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 P=.7 (for 70% success: items are selected so that the person achieves about 70% success on the administered items) and P=.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 of Success | 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 of Success | 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 |
When administering many items in a CAT test, it is also wise to consider item response times: "Utilizing Response Time Distributions for Item Selection in CAT," Zhewen Fan, Chun Wang, Hua-Hua Chang, and Jeffrey Douglas, Journal of Education and Behavioral Statistics, 2012.
John Michael Linacre
Computer-Adaptive Tests (CAT), Standard Errors and Stopping Rules, Linacre J.M. Rasch Measurement Transactions, 2006, 20:2 p. 1062
Forum | Rasch Measurement Forum to discuss any Rasch-related topic |
Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement
Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.
Coming Rasch-related Events | |
---|---|
Oct. 4 - Nov. 8, 2024, Fri.-Fri. | On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
Jan. 17 - Feb. 21, 2025, Fri.-Fri. | On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
May 16 - June 20, 2025, Fri.-Fri. | On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
June 20 - July 18, 2025, Fri.-Fri. | On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com |
Oct. 3 - Nov. 7, 2025, Fri.-Fri. | On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
The URL of this page is www.rasch.org/rmt/rmt202f.htm
Website: www.rasch.org/rmt/contents.htm