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

For dichotomies see www.rasch.org/rmt/rmt102t.htm

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)
 endif
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 www.rasch.org/rmt/rmt34e.htm

' 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)
 endif
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.




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

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