# [Rasch] Estimating Rasch Measures for Extreme Scores

Steven L. Kramer skramer1958 at verizon.net
Sat Apr 30 05:24:48 EST 2011

```Hi modelers:  please see the questions near the bottom of this email.  I'd really like to get answers!

I have done some checking, so here is a clarification to my idea and my questions
Instead of using the Maximum Likelihood Estimate student scores, your student could instead use a software program that outputs the Bayesian estimates I described in my previous email, which are called "Expected a posteriori"  (EAP) scores.  I think this might solve her ptoblems.  John Linacre wrote the WinSteps manual, and that manual notes that WinSteps does NOT produce EAP scores--but here is a website where John Linacre explains EAP:  http://www.rasch.org/rmt/rmt163i.htm

Here are my questions:

What is the downside to using EAP scores instead of the Maximum Likelihood scores?  Is the issue merely "We don't want to make the assumption that the ability scores in this population are normally distributed if we don't have to" or are there other problems?

Do the EAP estimates really produce reasonable ability estimates for the perfect scores?  Or are the estimates still outlandishly high?

I'd love to do an empirical study that ran your student's study three ways, with three alternate dependant variables: one using the EAP estimates; one using the WinSteps MLE method that assigns scores to kids with perfect results by treating them as if they got a fractional number (usually 0.3) wrong; and one that uses the raw dependant scores  (e.g. A+B+B=5+4+4=13).  Then, the analysis would investigate whether there was  really any practical difference in the results.  Has anyone out there done a study like this, using either real data or a Monte Carlo simulation?

Lastly, I believe a Tobit model assumes that scores are normally distributed, except for the top score, which has a censored "ceiling" score.  Is there an advantage to using Tobit or some other some kind of "censored data" model to analyze the results?  Does the Tobit make sense only in the raw score case (since the "perfect" scores are assumed to be at the ceiling, with the student's true score equal to the ceiling or higher) but not in the WinSteps or EAP cases (since the "perfect" scores are assumed to be somewhere near --above OR below--the estimated value output by the model)?

Steve Kramer
----- Original Message -----
From: Steven L. Kramer
To: rasch list
Sent: Friday, April 29, 2011 12:13 PM
Subject: Re: [Rasch] Estimating Rasch Measures for Extreme Scores

I hope these aren't a naive questions.

If I understand correctly, from an IRT perspective one can compute for each student either a Maximum Likelihood Estimate (MLE) score, or a Bayesian score.  The Bayesian score assumes that ability is normally distributed, and accounts for this by shrinking estimated student scores towards the mean.  For students with perfectly correct scores the MLE predicts infinite ability, but the Bayesian score predicts finite ability.  (Similarly for students who get perfectly "zero" scores.)

My questions:
Is the "Bayesian estimate"  the way WinSteps computes scores for extreme scores?
If so, couldn't your student just use the Bayesian estimate for everyone, thus incorporating a consistent measurement theory?
From a Rasch point of view, what would be the downside of using a Baysian instead of a MLE approach?

Steve Kramer
----- Original Message -----
From: Rense Lange
To: Iasonas Lamprianou
Cc: rasch list
Sent: Friday, April 29, 2011 10:46 AM
Subject: Re: [Rasch] Estimating Rasch Measures for Extreme Scores

Since getting better faculty seems out of the question, would this solve it? Re-run twice using Facets.
a.. Once use all items, including those giving "perfect" scores,
b.. Once with the perfect items removed for each student/teacher getting perfect ratings.

Then plot the two sets of students/teachers? estimated parameters - they should be very similar (Y = X). While you are at it, you could show the probably non-trivial effects of differences in rater leniency/ severity on the student/teacher evaluations.

Rense Lange

2011/4/29 Iasonas Lamprianou <liasonas at cytanet.com.cy>

Dear colleagues,
I rarely submit requests in this list unless it is urgent and important because I respect the time of the people who tend to reply most often. I would like to thank them. This time, it is important and that is why I politely request you to help me. The post is long, but it has to do with the problems Rasch-users face in the harsh world of academia. I think that the post concerns most of us.

