Rasch Estimation with Unobserved or Null Categories

With any set of polytomous items, where all response categories are logically possible for each item, it will sometimes happen that certain categories are unused in the immediate sample of data. This sample characteristic can not be allowed to interfere with a response framework which is a characteristic of the research design. The distinction between a logically null category and a category that is observed to be null in some sample is analogous to the distinction between a structural zero and a sampling zero in contingency table analysis (Fienberg 1985). When a category is not used for a particular item, we would not recommend omitting reference to that category by down-coding the categories above it, such as is the default in many Rasch analysis programs. This kind of down-coding can alter a person's ranking on a test. Of course, when a sampling zero occurs repeatedly for a particular item, it is wise to examine the item for an explanation.

A simple modification of the Partial Credit model retains all categories so that none are "collapsed" out (Wilson & Masters 1991). This reformulation has been incorporated into several Rasch programs. The partial-credit scale has categories 0 to m_i. If z is the category corresponding to a sampling zero (with local probability of zero) and categories z-1 and z+1 are present in the data, then, in the notation of Wright & Masters (1982),

P_nix = exp(sum j=0 to x (except z) (B_n - D_ij) /
      sum k=0 to m_i (except z) (exp (sum j=0 to k (except z) ((B_n - D_ij) )

with Pniz locally zero.

The full set of Andrich thresholds can be approximated by:
D' = D_{iz-1, iz+1} skipping over D_iz
D_i1, D_i2, ..., D_iz-1, D_iz=D'+40, D_iz+1=D'-40, ... D_mi
then Pniz ≈ 0

Example: if a dichotomous 0,1 observation is recoded as a polytomous 0,(1),2 observation, where 1 is unobserved, then
For 0-1 the Andrich threshold is 0 relative to the item difficulty.
For 0-(1)-2 the Andrich thresholds are 40, -40 relative to the same item difficulty.

See also: Unobserved categories: Estimating and anchoring Rasch Measures. RMT 17:2 p. 924-925

Rasch Estimation with Unobserved or Null Intermediate Categories, M Wilson … Rasch Measurement Transactions, 1991, 5:1 p. 128

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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