Sufficiency—The Significance of Total Scores

Abstract

The total score$r_{n}$ of person n on a set of items in the Rasch model is a sufficient 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 score. The Rasch model is a probabilistic form of the Guttman structure and the Guttman structure is a limiting deterministic case of the probabilistic Rasch model. Symmetrically, the total score of an item is a sufficient statistic for the item’s difficulty with the same implications as sufficiency of the total score for persons.

Exercises

Below is a table showing the estimated person location for three persons for a test with 42 dichotomous items. The three persons all have a total score of 21 and then the same estimate of −0.004. Below are their response patterns when items are ordered according to difficulty.

Person ID	Total score	Max score	Location
01	21	42	−0.004
02	21	42	−0.004
03	21	42	−0.004

01	010110110110100111010110011100001101000001
02	111111111111111111111000100001000000000000
03	111111011101111011110001000100000001001000

Given that the location estimates of persons with the same total scores are the same, what are the advantages of analysing the responses using the Rasch model? In your answer refer to the data fit and the response patterns for the persons above.