I have tried to use lme4 to analyze IRT like datasets, but now I am
confused. I have a data set with intelligence items (i.e. score 0 or 1), for
completely crossed subjects and items. Furthermore, the data contains some
personality scores on the subject level. Actually the data is more
complicated than this, but let's keep it simple for now. My research
question is whether a personality charcteristic, say extraversion, is
related to intelligence. My question is how I should incorporate the
extraversion variable in the analysis.
When I analyse this data using the Rasch model, I usually first fit the
model, then extract the 'latent trait scores', and relate these to the
extraversion scores. I could do the same with lmer:

myModel <- lmer(y~1+(1|item)+(1|subject),data=mydata, family=binomial);
intelligence <- ranef(myModel)$subject[[1]];
lm(intelligence~extraversion);

However, in the context of multilevel analysis, it is also possible to
incorporate the extraversion variable directly into the model:

myModel2 <- lmer(y~1+(1|item)+(1|subject)+extraversion,data=mydata,
family=binomial);

Conceptually both methods feel very similar, but they give different
results. What is the most appropriate method? What are the differences in
interpretation?

Thank you!

Jeroen

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