A Pointed Argument in Sherwood Forest!

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.

The Data
Arrow: 1 2 3 4 5 6 7 8 9 10 Score
John Hit H H H M M H H H H 8
Will Miss H M H H M M M H M 4

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 8 2 8/2 1.4±.8
Will 4 6 4/6 -.4±.6

"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
Will's Hits - 1
Will's Misses 5 -
---- =
(John's Hits &
Will's Misses)
(Will's Hits &
John's Misses)

"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

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
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