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
of Success
S.E. (Logits)

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
of Success
Logit Distance of Ability Estimate from Pass-Fail Point

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

To be emailed about new material on
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from welcomes your comments:

Your email address (if you want us to reply):


ForumRasch 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,

Coming Rasch-related Events
June 23 - July 21, 2023, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps),
Aug. 11 - Sept. 8, 2023, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),


The URL of this page is