# [Rasch] multidimensional vs two unidimensional models

Margaret Wu wu at edmeasurement.com.au
Sat May 3 07:40:40 EST 2014

```Hi Jason

Paul is precisely right. The correlation produced by ConQuest and TAM is
the latent correlation, while the correlation between estimated abilities
from two unidimensional runs is attenuated by measurement error (each
ability estimate has measurement error).

In much the same way as for classical test theory where you can
"disattenuate" to find the variance of the true scores (Var (T) =
reliability * Var(X)), you can disattenuate to find the latent correlation
from the correlation of estimated abilities:

Correlation of the latent construct = corr of estimated abilities /
sqrt(reliability 1 * reliability 2)

If you use the above formula, you should get from 0.39 to 0.75.

In the MML model, the population model has the variance (and mean) of the
abilities as parameters to estimate "directly" from the data, so the
variance is not attenuated. Similarly for two dimensional models, the
correlation between the two dimensions is also an explicit parameter in
the model, so it is directly estimated from the item response data.   The
estimation equations are derived with the correlation parameter as the
latent correlation.

I am not sure if attachments are allowed on the listserv. You can email me
(wu at edmeasurement.com.au) and I can send you an article on a comparison
decide whether you should use unidimensional models or multidimensional
models. Also I can provide some simulations in R to understand the
differences.

Margaret

-----Original Message-----
From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On
Behalf Of Swank, Paul R
Sent: Saturday, 3 May 2014 2:49 AM
To: rasch at acer.edu.au
Subject: Re: [Rasch] multidimensional vs two unidimensional models

I would guess that the two dimensional model is controlling for the
unreliability in the measures whereas when you look at the raw
correlations, it is not. Thus the .75 represents correlations between
latent constructs whereas .39 is the correlation between manifest or
measured variables.

Dr. Paul R. Swank, Professor
Health Promotion and Behavioral Sciences School of Public Health
University of Texas Health Science Center Houston

-----Original Message-----
From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On
Behalf Of Iasonas Lamprianou
Sent: Friday, May 02, 2014 11:37 AM
To: rasch at acer.edu.au
Cc: iasonas at ucy.ac.cy
Subject: Re: [Rasch] multidimensional vs two unidimensional models

Dear friends,

I am running an analysis on a very interesting dataset and I would like to
benefit from your experience with multidimensional Rasch models. I am
using a Conquest-like implementation in the open-source platform R (the
package is called TAM). Essentially, according to the authors, they have
moved the Conquest algorithms to R. For all practical intents and purposes
of this discussion, we can assume that I use (a flavour of) Conquest.

I have 22 dichotomous questions and 5000 persons. Twelve of the questions
tap on dimension 1 and 10 questions tap on dimension 2. I tried running
two independent unidimensional Rasch models with satisfactory results
(good fit statistics). I extracted the person estimates from the two
analyses and the correlation was r=0.39. It may be important to say that
for the second scale there is a very strong floor effect, more than 50%
got 0 (I know that this is a problem, but there is nothing I can do at
this point). Since I use MML estimation, the software gave some reasonable
estimates for those people, but it is likely hat this may have affected
the correlation between the two dimensions. However, when I run a
two-dimensional Rasch model, where each item taped only on its dimension,
the reported correlation of the two latent dimensions as reported by the
software was 0.75! But if I extract the two lists of estimates (on the two
dimensions), the correlation is 0.39 (the same as the  correlation from
the two independent analyses). I wonder why is that? Could the effect of
the two-dimensional model be so high?

Also, the "EAP reliability" of the second dimension (the one with the high
floor effect) is reported to be 0.268 when I run the analysis
individually, however, the same statistic becomes 0.449 when I run the
multidimensional model (the EAP reliability of the first scale is not
increased significantly).

My questions are:
(a) is it reasonable for the correlation between the two dimensions to be
twice as large (for the two-dimensional model) as the correlation of the
estimates of the independent analyses? Or am I doing something wrong? Did
anyone have such an experience before?
(b) why is the EAP reliability increased so much? Does the second
dimension "gain" information from the first dimension?

If I run a unidemensional model with all 22 items and compare it with the
two-dimensional model (two nested models) I get significant results but
not too impressive:
Model   loglike Deviance Npars      AIC      BIC   Chisq df       p
1 Model 1 -21435.62 42871.25    23 42917.25 43066.63 10.5914     0.00501
2 Model 2 -21430.33 42860.65    25 42910.65 43073.03      NA NA      NA

Would you be tempted to consider the 22 items to be unidimensional since
the correlation between the two latent dimensions is reported to be 0.75
and since there is only a modest reduction to the Deviance when I fit the
two-diemensional model?

Thank you for your patience to read this long email. I hope  one of you
may have similar experiences to share or comments to send for my
questions.

Have a nice weekend,

Jason
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