A popular "hierarchy" of psychometric models is
"1-parameter"
"2-parameter"
"3-parameter"
These models are said to be a "hierarchy" because when C_{i} = 0, then 3p becomes 2p, and when A_{i}=1, then 2p becomes 1p. This is sometimes interpreted to imply that the "best" model must be the one with the most parameters, i.e., 3p. The purpose of these models, however, is to construct scales on which measures can be made. The fundamental question, then, is do these models fulfill that purpose?
According to L.L. Thurstone (American Journal of Sociology, 1928, 33, 529-554, reprinted in The Measurement of Values, 1959, 228):
"The scale must transcend the group measured... One crucial experimental test must be applied to our method of measuring attitudes before it can be accepted as valid. A measuring instrument must not be seriously affected in its measuring function by the object of measurement. To the extent that its measuring function is so affected, the validity of the instrument is impaired or limited. If a yardstick measured differently because of the fact that it was a rug, a picture, or a piece of paper that was being measured, then to that extent the trustworthiness of that yardstick as a measuring device would be impaired. Within the range of objects for which the measuring instrument is intended, its function must be independent of the object of measurement."
What kind of model is necessary to meet Thurstone's requirement for a scale?
What model is required for the comparison and hence calibration of questions i and j to be independent of the persons used to elicit evidence of their relative standing on the scale? Questions i and j appear different only when they are answered differently. Realizing a comparison of i and j requires comparing how often i is answered "yes" when j is simultaneously answered "no" with how often the reverse happens.
1) The estimation of a comparison of i and j from these frequencies requires a comparison of their probabilities for some person n,
But this comparison must be the same regardless of which persons are involved. To satisfy Thurstone's requirement for scale validity, it must be the case that the comparison of questions i with j is always the same whoever the person.
2) Thus it must follow for any other person m that
3) For simplicity let j=o and m=o become the reference origins of the question and person scales so that the calibration of question i becomes its difference from "standard" item j=o and the measure of person n becomes its difference from "standard" person m=o, and align these scale origins so that P_{mj} = P_{oo} = ½, then
which defines a ratio scale for the measure of person n and the calibration of item i.
4) This ratio scale can be linearized by
which when solved for P_{ni} shows that
is the model necessary for Thurstone scale validity.
Step 2 can be rewritten to address Thurstone's concomitant 1926 requirement that the individual measure not depend on which particular items are used so that it becomes "possible to omit several test questions at different levels of the scale without affecting the individual score." This requires that the comparison of any pair of persons n and m be invariant with respect to the particular items employed as in
which is equivalent to step (2) and so leads to step (4).
Thus of the popular "hierarchy", only the "1-parameter" Rasch model meets Thurstone's requirement for scale validity. The "2-parameter" and "3-parameter" models do not. Their use does not, and cannot, produce a scale which is independent of the persons used to obtain it.
Benjamin D. Wright
Rasch model derived from Thurstone's scaling requirements. Wright B.D. … Rasch Measurement Transactions, 1988, 2:1 p. 13-4.
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