I am trying to help a student with her PhD thesis (so I am writing on her behalf). She submitted her thesis and her examiners spotted some problems and she has to address them.

The problem: The PhD thesis is about the performance of students.  For each student participating in the study (N>1000), the researcher has his/her score on four subjects: language, science, maths and history. For each subject, each student has three teacher assessments which were awarded in January, March and June. Each score runs from E (Failure) to A (Excellent). So, overall, each student has three ordinal teacher assessment measures for each of four subjects. It is a typical repeated measures case for four variables/subjects with three measures per variable/subject.

Design: Since the data are ordinal (E=1=Failure to A=5=Excellent) the researcher used a Partial Credit Rasch model with three  items  to build four Ability scales, one for each subject (the Rating Scale did not have good fit). Also, the student used all 12 scores (4 subjects X 3 measures) to produce one overall Ability  Academic Performance  measure. Then, the researcher used these Rasch ability measures as dependent variables to run OLS regressions.

Issue 1:
A serious problem spotted by the examiners is that a large proportion of students (around 20%) has perfect scores (three  A s) on some of the four subjects. The researcher used a Winsteps routine to find measures of ability for those students with extreme scores. The examiner has major reservations about the validity of this decision and asks whether these data (extreme scores) should be dropped. The examiner says:  If a Rasch analysis is to be used to derive attainment scores, the final distribution must provide a realistic representation of attainment. This means that the large group of candidates who achieve perfect scores (on the extreme right of the histograms) need to be properly represented. These scores need to be appropriately dealt with by Rash (if this is possible), or they need to be removed from the analysis (with an
assessment made about the impact of the resulting loss of data).
To the defense of the researcher, the distance between the  perfect score  and the  perfect-1  estimate is neither huge nor unreasonable: it is around 1.4 logits on a scale which extends from around -11 to 11 logits. When the researcher draws the scatterplot between raw scores and logits, the sigma-curve looks beautifully smooth and the estimates of the extreme scores look neither  too extreme  nor out of tune with the rest data points on the scatterplot. The distance between the  perfect score  and the  perfect-1  estimate is not grossly out of line compared to the other distances between raw scores estimates (for example, the distance between the  perfect-1  and the  perfect-2  scores is only around 0.3 logits smaller).
(a) The researcher needs strong references to defend her decision NOT to drop the extreme data estimates. Can anyone please provide strong peer-reviewed papers to support the decision to keep the extreme score estimates as valid representations of the ability of the participants?

Issue 2:
Stemming from the previous comment, one of the suggestions of the examiners is that the researcher could ditch the Rasch model and instead sum the three measures in one subject (e.g. A+B+B=5+4+4=13) and then use this sum for an OLS regression. The examiner says  A serious discussion needs to be held about the benefits, if any, the Rasch analysis provides over a more direct analytical path (e.g.   a linear regression of results averaged over three   [teacher assessments] . We all know that this is simply wrong to do because we cannof average ordinal measures and the student already explains this in her Methodology section, but she probably needs more references.
(b) Can anyone please provide a list of (recent, if possible) papers in good peer-reviewed journals which explain that this is not the right thing to do?

Issue 3:
Another suggestion of the examiners is that the researcher could ditch the Rasch model and just use the ordinal measure (E=1=Failure to A=5=Excellent) as a dependent variable in a proportional odds models. This means that the researcher should run three different models for each subject (for the Teacher Assessment awarded in January, March and June).
(c) Can anyone pleased provide a list of (recent, if possible) papers in good peer-reviewed journals which explain that this is NOT better than using the Rasch model to get one linear measure instead of three ordinal?

I feel that the examiners did a very good job overall and were very fair and consistent. They spent too much time to read every little detail in a long thesis, they spotted some important issues and we need to credit them for this. I feel that we may want to help the student address these interesting issues to the full satisfaction of the examiners.

Thank you for your time

In anticipation of your help
Jason Lamprianou
University of Cyprus

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Rense Lange, Ph.D.
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