Estimating Rasch Measures with Known
Polytomous (or Rating Scale) Item Difficulties
Anchored Maximum Likelihood Estimation (AMLE)

For dichotomies see

Once item and step difficulties have been calibrated, we can administer some or all of the calibrated items to further examinees and measure them:

John Michael Linacre
with typesetting assistance from Stacie Hudgens

This estimation is implemented in the Excel Spreadsheet for polytomous estimation.

Estimating Rasch measures with known polytomous item difficulties.Linacre J.M. … Rasch Measurement Transactions, 1998, 12:2 p. 638.

Warm's (Weighted Mean) Likelihood Estimates (WLE)

For an explanation of WLE, see RMT (2009), 23:1, 1188-9

Warm's bias correction is applied to each MLE estimate, M, to produce a Warm's Mean Likelihood Estimate (WLE), MWLE, which is almost always closer to the mean item difficulty than M.

person n's WLE estimate = MWLE = M + ( J / ( 2 * I2 ) )
where, for polytomous Rasch items,
J = Σ( (Σ j³Pnij ) - 3(Σ j²Pnij )(Σ jPnij ) + 2(Σ jPnij )³ ) summed over i = 1,L and j = 0,m
I = test information = Variance

Visual Basic Code to do some of the above.

' Step 1) above

' for the responses
Dim itemcount&
itemcount = 50 ' the number of items

ReDim observedrating&(itemcount) ' for your data for one person

' Collect your data here and compute raw scores here
' code missing data as -1 in observedrating&() and exclude from the raw score

Dim ObservedScore&

' ObservedScore& = The raw score

' Step 2) above

' For the items
ReDim itemdifficulty!(itemcount)
itemdifficulty(1) = 1.23 ' your item difficulties in logits
' all the other items
itemdifficulty(itemcount) = 3.45

' for the ratings

Dim bottom&, top&
bottom& = 1 ' the score for your lowest rating-scale category
top& = 5 ' the score of your highest rating-scale category

Redim stepdifficulty!(top&) ' Rasch-Andrich thresholds of your rating scale
stepdifficulty(bottom&) = 0 ' this is always 0.0
stepdifficulty(bottom&+1) = -3 ' from bottom category to 2nd category
stepdifficulty(bottom&+2) = -1 ' your values go here
stepdifficulty(bottom&+3) = 1
stepdifficulty(bottom&+4) = 3 ' step difficulty into top level

' for the person

' Steps 3) and 4) above

Dim ability!
ability = 2.34 ' an initial logit estimate of ability

' Step 5) above

Dim ExpectedScore!, ModelVariance!

ExpectedScore! = 0
ModelVariance! = 0

ReDim expectation!(itemcount), variance!(itemcount)
Dim item&, logit!, cat&, normalizer!, currentlogit!
Dim value!, expect!, sumsqu!

For item = 1 To itemcount
  if observedrating&(item) > -1 then

   logit! = ability - itemdifficulty(item)

' compute the category probabilities
' and rating expectation
   normalizer = 0 ' this will force the sum of the probabilities = 1.0
   expect = 0
   sumsqu = 0
   currentlogit = 0
   For cat = bottom& to top&
     currentlogit = currentlogit + logit - stepdifficulty(cat)
     value! = Exp(currentlogit)
     normalizer = normalizer + value
     expect = expect + cat * value
     sumsqu = sumsqu + cat * cat * value
   Next cat
   ' expected rating on the item
   expect = expect / normalizer
   expectation(item) = expect ' matches observed rating
   ' model variance on the item
   variance(item) = sumsqu / normalizer - expect ^ 2
   ExpectedScore! = ExpectedScore! + expectation(item)
   ModelVariance! = ModelVariance! + variance(item)
Next item

' Steps 6), 7) go here

' they are an elaboration of ...
' ability = ability + (ObservedScore& - ExpectedScore!)/ModelVariance!
' Loop back to step 5) until the change in ability is too small to matter

' Step 8)
' Final ability estimate is reported
' Standard error of ability estimate = 1 / sqrt(ModelVariance!)

