Sufficiency and Conditional Estimation of Person Parameters in the Polytomous Rasch Model

Autor/in	Andrich, David
Titel	Sufficiency and Conditional Estimation of Person Parameters in the Polytomous Rasch Model
Quelle	In: Psychometrika, 75 (2010) 2, S.292-308 (17 Seiten) PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0033-3123
DOI	10.1007/s11336-010-9154-8
Schlagwörter	Simulation; Computation; Statistics; Item Response Theory; Scores; Equated Scores; Algebra; Geometric Concepts; Models + Suchen Sie Ihr Suchwort? Simulation program; Simulationsprogramm; Statistik; Item-Response-Theorie; Elementare Geometrie; Analogiemodell
Abstract	Rasch models are characterised by sufficient statistics for all parameters. In the Rasch unidimensional model for two ordered categories, the parameterisation of the person and item is symmetrical and it is readily established that the total scores of a person and item are sufficient statistics for their respective parameters. In contrast, in the unidimensional polytomous Rasch model for more than two ordered categories, the parameterisation is not symmetrical. Specifically, each item has a vector of item parameters, one for each category, and each person only one person parameter. In addition, different items can have different numbers of categories and, therefore, different numbers of parameters. The sufficient statistic for the parameters of an item is itself a vector. In estimating the person parameters in presently available software, these sufficient statistics are not used to condition out the item parameters. This paper derives a conditional, pairwise, pseudo-likelihood and constructs estimates of the parameters of any number of persons which are independent of all item parameters and of the maximum scores of all items. It also shows that these estimates are consistent. Although Rasch's original work began with equating tests using test scores, and not with items of a test, the polytomous Rasch model has not been applied in this way. Operationally, this is because the current approaches, in which item parameters are estimated first, cannot handle test data where there may be many scores with zero frequencies. A small simulation study shows that, when using the estimation equations derived in this paper, such a property of the data is no impediment to the application of the model at the level of tests. This opens up the possibility of using the polytomous Rasch model directly in equating test scores. (Contains 3 tables.) (As Provided).
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Update	2017/4/10