Rosenthal and Rubin (1979) deliberately challenge conventional statistical wisdom by means of a paradoxical interpretation of a simple data set. They suppose that 100 patients in a medical study are randomly assigned to new treatment N and 100 to standard treatment S. At the end of one year, 70% of the patients under treatment N are still alive, but only 30% under treatment S. Here is their data matrix:
Treatment Survival after Assignment Dead one year New 30 70 Standard 70 30
How much better is Treatment N than Treatment S?
1) The Pearson product-moment correlation, r, between surviving and receiving treatment N is 0.4. Not an impressive value.
2) The variance in "survival explained" by assignment to treatment is r^2 = 0.4^2 = 0.16, i.e., only 16%. This is less than 1/6th of the total variance. Obviously a small and, therefore, obviously an "unimportant" amount.
But, when considering one's own medical treatment, Treatment N is clearly preferable to Treatment S. Rosenthal and Rubin (1979) stop here, having made their point that the relationship between statistics and substance is not always obvious. Let us continue.
3) The standard error of r for testing whether r=0 is approximately 1/sqrt(200) = 0.07. t = 0.4/0.07 = 5.7. Thus the probability that there is no correlation between treatment and survival is <.0001. This is statistically very significant, but it enables us only to declare that N and S are "not the same". It does not say how much better N is than S.
4) The logit distance between the effects of treatments N and S is the difference between the log-odds for survival with N and the log-odds for survival with S = loge(70/30) - loge (30/70) = 1.7 logits, with a standard error of sqrt(200/(30*70)) = 0.3 logits. Testing the hypothesis that the distance is zero, t = 1.7/0.3 = 5.7 (as expected).
5) Let us take this extremely simple logit analysis a step further. The analysis provides a useful 95% confidence interval for the survival advantage of the New treatment over the Standard:
(New - Standard) = 1.7 +- (2 * 0.3) = 1.7 +- 0.6 logits
Now we can conclude, with 95% confidence, that the survival advantage of New over Standard is:
1.1 logits < (New - Standard) < 2.3 logits
6) Finally, what are these advantages of 1.1, 1.7 and 2.3 logits on New treatment benefit worth to you, a prospective patient? The advantage to you depends on your Standard probability of survival.
Your Your New survival probability Standard for various logit advantages Survival At least Average At most Probability 1.1 1.7 2.3 .10 .25 .38 .50 .20 .43 .58 .71 .30 (in example) .56 .70 .81 .40 .67 .78 .87 .50 .75 .85 .91
Despite the low correlation of 0.4 and low "explained variance" of only 16%, are you in any doubt as to which treatment is best for you? These data, when understood from the Rasch measurement point of view, are in no way paradoxical. They provide an entirely decisive answer to your important question.
By choosing to think with measures rather than correlations and variances, the conflict between the substantive and the statistical findings disappears.
Rosenthal R, Rubin DR (1979) A note on percent variance explained as a measure of the importance of effects. Journal of Applied Social Psychology 9:5, 395-396.
Measures, correlations and explained variances. Rosentahl R, Rubin DR, Linacre Jm, Wright BD. Rasch Measurement Transactions, 1995, 9:2 p.435
|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|
|March 21, 2019, Thur.||13th annual meeting of the UK Rasch user group, Cambridge, UK, http://www.cambridgeassessment.org.uk/events/uk-rasch-user-group-2019|
|April 4 - 8, 2019, Thur.-Mon.||NCME annual meeting, Toronto, Canada,https://ncme.connectedcommunity.org/meetings/annual|
|April 5 - 9, 2019, Fri.-Tue.||AERA annual meeting, Toronto, Canada,www.aera.net/Events-Meetings/Annual-Meeting|
|April 12, 2019, Fri.||On-line course: Understanding Rasch Measurement Theory - Master's Level (G. Masters), https://www.acer.org/au/professional-learning/postgraduate/rasch|
|May 24 - June 21, 2019, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|May 22 - 30, 2019, Wed.-Thu.||Measuring and scale construction (with the Rasch Model), University of Manchester, England, https://www.cmist.manchester.ac.uk/study/short/intermediate/measurement-with-the-rasch-model/|
|June 17-19, 2019, Mon.-Wed.||In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Introduction to Rasch measurement (Trevor Bond, Winsteps), Announcement|
|June 20-21, 2019, Thurs.-Fri.||In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Advanced Rasch measurement with Facets (Trevor Bond, Facets), Announcement|
|June 28 - July 26, 2019, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
|July 2-5, 2019, Tue.-Fri.||2019 International Measurement Confederation (IMEKO) Joint Symposium, St. Petersburg, Russia,https://imeko19-spb.org|
|July 11-12 & 15-19, 2019, Thu.-Fri.||A Course in Rasch Measurement Theory (D.Andrich), University of Western Australia, Perth, Australia, flyer - http://www.education.uwa.edu.au/ppl/courses|
|Aug 5 - 10, 2019, Mon.-Sat.||6th International Summer School "Applied Psychometrics in Psychology and Education", Institute of Education at HSE University Moscow, Russia.https://ioe.hse.ru/en/announcements/248134963.html|
|Aug. 9 - Sept. 6, 2019, Fri.-Fri.||On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com|
|August 25-30, 2019, Sun.-Fri.||Pacific Rim Objective Measurement Society (PROMS) 2019, Surabaya, Indonesia https://proms.promsociety.org/2019/|
|Oct. 11 - Nov. 8, 2019, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|Jan. 24 - Feb. 21, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|May 22 - June 19, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 26 - July 24, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
|Aug. 7 - Sept. 4, 2020, Fri.-Fri.||On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com|
|Oct. 9 - Nov. 6, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 25 - July 23, 2021, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
The URL of this page is www.rasch.org/rmt/rmt92m.htm