Success Rates or Log-Odds? What Makes Sense?

Fred Mosteller and John Tukey (M&T) untangle many problems in "Data Analysis and Regression" (2nd Ed. Reading, MA: Addison-Wesley, 1977). Occasionally, however, they lose sight of meaning. M&T's Exhibit 11, Chapter 11, also printed on the book's back cover, is an example.

Exhibit 11 plots the success rates of two treatments on non-equivalent "Easy-to-treat" and "Hard-to-treat groups. M&T describe techniques for equating the groups that result in locating the four treatment groups on an infinite linear "standard" scale of treatment difficulty, the X-axis in M&T's Exhibit 11.

M&T's plotting of "Rate of success %" on the Y-axis, however, clouds the meaning:

1) M&T extrapolate Treatment II's regression line beyond 100% into the unobservable. What can this regression line mean?

2) M&T plot a straight regression line that maps a finite "Rate of Success %" scale onto an infinite "Standard difficulty" scale. This implies that, as patient groups get "harder", they soon intercept zero and so have no chance of successful treatment. Also, as patient groups get "easier", they become certain to be treated 100% successfully, but this is clearly unachievable. Indeed, are we to infer from this plot that no patients in the "hard" group of Treatment II could have been successfully treated by Treatment I?

Rescaling "Rate of success %" to "Log-odds of success" in the "Log- Odds" plot solves both problems. This plot looks like M&T's Exhibit 11, but has infinite range on both axes. Now the regression lines are meaningful everywhere.

This can be seen in the "Ogival" plot that transforms the linear log-odds Y- axis back to the ogival "Rate of success %". The ogives do not exceed the 100% success rate limit. They do not predict unrealizable treatment success or failure. A meaningful rate of success with Treatment I can now be predicted for the "hard" group of Treatment II.

Success Rates or Log-Odds? What Makes Sense?, Linacre JM, Mosteller DCF, Tukey JW … Rasch Measurement Transactions, 1993, 6:4 p. 264

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