These letters illustrate how Lord's and Wright's explorations intersected and then diverged, with Lord basing his thinking on the normal ogive and the frailties of empirical data, but Wright basing his on objective measurement and the demands of an ideal model.
Letter from Frederic M. Lord to Benjamin D. Wright, November 18, 1965: "Rasch's model for unspeeded tests [the Rasch dichotomous model] can be considered as a special case of the normal-ogive model, as Rasch himself points out extremely briefly at the end of Section 8 of his Chapter VII. The usual normal-ogive model has two parameters for each item, whereas Rasch uses only one of these. Rasch's model is thus a somewhat special case. Birnbaum's [2-PL] logistic model seems to provide a very satisfactory approximation to the normal-ogive model with two parameters per item. Altogether, we are devoting six chapters to the normal-ogive model and to Birnbaum's logistic model in our book ["Statistical Theories of Mental Test Scores"]."
Ben Wright to Fred Lord, November 23, 1965: "About Rasch's item analysis model as described in the latter part of his book, I think he would be horrified to learn that you regard his model as a special case of the normal-ogive model. The special feature of his model is that it allows for separating parameters of objects and agents, that is of children and test items. This is not possible with the normal-ogive model, and, in fact, if one sets down a few reasonable characteristics of objectivity, it can be proven that in the special case where observations are limited to ones and zeros, that the Rasch item analysis model is the only model which retains parameter separability. From Rasch's point of view this separability is a sine qua non for objective measurement."
Fred Lord of Ben Wright, November 26, 1965: "I am aware of the virtue of Rasch's model, which he elucidates very well in his chapter in the Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. On the other hand, it is quite clear that his model cannot really apply to the types of test items usually used in our tests. We all know that test items can have the same difficulty level and still differ very much in discriminating power -- some items have high discriminating power and some have none at all. This means that the item characteristic curves of typical test items frequently cross each other. In Rasch's model, it is impossible for the characteristic curves to intersect (except, of course, at the extreme ends where all curves meet in the same points).
"This leaves us with a dilemma. Shall we have objective measurement, which does not really hold for the test items we use? Or shall we allow the term measurement to include what we get from actual test items? I suppose one possible solution would be to discard all of those items that violate Rasch's assumptions. This possibility would certainly be an interesting one to explore."
Melvin R. Novick to Ben Wright, November 30, 1965: "I enjoyed reading your comment [in the letter to Lord] on Rasch's work as I too share a certain enthusiasm for it despite certain reservations and qualifications. ... More to the point, however, is Birnbaum's demonstration (see page 15 of part V of our text that the third Rasch model is a special case of his more general logistic model which obtains when all items have the same discriminating power. Since few tests are composed of items all having the same discriminating power, the practical utility of the third Rasch model would seem to be limited."
Ben Wright to Fred Lord, December 3, 1965: "If you write out Rasch's model for the binary case, that is where the alternative answers are right and wrong, and introduce a second item parameter, ... you can then take account of the variation and discriminating power of the items. This puts the model into the situation of there being one person parameter and two item parameters. The situation has a slightly unfortunate consequence as far as the estimation of item parameters are concerned. At least at present it seems to me that they now cannot be estimated entirely independently of the standardizing population.
"The other line, of course, is the one that you end up with and that is to only to accept items which conform to the simpler model, that is where the second parameter ... are all the same, let us say all one. Rasch believes that this is the only case where full objectivity can be reached. He has developed a proof which shows that only models of his kind, or models which reduce in a trivial way to his kind, allow for the specific objectivity in which he is interested.
"Should this proof stand the test of other people's scrutiny, well then I think the solution to discard all items that violate the Rasch assumptions may be the most attractive one and may even come to define the domain in which objective measurement is possible."
Ben Wright to Fred Lord, June 12, 1967: "Is there any reason for working for a normal ogive rather than a logistic ogive, or to put it in another way, is there a reason worth the added computing difficulty of working with the normal ogive?"
Fred Lord to Ben Wright, June 20, 1967: "You asked about the relative merits of the normal-ogive and logistic models. It is true that there is better a priori reason to use the normal ogive than the logistic; on the other hand, the difference between the two is so small that it would be very difficult to prove that one model was better than the other. The real answer to the dilemma is surely both models are wrong. Since they are so much alike, it seems futile to wonder whether one is slightly more wrong than the other. For this reason, I would use whichever is most convenient, until such time as we know a better model to use."
Fred Lord and Ben Wright discuss Rasch and IRT Models, F. Lord & B.D. Wright ... Rasch Measurement Transactions, 2010, 24:3 p. 1289-90
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