Rasch with lme4

On Tue, 9 Jun 2009, Reinhold Kliegl wrote:

On 09.06.2009, at 00:09, David Duffy wrote:

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.

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

versus

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

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.

Always interested to be corrected...

My comment was referring to the use of conditional modes (formerly known
as BLUPs)--extracted with ranef()--not bootstrap or SEM. Not sure to what
degree these alternatives correspond to the use of conditional modes. ?As
far as conditional modes are concerned, we (Kliegl, Masson, & Richter) ?have
a paper in press (available at my publication page), in which we write the
following:

To this end, we generated 100,000 sets of data for a simple LMM model
including 30 ?subjects? and a predictor with 10 levels, conforming to a
known variance for intercept and slope across subjects and varying the true
correlation between these parameters from -0.9 to +0.9 in 2,000 steps (i.e.,
each simulation used a different correlation). ...
... [C]onditional means underestimate variances and exaggerate covariances
and correlations. The shrinkage of variance reflects the contribution of the
likelihood in the computation of conditional means. Shrinkage correction for
predictions leads to dampening of the variance components, but, as we have
shown in this section, not of the associated covariance component. The
shrinkage of variance prevents overfitting of unreliable data but, as a
curious side effect, the "correlations" based on conditional means for
individual subjects are larger in absolute value than the corresponding LMM
estimates of the correlation.

Well I'm not sure how much that reflects the specific model you have
simulated, and I don't have the time right now to do simulations based on
the original poster's setup. ?(I have experienced your problems with
shrinkage etc in a simple minded attempt to carry out genetic linkage
analysis of breeding values (BLUPs) estimated from the same pedigree). And I
would concur with your postscript. ?And I will study your paper with great
interest.

However, the impression I have is that usually the effects of just plugging
in the factor scores when they are based on, say, 20 or 30 individual items
with a straightforward structure are not too misleading, and are just what
people have been doing for the last 50 years. I am currently comparing
results from a two-stage mixed model analysis of BLUPs from an IRT (carried
out in BUGS) analysis of multiple ordinal measures adjusting for multiple
fixed covariates to results of analyses I am performing on the original
variables. ?I have not seen any major inconsistencies, but I will look for
effects of the type you have described.

Colleagues have examined the multitrait mixed model analysis of pedigree
data using the full analysis and compared it to using BLUPs:

Dorret I. Boomsma and Conor V. Dolan (1998). ?A Comparison of Power to
Detect a QTL in Sib-Pair Data Using Multivariate Phenotypes, Mean
Phenotypes, and Factor Scores. Behavior Genetics 28: 329-340

They found in their simulations that there was "negligible overestimation"
of the genetic covariance in models where they split the sample in two,
using one-half to generate the prediction model, which was applied to the
other half to generate the BLUPs, and then vice-versa.

My specific comment was based on an impression that that the second model
with Extraversion as a fixed effect doesn't give the original poster what he
is interested in, viz an assessment of the relationship between two
(imperfectly measured) psychological traits: IQ and E.

I find it a bit confusing, but the test of the single regression coefficient
for Extraversion in the second model seems to me to be different from the
that in the first. ?Specifically, the E->IQ->item(1..N) model constrains the
pattern of expected covariation between E and any one IQ item differently
from the fixed effects model.

Finally, multiple imputation type methods are one way people get around
these types of problems, as a full maximum likelihood analysis is often
expensive computationally (with a multimodal likelihood surface). I don't
think a simple bootstrap resampling repeatedly calling lmer and raneff would
get around the biases you have noted.

Cheers, David Duffy

PS The OP may be aware of work of my colleagues on this particular topic:
http://genepi.qimr.edu.au/contents/p/staff/CV516.pdf

--
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