Twice as hard...

Question: If one has, for example, three items. Item Two is one logit more difficult than Item One. Item Three is two logits more difficult than Item One. Can one say that Item Three is three times as hard as Item One, and Item Two is twice as hard?

An answer: Thinking through parallel situations in physical measurement almost always clarifies problems like this. Suppose that, instead of items, we are investigating the heights of mountains. Mountain Two is 1 kilometer higher than Mountain One. Mountain Two is 2 kilometers higher than Mountain One. What is the ratio of the heights of the mountains? We don't know until we have chosen a zero reference point, an origin. The usual choice is "sea-level". Then, if Mountain One is 1 kilometer high, Mountain Two is twice as high, etc. So in your example, if Item One is one logit more difficult than some reference origin, then Item Two would be twice as hard, etc.

But how do we choose an origin? Even for mountains a choice must be made, because there are mountains under the sea! Where is the origin of your scale? You must choose a useful location. Typical locations are the average difficulty of all your items. Or the average ability of all persons in your sample. Or the lowest ability you plan to measure. Or the difficulty of the easiest item. But other choices could be as good, or better, for you to make best use of your measures.

Twice as hard... Linacre J.M. … Rasch Measurement Transactions, 2002, 15:4 p. 855

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
Oct. 6 - Nov. 3, 2023, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Facets),
Oct. 12, 2023, Thursday 5 to 7 pm Colombian timeOn-line workshop: Deconstruyendo el concepto de validez y Discusiones sobre estimaciones de confiabilidad SICAPSI (J. Escobar, C.Pardo)
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden
Aug. 9 - Sept. 6, 2024, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),


The URL of this page is