Model fit in CFA
PSY544 – Introduction to Factor Analysis
Week 12

CFA Review
•Think of CFA as a succession of three steps:
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•1) Specify a model
•2) Fit the model to data (estimate the free parameters)
•3) Evaluate the result

CFA Review
•1) Specify a model
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•We know what this is – either specify the model through specification of the model matrices or
through path diagrams
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CFA Review
•2) Fit the model to data
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•Obtain estimates of the free parameters that minimize the discrepancy function
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•I haven’t mentioned one thing yet, but it’s very important – do not carry out an EFA with a blind
rotation to decide on the number of factors and the position of zero loadings and then fit an
analogous CFA model using the same data.

CFA Review
•2) Fit the model to data
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•That would be capitalizing on chance – what you would be effectively doing is “confirming” a
solution by CFA that was suggested by EFA.
•First of all, that’s kind of running in circles, second of all, it invalidates statistical tests
and standard errors obtained by fitting the confirmatory model

CFA Review
•2) Fit the model to data
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•What you can do is exploring the factor structure using one half of the data (split randomly) and
then “cross-validating” your hypothesized model using the other half of the data.
•This is an example of a good practice. You want to formulate a model that holds for the
population, so being able to replicate the model on different data tells you good things about your
model.

CFA Review
•3) Evaluate the result
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•That’s what we’ll be talking about today.
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•Generally:
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•A) Evaluate the parameter estimates
•B) Evaluate the overall model fit

Evaluating the result
•First, make sure the solution looks reasonable enough.
•Did the software produce any warnings?
•Examine the parameter estimates – the point estimates and the standard errors.
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•Are the point estimates within their logical bounds? Look for correlations greater than 1,
variances less than or equal to 0. These are signs that things are breaking down.

Evaluating the result
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•Such “broken down” things are called inadmissible or improper solutions. You should be
super-careful when interpreting such solutions, as *something* is most likely wrong in your data or
with your model. Treat this as a warning and try to figure out what’s wrong.
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Evaluating the result
•Even if the point estimates look reasonable, do they make sense to you?
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•Were you expecting a positive correlation where a negative correlation is?
•Were you expecting a high loading where you got a small one?
•Were you expecting a negative loading on an MV, but got a positive one?

Evaluating the result
•Standard errors.
•Standard error is the estimate of the standard deviation of the parameter estimate over repeated
sampling.
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•How much would a parameter estimate vary over different samples?
•High value = unstable estimate
•Low value = stable estimate

Evaluating the result
•Treat the standard errors like you would treat them with any other parameter estimates – say, in
linear regression.
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•You can use the standard error of a point estimate to compute a t-statistic [point estimate / its
standard error] to test difference from zero.
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•Typically, the absolute value of these t-statistics will be much, much larger than 2

Evaluating the result
•Large standard errors can be an indication of trouble.
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•If all standard errors are large, then your N is probably too small
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•If only some standard errors are large while most are small, then you probably have some other
problem that is more specific – investigate the respective MVs or LVs that are involved.

Model fit
•We’ll cover only a couple of fit indices, but there are many many maaaaaaaaaaaaaaaany more.
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•Good fit doesn’t mean your model is “true” or “correct” or “best”. There are probably other models
that fit just as well or even better (models with more free parameters, for instance)
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•Does your model fit “well enough”, though?

Model fit



Model fit
•We already covered RMSEA (Root Mean Square Error of Approximation) and TLI (Tucker-Lewis Index)
before. They can be used in the same way as in EFA, and they have identical meaning.
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•TLI is sometimes called NNFI (Non-normed Fit Index) in CFA context.

Model fit



Model fit
•GFI (Goodness-of-fit index) and AGFI (Adjusted Goodness-of-fit index) are affected by sample size
and have been shunned by the community. Don’t use them, don’t report them, ignore them.
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Model fit
•Residual-based indices:
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•The residual matrix is a good source of information on model fit (i.e., how well does the model
reproduce the observed correlation / covariance matrix).
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•Some fit indices directly work with the elements in the residual matrix.
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Model fit
•RMSR (Root mean squared residual) – a zero indicates perfect fit
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•SRMR (Standardized root mean [squared] residual) – again, a zero indicates perfect fit. Typically
a value of less than .08 is considered good fit.
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•Keep in mind that most residual-based fit indices incorporate no penalty for model complexity.
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