Your Handy Guide to Fit Statistics in SEM

We've all been there... you've just completed your analyses and now have your output, representing not only your hopes and dreams of academic stardom, but also weeks of entering data, cleaning data, creating datasets, drawing models, scrapping models, reformatting datasets, crying to your adviser, writing syntax, cursing the stats gods, rewriting syntax... and so on.  You look down at your beautiful, hard earned output with a tear in your eye and think "What does it mean??".

Well not to worry! DASAL is here to help (at least the top portion of your output) with our handy guide to model fit statistics!  

Model fit refers to how well the relations in you data are represented by a particular model (or, to be more "statistic-y": how well all observed covariance and variance in the data are explained by an entire model - including errors), and can be inferred from the model fit statistics generated by your statistical package of choice.  Although fit statistics have come under fire in recent years (Barrett, 2007), understanding and accurately reporting them is a crucial task for the diligent researcher.  The following are a list of the most common fit statistics and their cutoff criteria for "good model fit".

Absolute Fit Indices
These indices provide a measure of how well an a priori model fits your sample data (McDonald & Ho, 2002) by comparing the observed and model-implied variance and covariance matrices. It can also be used to determine the best fitting model among many.  In general, these indices improve as more parameters are added to the model.  Good fit indicates that most of the (co)variance was explained by the model, or that there wasn't much to explain!

Chi-Squared
This is the most traditional measure of model fit and "asses the magnitude of discrepancy between the sample and the fitted covariance matrices" (Hu & Bentler, 1999).  Chi-square is very sensitive to sample size, and will generally indicate bad fit for larger samples.

Smaller values, relative to the degrees of freedom, indicate better fit.  A nonsignificant chi-square value indicates good model fit.

Goodness-of-Fit Index (GFI)
This is designed to be an R square type index.

Larger values indicate better fit. In general, values above .90 indicate good fit.

Standardized Root Mean Squared Residual (SRMR)

Smaller values indicate better fit. In general, values below .08 indicate good fit.

Parsimonious Fit Indices
These indices evaluate the overal discrepancy between the observed and model-implied variance and covariance matrices, but also take into account the simplicity of the model.  In general, parsimonious indices improve as more parameters with useful contributions are added to the model. Good fit indicates that a good amount of the (co)variance was explained by the model, relative to its parsimony.  This does not mean that the amount explained is good overall - for that information, you must look to absolute fit indices.

Akaike Information Criterion (AIC)
Values generated are relative, and AIC is often used for model comparison.

Smaller values indicate better fit.

Root Mean Squared Error of Approximation (RMSEA)
These fit indices are presented with a confidence interval, usually a 90% CI.

Smaller values indicate better fit. In general, values below .06 indicate good fit.

Adjusted Goodness-of-Fit (AGFI)

Larger values indicate better fit. In general, values above .90 indicate good fit.

Incremental Fit Indices
These indices compare the model's absolute or parsimonious fit with that of a baseline model (usually the null model).  In general, larger values indicate better fit.  Good fit in this sense means that much of the (co)variance was explained by the tested model in comparison to the amount that would have been explained by the null model.

They include:

Comparative Fit Index (CFI)
In general, values above .95 indicate good fit.

Normed Fit Index (NFI)
In general, values above .90 indicate good fit.

Nonnormed Fit Index (NNFI)
In general, values above .95 indicate good fit.

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SO NOW WHAT?? We have so many measures, which ones do we choose?  People generally look to the chi-squared first.  However, in cases where sample size is small, or when the chi-square is borderline, Hu and Bentler (1999) have proposed some joint criteria as follows:

A model has good model fit if:

NNFI and CFI are greater than or equal to .96 AND SRMR is less than or equal to .09.

or

SRMR is less than or equal to .09 AND RMSEA is less than or equal to .06.

But how do I interpret it?

If you have a good fit to your data, you may not say that you have confirmed your underlying theory as true, but rather that you fail to reject the proposed model as one viable representation of the true relations underlying the data.