' Next step ....

' This computes fit statistics: see

' we now have the expected ratings for the items and their model variances
' the observed ratings are observedrating&()

Dim ability!, outfitmeansquare!, infitmeansquare!
ReDim standardizedresidual!(itemcount), residual!(itemcount)
Dim infitmeansquaredivisor!, , activeitem&

outfitmeansquare = 0
infitmeansquare = 0
infitmeansquaredivisor = 0
activeitem& = 0
For item = 1 To itemcount
  if observedrating&(item) > -1 then
    activeitem = activeitem + 1
    residual(item) = observedrating&(item) - expectation(item)
    standardizedresidual!(item) = residual(item) / Sqr(variance(item))
    If standardizedresidual(item) > 2 Then
        ' report unexpectedly high rating
    ElseIf standardizedresidual(item) < 2 Then
      ' report unexpectedly low rating
    End If
    outfitmeansquare = outfitmeansquare + standardizedresidual(item) ^ 2
    infitmeansquare = infitmeansquare + residual(item) ^ 2
    infitmeansquaredivisor = infitmeansquaredivisor + variance(item)
Next item
' fit for the person
outfitmeansquare = outfitmeansquare / activeitem
infitmeansquare = infitmeansquare / infitmeansquaredivisor
' if outfitmeansquare or infitmeansquare are > 1.5 there is noticeable noisy misfit.

Estimating Rasch measures with known polytomous (or rating scale) item difficulties: Anchored Maximum Likelihood Estimation (AMLE), Linacre J.M. … 1998, Rasch Measurement Transactions 12:2 p. 638.

Please help with Standard Dataset 4: Andrich Rating Scale Model

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

To be emailed about new material on
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from welcomes your comments:

Your email address (if you want us to reply):


ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website,

Coming Rasch-related Events
July 31 - Aug. 3, 2017, Mon.-Thurs. Joint IMEKO TC1-TC7-TC13 Symposium 2017: Measurement Science challenges in Natural and Social Sciences, Rio de Janeiro, Brazil,
Aug. 7-9, 2017, Mon-Wed. In-person workshop and research coloquium: Effect size of family and school indexes in writing competence using TERCE data (C. Pardo, A. Atorressi, Winsteps), Bariloche Argentina. Carlos Pardo, Universidad Catòlica de Colombia
Aug. 7-9, 2017, Mon-Wed. PROMS 2017: Pacific Rim Objective Measurement Symposium, Sabah, Borneo, Malaysia,
Aug. 10, 2017, Thurs. In-person Winsteps Training Workshop (M. Linacre, Winsteps), Sydney, Australia.
Aug. 11 - Sept. 8, 2017, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),
Aug. 18-21, 2017, Fri.-Mon. IACAT 2017: International Association for Computerized Adaptive Testing, Niigata, Japan,
Sept. 15-16, 2017, Fri.-Sat. IOMC 2017: International Outcome Measurement Conference, Chicago,
Oct. 13 - Nov. 10, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
Oct. 25-27, 2017, Wed.-Fri. In-person workshop: Applying the Rasch Model hands-on introductory workshop, Melbourne, Australia (T. Bond, B&FSteps), Announcement
Jan. 5 - Feb. 2, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
Jan. 10-16, 2018, Wed.-Tues. In-person workshop: Advanced Course in Rasch Measurement Theory and the application of RUMM2030, Perth, Australia (D. Andrich), Announcement
Jan. 17-19, 2018, Wed.-Fri. Rasch Conference: Seventh International Conference on Probabilistic Models for Measurement, Matilda Bay Club, Perth, Australia, Website
April 13-17, 2018, Fri.-Tues. AERA, New York, NY,
May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps),
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
The HTML to add "Coming Rasch-related Events" to your webpage is:
<script type="text/javascript" src=""></script>


The URL of this page is