Statistical information is a concept originated by Ronald Fisher (1922, 1925). The information in an observation quantifies how much we learn about the modeled variable from that observation.
There are many types of information in an observation. If we discover from an observation that an MCQ item has an incorrect scoring key, then we have acquired substantive information that leads us to reconsider all statistical information acquired for that item to date. Similarly, if we conclude from an observation that a person is applying a response set, mechanically marking the same response option to several items, then that behavioral information leads us to reconsider the statistical information acquired for that person.
Suppose the data do fit the model. Then we would expect that each extra observation would give us more statistical information, making our conclusions more secure. For the Rasch model, this is always true. The statistical information, I_{ni},in the dichotomous response (right or wrong) of person n to item i is:
where P_{ni} is the
probability of success. If the item is very hard then P_{ni} is
small, and there is little statistical information (but the opportunity
for plenty of misfit if the person succeeds!). Similarly, if the item
is very easy, then (1-P_{ni}) is small, also yielding little
information. I_{ni} is largest when P_{ni}=0.5, i.e.,
when the item is targeted on the person. It is with this targeting that
we have the greatest uncertainty about how the person will answer, and
so the greatest opportunity to learn more about the relationship between
the person's ability and the item's difficulty.
When an observation contains negative information, then we know less about the variable after collecting the observation than we did before, even when the data fit the model. Though this paradox cannot occur with the Rasch model, it occurs in most 3PL (3-PL, three parameter logistic model) analyses.
Imagine an ideal adaptive test. For both 3PL and Rasch, when a person fails an item, then it is likely that the item is hard for that person, and so an easier item is administered next.
But what if the person succeeds? Under Rasch model conditions, it is likely the item is easy for that person, and so a harder item is administered next. Under 3PL conditions, however, a quandary arises. The person succeeds, but why? If success is because the item is easy for that person, then a harder item is administered next. But if success is due to lucky guessing, then the item is too hard for the person. (There's no point to guessing on easy items!) This means that an easier item should be administered next. So 3PL requires us to administer both a harder and an easier item next leaving us more uncertain, i.e., less statistically informed, about the ability of the person relative to the difficulty of the item than before the observation was made! We have lost information, i.e., we have acquired negative statistical information! (See Bradlow, 1996, for a mathematical formulation of this paradox.)
Of course, the quandary can be avoided by arranging targeting and test conditions so that respondents have little incentive to guess. This also removes any justification for including a guessing component in the statistical model. The few remaining lucky guesses are readily identifiable as misfits to a guessingless-model. These can then be dropped from the data for the purposes of calibrating the items, or included for the purposes of measuring and diagnosing person performance.
Benjamin D. Wright
Bradlow E.T. 1996. Teacher's corner: Negative information and the three-parameter logistic model. Journal of Educational Statistics 21:2 179-185.
Fisher R.A. 1922. On the mathematical foundations of theoretical statistics.Proceedings of the Royal Society 222: 309-368.
Fisher R.A. 1925. Theory of statistical estimation. Proceedings of the Cambridge Philosophical Society, 22: 700-725. Also in Collected Papers of R.A. Fisher, ed.J.H. Bennett. Adelaide: The University of Adelaide.
Wright B.D. (1996) Negative information. Rasch Measurement Transactions 10:2 p. 504.
Negative information. Wright B.D. … Rasch Measurement Transactions, 1996, 10:2 p. 504
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