Sequential classification on partially ordered sets
by Curits Tatsuoka & Thomas Ferguson (2003)
Partially ordered sets (posets) can be used in cognitive diagnosis and computerized adaptive testing (i.e., Tatsuoka, 1995). The basic problem is to choose a sequence of experiments sequentially and a stopping rule to determine the true state as quickly as possible. In Tatsuoka (2002), the partially ordered classification model was used for cognitive diagnosis. The result shows that comparison to non-sequentially classifications, sequentially classifications can reduce the amount of administered items to achieve the similar accuracy. One important property of the poset classification model is that the optimal rates of the posterior probability convergence are exponential. The stopping rule is that the posterior probability of the true state converge to 0.8, it is imply that the more peak of the posterior probability the more reliable of the estimate. Following Tatsuoka (2002), the purposes of this paper are that, (1) exploring the rate of convergence and find the optimal rate of convergence of the posterior probability of the true state to one; (2) introducing a class of experiment selection procedures. For the first purpose, it is find that the posterior probability of the true state converge to 1 at the exponential rate. For the second purpose, the halving algorithm and the Shannon entropy were introduced.
Comments
The cognitive diagnostic models (CDMs) can be considered as the partially ordered sets model, because the probabilities of some of latent states (or patterns) can be distinguished which is larger (smaller) than the other, but some are not. For example, suppose a 5-attribute test, the probability of possesses 4 attributes is larger than that of possesses 3 attributes, but which probability is the biggest among the latent states which possess 4 attributes are not clear.
Future study can investigate the relationship (i.e., exponential rate) between the number of administer items and the probability of the true state in computerized adaptive testing for cognitive diagnosis when using CDMs.