Unobserved or null categories

There are two types of unobserved or null categories: structural zeroes and incidental/sampling zeroes.


Structural null categories occur when rating scale categories are numbered 10, 20, 30,... instead of 1,2,3. Facets ordinarily eliminates unobserved categories.


incidental/sampling zeroes occur when occur when rating scale categories are numbered 1, 2, 3, ... but a category such as 2 is not observed this time. Since Facets ordinarily eliminates unobserved categories, the unobserved categories must be "kept". Extreme unobserved categories can only be kept by anchoring.


For intermediate incidental or sampling null zeroes, imagine this scenario: The Wright & Masters "Liking for Science" data are rescored from 0,1,2 to 0,1,3 with a null category at 2. the categories now mean "disagree, neutral, somewhat agree, agree". We can imagine that no child in this sample selected the half-smile of somewhat agree.

The category frequencies of categories 0,1,2,3 are 378, 620, 0, 852

The three Rasch-Andrich threshold parameters are -.89, +infinity, -infinity.

The +infinity is because the second parameter is of the order log(620/0). The -infinity is because the third parameter is of the order log(0/852).

Mark Wilson's 1991 insight was that the leap from the 2nd to the 4th category is of the order log(620/852). This is all that is needed for immediate item and person estimation. But it is not satisfactory for anchoring rating scales. In practice however, a large value substitutes satisfactorily for infinity. So, a large value such as 40 logits is used for anchoring purposes. Thus the approximated parameters become -.89, 40.89, -40.00 for the Anchorfile=. With these anchored threshold values, the expected category frequencies become: 378.8, 619.4, .0, 851.8. None of these are more than 1 score point away from their observed values, and each represents a discrepancy of .2% or less of its category count. To force unobserved intermediate categories into the analysis, use:

Models = ?,?,?, R9K


Models = ?,?,?, myscale

Rating scale = myscale, R9, Keep


Extreme incidental null categories (unobserved top or bottom categories) are essentially out of range of the sample and so the sample provides no direct information about their estimates. To estimate those estimates requires us to make an assertion about the form of the rating scale structure. The Rasch "Poisson" scale is a good example. All its infinitude of thresholds are estimable because they are asserted to have a specific form. .


Our recommendation is that structural zeroes be rescored out of the data.


Unobserved extreme categories:

If these categories may be observed next time, then it is better to include dummy data records in your data file which include observations of the missing category and reasonable values for all the other item responses that accord with that missing category. Give the data dummy records a very small weight using R weighting. These few data records will have minimal impact on the rest of the analysis.

If there are too many unobserved categories, then it may be better to impute a rating-structure using anchor values in Rating scale=, or model the rating scale as a binomial trial.



Question 1: The items in my data have different scoring systems. For example,

Items 1 and 2 have categories: 0, 3, 6, 8

Items 3 and 4 have categories: 0, 4, 8, 12


Answer: Modeling these depends on how you conceptualize these scales. Unobserved categories are always difficult to model.


A. If 0,3,6,8 really mean 0,1,2,3

and 0,4,8,12 really means 0,1,2,3, then, if items are facet 2


?, 1-2, ?, R8

?, 3-4, ?, R12



B. Or if 0,3,6,8 really mean 0,1,2,3,4,5,6,7,8

and 0,4,8,12 really means 0,1,2,3,4,5,6,7,8,9,10,11,12, then


?, 1-2, ?, scalea

?, 3-4, ?, scaleb


rating scale=scalea,R8,K

rating scale=scalea,R12,K


C. Or if 0,3,6,8 really mean 0,1,2,3,4,5,6,7,8, 9, 10,11,12

and 0,4,8,12 really means 0,1,2,3,4,5,6,7,8,9,10,11,12, both on the same 0-12 scale, then


?, ?, ?, scalec


rating scale=scalec,R12,K


D. Or if 0,3,6,8 and 0,4,8,12 correspond to 0,3,4,6,8,12 and really mean 0,1,2,3,4,5, then






Question 2: I am using the Partial Credit Model, #, and my raters have used different parts of the rating scale. How can I get the full range of the rating scale reported for every rater?


Answer: Include a dummy examinee (person) rated in the highest and lowest category by every rater. Also include "K" (for Keep) next to the rating scale specification.


Example: the rating scale should go from 1 to 7, but some raters have missing categories:

Models = ?,?,#, R7K ; raters are facet 3


1, Examinees

9999=Dummy,,,, 0.0001 '; a dummy examinee with a very small weight

(other examinees unchanged)


Data =

(your data, then)

9999, 0, 1, 1  ; rater 1 gives a rating of 1 to examinee 9999

9999, 0, 1, 7  ; rater 1 gives a rating of 7 to examinee 9999

and so on for all raters

Replace 0 by relevant element numbers for facet 2 if needed.

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Rasch Publications
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Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments, George Engelhard, Jr. & Stefanie Wind 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
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