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 Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

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