How to compare two performances? The Merry Men of Sherwood Forest dispute again over archery proficiency. Little John and Will Scarlet shoot 10 arrows at a target. To make the comparison fair, they shoot the same 10 arrows in the same order.
Robin Hood insists "Those 10 arrows were crafted to be equivalent. The comparison should be:"
|Robin Hood's Analysis|
|Hits||Misses||Odds of Success||Log-Odds|
"John is 6 times better than Will!"
Maid Marian is skeptical. "No two arrows are identical. Some fly true. Some less so. If John and Will miss the target with the same arrow, perhaps that arrow is defective. If John and Will hit the target with the same arrow, perhaps that arrow is exceptionally well made. The comparison should be:"
|Maid Marian's Analysis|
|John' Hits||John's Misses|
|(John's Hits &
(Will's Hits &
"John is still better than Will, but only 5 times."
Robin Hood objects: "But you are throwing away almost half of the data! My answer must be better."
Friar Tuck intercedes. "Those answers mean the same! You have forgotten that meaning is in the probabilities, not data! Write both statements in probability terms counting only the arrows you counted! Let PJ be John's probability of hitting the target. PW be Will's. Here is Robin Hood's computation, considering the arrows to be equivalent:"
|Friar Tuck's Version of Robin Hood's Analysis|
|Hits||Misses||Odds of Success on Target|
|John||PJ||1-PJ||PJ / (1-PJ)|
|Will||PW||1-PW||PW / (1-PW)|
|PJ / (1-PJ)
PW / (1-PW)
|PJ * (1-PW)|
PW * (1-PJ)
"and here is Maid Marian's, considering the arrows to be heterogeneous:"
|Friar Tuck's Version of Maid Marian's Analysis|
|John' Hits||John's Misses|
|Will's Hits||PW * (1-PJ)|
|Will's Misses||PJ * (1-PW)|
John (John's Hits & Will's Misses) PJ * (1-PW)
------ = ------------------------------------- = ---------------
Will (Will's Hits & John's Misses) PW * (1-PJ)
"Algebraically, both numbers are attempts to estimate the same unknowable truth! Now look at those numbers in log-odds terms. The difference between the measures is 1.6 - 1.0 = 0.6, but the precision of a measure is ±1.0 at best. Numerically, we are talking about the same number. 5 times better or 6? All we are arguing over is the imprecision in the estimates. These results provide no firm evidence about the quality of the arrows one way or the other, so we don't know which estimate is better, but we do know that regarding the arrows as heterogeneous is the more conservative option. So, Maid Marian's 5 times is the more defensible result!"
John Michael Linacre
A Pointed Argument in Sherwood Forest. Linacre J.M. Rasch Measurement Transactions, 1997, 11:3 p. 574
Please help with Standard Dataset 4: Andrich Rating Scale Model
|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|
|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|
|Oct. 25-27, 2017, Wed.-Fri.||In-person workshop: Applying the Rasch Model hands-on introductory workshop, Melbourne, Australia (T. Bond, B&FSteps), Announcement|
|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|
|April 13-17, 2018, Fri.-Tues.||AERA, New York, NY, www.aera.net|
|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|
|The HTML to add "Coming Rasch-related Events" to your webpage is:|
The URL of this page is www.rasch.org/rmt/rmt113c.htm