# Practical Adaptive Testing CAT Algorithm

Here are the core steps needed for practical adaptive testing with the Rasch model.

0. Request next candidate: Set D=0, L=0, H=0, and R=0.
1. Find next item near difficulty (D).
2. Set D at the actual calibration of that item.
4. Obtain a response.
5. Score that response.
6. Count the items taken: L = L + 1
7. Add the difficulties used: H = H + D

If response not correct,
8. Target next item difficulty: D = D - 2/L

If response correct,
9. Count right answers: R = R + 1
10. Target next item difficulty: D = D + 2/L

If not ready to decide to pass/fail,
11. Go to step 1.

12. Calculate wrong answers: W = L - R
13. Estimate measure: B = H/L + loge(R/W)
If W = 0 then B = H/L + loge(R-0.5 / W+0.5)
If R = 0 then B = H/L + loge(R+0.5 / W-0.5)
For a more precise estimate of B, see Estimating Measures with Known Item Difficulties
14. Estimate Error: S = sqrt[L/(R*W)]
If W = 0 then S = sqrt[L/(R-0.5 * W+0.5)]
If R = 0 then S = sqrt[L/(R+0.5 * W-0.5)]

15. Compare measure B with pass/fail standard T.
16. If (T - S) < B < (T + S), go to step 1.
17. If (B - S) > T, then pass.
18. If (B + S) < T, then fail.
20. Go to step 0.

Practical adaptive testing. Wright BD. … Rasch Measurement Transactions 2:2 p.24

Practical adaptive testing. Wright BD. … Rasch Measurement Transactions, 1988, 2:2 p.24

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

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