The standard error of measurement (S.E.) is widely used for stopping a computer-adaptive test. For instance, if the current measure estimate is more than 1.96 S.E.s from the pass-fail measure, then there is 95% confidence in the pass-fail decision. Or 2.58 S.E.s for 99% confidence. But how many items are needed to reach a desired S.E.?
If a person has probability, P, of succeeding on a dichotomous item (such as a multiple-choice question), then the statistical information in the response is P*(1-P). The standard error of the estimated measure is
S.E. = 1/sqrt(information) = 1/ sqrt(sum(P*(1-P)))
The largest information, and so the smallest standard error, occurs when P=0.5, i.e., when the CAT items are targeted exactly on the persons. But this can produce an unsatisfactory testing experience for the examinee so higher probabilities of success are targeted, such as .7 (for 70% success) and .8 (for 80% success). Here is a Table showing the targeting, standard error, and minimum number of items administered for a specific S.E.:
| Minimum number of CAT Items Administered | ||||||
|---|---|---|---|---|---|---|
| Targeting Probability | S.E. (Logits) | |||||
| 0.5 | 0.4 | 0.3 | 0.2 | 0.15 | 0.1 | |
| P=0.5 | 16 | 25 | 45 | 100 | 178 | 400 |
| 0.6 | 17 | 27 | 47 | 105 | 186 | 417 |
| 0.7 | 20 | 30 | 53 | 120 | 212 | 477 |
| 0.8 | 25 | 40 | 70 | 157 | 278 | 625 |
| 0.9 | 45 | 70 | 124 | 278 | 494 | 1112 |
It is seen that the penalty for going from P=0.5 to P=0.6 targeting is the administration of about 5% more items. From P=0.5 to P=0.7 is about 20% more items. From P=0.5 to P=0.8 is 60% more items. P=0.9 almost triples the test length. An S.E. of 0.15 logits requires about 10 times as many items as an S.E. of 0.5 logits.
| Minimum Number of Items for 95% Confidence (|t|>=1.96) in Pass-Fail Decision | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Targeting Probability | Logit Distance of Ability Estimate from Pass-Fail Point | |||||||||
| 1 | 0.9 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 | |
| P=0.5 | 16 | 19 | 25 | 32 | 43 | 62 | 97 | 171 | 385 | 1537 |
| 0.6 | 17 | 20 | 26 | 33 | 45 | 65 | 101 | 178 | 401 | 1601 |
| 0.7 | 19 | 23 | 29 | 38 | 51 | 74 | 115 | 204 | 458 | 1830 |
| 0.8 | 25 | 30 | 38 | 49 | 67 | 97 | 151 | 267 | 601 | 2401 |
| 0.9 | 43 | 53 | 67 | 88 | 119 | 171 | 267 | 475 | 1068 | 4269 |
John Michael Linacre
Computer-Adaptive Tests (CAT), Standard Errors and Stopping Rules, Linacre J.M. … Rasch Measurement Transactions, 2006, 20:2 p. 1062
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