Sufficiency—The Significance of Total Scores

Abstract

The total scoreTotal score$$ r_{n} $$of person n on a set of items in the Rasch model is a sufficient statisticSufficient statistic. Sufficiency implies that there is no further information about the person’s proficiency $$ \beta_{n} $$in the pattern of the person’s responses. If the response patterns fit the Rasch model, then they are likely to be close to the Guttman pattern (but not perfectly) and in the case of patterns close to the Guttman pattern, there is no further information in the profile other than that in the total scoreTotal score. The Rasch model is a probabilistic form of the Guttman structureGuttman structureand the Guttman structureGuttman structureis a limiting deterministicDeterministic modelcase of the probabilistic Rasch model. Symmetrically, the total scoreTotal scoreof an item is a sufficient statisticSufficient statisticfor the item’s difficulty with the same implications as sufficiency of the total scoreSufficiency of the total scorefor persons.