# Computer Adaptive Tests (CAT), Item Selection, Standard Errors and Stopping Rules

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)
0.50.40.30.20.150.1
P=0.5162545100178400
0.6172747105186417
0.7203053120212477
0.8254070157278625
0.945701242784941112

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
10.90.80.70.60.50.40.30.20.1
P=0.5161925324362971713851537
0.61720263345651011784011601
0.71923293851741152044581830
0.82530384967971512676012401
0.94353678811917126747510684269

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

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

 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
March 31, 2017, Fri. Conference: 11th UK Rasch Day, Warwick, UK, www.rasch.org.uk
April 2-3, 2017, Sun.-Mon. Conference: Validity Evidence for Measurement in Mathematics Education (V-M2Ed), San Antonio, TX, Information
April 26-30, 2017, Wed.-Sun. NCME, San Antonio, TX, www.ncme.org - April 29: Ben Wright book
April 27 - May 1, 2017, Thur.-Mon. AERA, San Antonio, TX, www.aera.net
May 26 - June 23, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 30 - July 29, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
July 31 - Aug. 3, 2017, Mon.-Thurs. Joint IMEKO TC1-TC7-TC13 Symposium 2017: Measurement Science challenges in Natural and Social Sciences, Rio de Janeiro, Brazil, imeko-tc7-rio.org.br
Aug. 7-9, 2017, Mon-Wed. In-person workshop and research coloquium: Effect size of family and school indexes in writing competence using TERCE data (C. Pardo, A. Atorressi, Winsteps), Bariloche Argentina. Carlos Pardo, Universidad Catòlica de Colombia
Aug. 7-9, 2017, Mon-Wed. PROMS 2017: Pacific Rim Objective Measurement Symposium, Sabah, Borneo, Malaysia, proms.promsociety.org/2017/
Aug. 10, 2017, Thurs. In-person Winsteps Training Workshop (M. Linacre, Winsteps), Sydney, Australia. www.winsteps.com/sydneyws.htm
Aug. 11 - Sept. 8, 2017, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Aug. 18-21, 2017, Fri.-Mon. IACAT 2017: International Association for Computerized Adaptive Testing, Niigata, Japan, iacat.org
Sept. 15-16, 2017, Fri.-Sat. IOMC 2017: International Outcome Measurement Conference, Chicago, jampress.org/iomc2017.htm
Oct. 13 - Nov. 10, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 5 - Feb. 2, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 10-16, 2018, Wed.-Tues. In-person workshop: Advanced Course in Rasch Measurement Theory and the application of RUMM2030, Perth, Australia (D. Andrich), Announcement
Jan. 17-19, 2018, Wed.-Fri. Rasch Conference: Seventh International Conference on Probabilistic Models for Measurement, Matilda Bay Club, Perth, Australia, Website
May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com