Expected Cross Validation Index (ECVI): measures how well a model can predict future sample covariances (Brown & Kudeck, 1993).
- Root Mean Square Residual (RMR) and Standardized Root Mean Square Residual (SRMR): calculated by the square root of the difference between the residuals of the sample covariance matrix and the hypothesized model for the covariance.
- Root Mean Square Error of Approximation (RMSEA): based on the non-centrality parameter.
If an absolute fit index takes on values above 0.9 that is generally considered a good fit. Note that residuals matrix based indexes like the SRMR are almost the opposite; An SRMR of zero is a perfect fit, but well fitting models have values less than .05 (Byrne, 1998; Diamantopoulos and Siguaw, 2000).
Relative Fit Indices
Relative Fit Indices, also called the incremental fit, includes a factor that represents deviations from a null model; so these are sometimes called comparative indices. The null model, also called the baseline model, should always have a poor fit (a very large Chi-square) (Ching et. al, 2014).
The comparative fit index, can be generalized as 1 – δM/δB; it compares the performance of your proposed model with the performance of a null or baseline model in which there was no correlation between observed variables.
Relative Fit Indices include:
- Bollen’s (1989) Incremental Fit Index (IFI): adjusts the Normed Fit Index for sample size and degrees of freedom. Over .9 is a good fit, but the index can exceed 1.
- Normed Fit Index (NFI): also called the Bentler-Bonett NFI. Ranges from 0 to 1, where 1 is a perfect fit. The NFI is difference between the null model’s chi-square and the target model’s chi-square, divided by the null model’s chi-square.
- Tucker-Lewis Index (TLI): unlike the Bentler-Bonnett, this index, also called the non-normed fit index, penalizes for adding model parameters.
- Bentler-Bonett Normed Fit Index (NFI): Generally not recommended because it doesn’t penalize for adding model parameters (Kenny, 2015).
Strengths and Weakness in Using an Index of Fit
In general, fit indices are descriptive and can be intuitively interpreted. That is one of their strong points.
But it’s important to remember that they only measure average or overall fit. What this means is that the index of fit may give ‘good’ values even if, in one portion of your model, the fit is actually quite bad. Good values don’t mean that your model makes theoretical sense, and do not guarantee in any way that your model is correct. They aren’t meant to prove or disprove a null hypothesis. Instead, they give you a number that summarizes the fit.
It’s often useful to look at (and record) more than one index of fit, as each has its own weaknesses and its own strong points. All your indices of fit should lead to the same general conclusion. If they don’t, you may need to find out why, or reject your model.
References
Bollen, K. (1989). A New Incremental Fit Index for General Structural Equation Models. Sociological Methods and Research. Volume: 17 issue: 3, page(s): 303-316. SAGE.
Browne, M.W. & Cudeck, R. (1993). Alternative ways of assessing model fit. In Bollen, K.A. & Long, J.S. [Eds.] Testing structural equation models. Newbury Park, CA: Sage.
Byrne, B.M. (1998), Structural Equation Modeling with LISREL, PRELIS and SIMPLIS: Basic Concepts, Applications and
Programming. Mahwah, New Jersey: Lawrence Erlbaum Associates.
Ching, G. et. al. (2014) Developing a scale to measure the situational changes in short-term study abroad programs. International Journal of Research Studies in Education. December, Volume 3 Number 5, 53-71
Diamantopoulos, A. and Siguaw, J.A. (2000), Introducing LISREL. London: Sage Publications.
Hooper, G. et. al. (2008). Structural Equation Modelling: Guidelines for Determining Model Fit. Retrieved December 5, 2017 from: www.ejbrm.com/issue/download.html?idArticle=183
Hoyle, R. H. (1995a). The structural equation modeling approach: Basic concepts and fundamental issues. In R.
H. Hoyle (Ed.), Structural equation modeling: Concepts, issues, and applications (pp. 1-15). Thousand
Oaks, CA: Sage.
Kenny, D. (2015). SEM: Fit. Retrieved December 6, 2015 from: http://davidakenny.net/cm/fit.htm
Leung, S. Index of fit for identifying statistical distributions. Retrieved December 5, 2017 from: from
http://ira.lib.polyu.edu.hk/handle/10397/2641
Newsom, J. (2017). Some Clarifications and Recommendations on Fit Indices. Retrieved December 5, 2017 from: http://web.pdx.edu/~newsomj/semclass/ho_fit.pdf
Tabachnick, B.G. and Fidell, L.S. (2007), Using Multivariate Statistics (5th ed.). New York: Allyn and Bacon.