A naive question...

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Justine Fraize

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Jul 1, 2019, 6:30:22 AM7/1/19
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Dear users,

we're interested in a strutural model where variables are better related after logarithmic transformation.

Our modeling would look like:

###
lX <- log10(X)
lY <- log10(Y)
lZ <- log10(Z)
dat<-data.frame(lX,lY,lZ)
 
T_Model_Regression <- 'T =~ NA*lX + lY + lZ
T ~~ 1*T'

fit_T_Model<- cfa(T_Model_Regression, data=dat)
###

We wonder wether the latent variable T will eventually correpond to (logX + logY + logZ) or to log(X+Y+Z), which changes the interpretation of the loadings...

Terrence Jorgensen

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Jul 2, 2019, 4:14:02 AM7/2/19
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we're interested in a strutural model where variables are better related after logarithmic transformation.

That sounds like dubious data-driven reasoning.  Is there a theory behind why a factor has a log-linear relationship with its indicators, instead of a linear one?

We wonder wether the latent variable T will eventually correpond to (logX + logY + logZ) or to log(X+Y+Z)

Neither.  Summing indicators implies a formative construct.  Your model implies a reflective construct.


which changes the interpretation of the loadings...

The loadings are now linear regression slopes on the transformed outcomes.  So when the factor T increases by 1 unit (here, 1 SD), a log10-transformed indicator's expected value increases by its factor loading.  Like any log-linear model, you can transform that loading/slope by exponentiating out the log (so here, raise 10 to the power of the loading) in order to interpret how the expected value of the original variable changes as a function of T.  But your SEs and tests only apply to the log10 scale because that is how the parameters were estimated.

Terrence D. Jorgensen
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam

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