Computer-adaptive tests are usually designed to have minimum and maximum possible test lengths. A question arises: "How far can examinees fall below a criterion level, before there is no chance that they could achieve that level, even if they succeeded on every subsequent CAT item administered?"
The Nomograph overleaf shows what can happen. Along the x-axis are the number of CAT items administered so far. Two targeting rules are illustrated. One rule, designed for maximum test efficiency, selects the next item such that the examinee has a 50% probability of success on it. The other rule, designed to maximize the examinee's "flow", selects an item with an 80% probability of success.
The "baseline ability" corresponds to the examinee's ability at any point in the CAT test. From this point onwards, the examinee succeeds on every administered item. This causes the examinee's ability estimate to increase and ever harder items to be administered, but according to the targeting rule. The 50% rule produces the solid lines, the 80% rule the dotted lines.
For example, imagine that, after the administration of 160 items, the examinee is 1 logit below the criterion level. Can that examinee pass the test before a maximum of 200 item administrations? On the Nomograph, follow the diagonal lines upward from 160 on the x-axis. 40 extra successes, i.e., 200 items administered in total, only raise the examinee's ability ½ logit. If the test continues to 260 items, then the examinee could just pass under the 50% success rule, but would be ¼ logit low under the 80% success rule.
At what point does it become impossible for an examinee to pass a test? Imagine that the maximum number of items to be administered is 100. The examinee is 1 logit below the pass-fail point. On the Nomograph, locate the point at 100 items and 1 logit. Follow down the solid diagonal line immediately to its right. It intercepts the x-axis at 70 items. Under 50% targeting, any examinee estimated to be one logit below the criterion level (after the administration of 70 items) cannot pass within 100 items. Now follow down the dotted line immediately to the right of the point at 100 items and 1 logit. This dotted line intercepts the x-axis at 60 items. Under the 80% success rule, after 60 items an examinee one logit below a criterion level cannot pass it within 100 items.
John M. Linacre
CAT: Maximum possible ability.Linacre J.M. Rasch Measurement Transactions, 1998, 12:3 p. 657-8.
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