[R-sig-ME] Rasch with lme4

[David Duffy]

On Mon, 8 Jun 2009, Reinhold Kliegl wrote:

> Conditional modes (generated from the model parameters and the data) are not 
> independent observations. Therefore, only the second method is valid.
>
> Reinhold Kliegl
>
> On 08.06.2009, at 13:04, Jeroen Ooms wrote:
>
>> 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?
>

Aside from Reinhold's comment, which is not a showstopper (you could 
bootstrap etc), they are quite different models. In the first model, the 
estimated IQ-extraversion correlation is disattenuated for measurement 
error -- the equivalent SEMs are something like (I think ;)):


IQ <-----> E  v.   E
| \                | \
v  v               v  v
i1 i2              i1 i2
                    ^  ^
                    | /
                    IQ

Most people would prefer something like the first model, and in fact would
estimate the correlation between IQ and E estimated (as if without error) 
from two measurement models given by the scoring rules for the 
instruments (these are essentially BLUPs).  Incorporating measurement 
error for both measures is the truest way to do it.

You could compare results using the sem and polycor packages to those from 
your lmer model.

my 2c, David Duffy.

-- 
| David Duffy (MBBS PhD)                                         ,-_|\
| email: davidD at qimr.edu.au  ph: INT+61+7+3362-0217 fax: -0101  /     *
| Epidemiology Unit, Queensland Institute of Medical Research   \_,-._/
| 300 Herston Rd, Brisbane, Queensland 4029, Australia  GPG 4D0B994A v