CFA: Basics

Beaujean Chapter 3

Other readings

• Kline 9 – a good reference, but lumps this entire section into one chapter.

How things are related:

SEM

Traditional Multivariate

Canonical Correlation

CFA

IRT EFA

Path Analysis

Multiple Regression

ANOVA t-tests Correlation

CFA Models

• EFA models– You have a bunch of questions– You have an idea (or sometimes not!) of how

many factors to expect– You let the questions go where they want– You remove the bad questions until you get a good

fit

CFA Models

• CFA models– You set up the model with specific questions onto

specific factors• Forcing the cross loadings be zero• (draw)

– You test to see if that model fits– (so the C = Confirming the EFA).

CFA Models

• Reflective – the latent variable causes the manifest variables scores– Purpose is to understand the relationships

between the measured variables– Same theoretical concept as EFA.

CFA Models

• Formative – latent variables are the result of manifest variables– Similar to PCA theoretical concept– Demographics?

CFA Models

• The manifest variables in a CFA are sometimes called indicator variables– Because they indicate what the latent variable

should be since we don’t directly measure it.

CFA Models

• General rules:– The latents will be correlated• Similar to an oblique rotation

– Each factor section has to be identified– Arrows go from latent to measured (reflexive)• We think that latent caused the measured answers

– Error terms on the measured variables.

CFA Models

• Generally, you leave the error terms uncorrelated– BUT!– These questions all measure the same factor

right? So their answers on some will be tied to answers on another. So the errors may also be correlated.

– You can get away with adding those here, if you have strong modification indices or theoretical reasons.

CFA Models

• Factor loadings – same idea as EFA, you want the relationship between the latent variable and manifest variable to be strong– Otherwise why you are using that item/scale as an

indicator of the latent?

CFA Models

• Pattern coefficients versus structure coefficients

• Pattern – regression coefficient, how much does the manifest variable increase for each one unit of the latent variable

• Structure – correlation between latent and manifest variable

CFA Models

• Pattern = structure when there is only one latent variable.– Not equal when there are multiple latent

variables.

CFA Models

• Identification rules of thumb:– Latent variables should have four indicators– Latent variables have three indicators AND• Error variances do not covary

– Latent variables have two indicators AND• Error variances do not covary• Loadings are set to equal each other.

CFA Models

• Scaling – setting the scale for latent variable– This scaling also helps with identification issues.– Now, we’ll explain why you’ve been using the 1 to

set a particular path and explore other options.

CFA Models

• Scaling options:– Standardized latent variable – constrain the latent

variable variance to 1.– What does that do?• Sets the scale to z-score.• Makes double headed arrow between latents

correlation.• Use the unstandardized solution or you’ve double

standardized.

CFA Models

• Scaling options:– Marker variable – sets one of the factor loadings

to one (what we’ve been doing)– Gives the items a scale.

CFA Models

• Scaling options:– Effects coding – estimates all loadings but

constrains them to averaging 1.0 across a latent variable.

CFA Models

• These options will give you different loadings but not different model fit.

CFA Models

• Empirical underidentification– When the factor loading for a question is very

close to 0 or factor correlations are very close to 1.

Let’s try it out!

• New stuff:– Translate a correlation matrix to covariance

matrix. – Something not super clear from before:• If you use a correlation matrix as the input:

– The Unstandardized output is technically standardized.

• If you use a covariance table or raw data:– The unstandardized output is unstandardized.

Example 1

• CFA with one latent variable • New function:– cor2cov(correlations, SDs)– Converts correlation tables to covariance tables– You need a correlation matrix AND a vector of SDs

Example 1

• =~ symbol in the model description • Before we did– Y ~ X– Because all the variables were manifest

• Now we use =~ to tell lavaan that the Y is a latent variable.

Example 1

• Also new code:• parameterEstimates(wisc4.fit,

standardized=TRUE)

You get the same output as the summary function (parameter wise).This version will give you CIs though!

Example 1

• Std.lv = FALSE– False is the default– Makes the first variable the indicator (i.e. sets it to

1).• TRUE– Sets the latent as the indicator, estimates all

loadings and constrains the variance to 1.

Example 1

• What are the new Standardized Columns?– Std.lv = standardizes the latent variable but leaves

manifest in the scale– Std.all = standardizes everything.

Example 1

• fitted() function– Gives you the recreated covariance table

• residuals() function– Gives you the difference between actual and

reproduced correlation table

Example 1

• fitMeasures()– How you can get ALL the fit indices!

• modificationIndices()– Get the modification indices separately from all

the output.

Example 2

• Switching to estimating all variables, constraining the variance instead of an indicator.– Use std.lv=TRUE

Example 3

• Let’s specify a two-factor model!

Example 4

• A fully latent model!• Use ~ for latent to latent

Things to check out

• Heywood cases/logical solution– Are our variances positive + SMCs ok?– Are there any crazy SEs?

• Estimates – Did our questions load?

• Model fit– Are the fit indices any good?

Parameters

• Remember – you want parameters that make sense– You can check out the standardized parameters to

determine if questions are still loading like they would in an EFA.

– Z = parameter / SE• Sometimes called a critical ratio.

Parameters

• Standard errors are tricky– They are based on the scale of the variable– You do not want them to be zero• Estimating no variance is bad … some variance is always

good!

– You do not want them to be large• That means you are not estimating very well

Model Fit

• See previous notes but here’s a quick reminder:– X2 nonsignificant (ha!)– RMSEA, SRMR = small numbers– CFI/TLI = large numbers

Model Fit

• We talked about modification indices in the last section.– In this section – think about what they mean

before adding paths. – Usually CFA is meant to test that specific

question/latent combination, so it may not help to add the paths or correlate errors on two different latents.

Model Fit

• Overfitted model – when you add parameters that help model fit, but do not help with theory (and probably won’t replicate)

Compare the models!

• Make a chi-square difference table– Chi square difference, df difference, critical chi

square, reject?• Look at differences in CFI– Is the change greater than .01?

• Look at the AIC/ECVI– Which one is lower?