[Rasch] More on the Rasch model

Mike Linacre mike at winsteps.com
Fri Nov 2 12:55:29 AEDT 2018

Hi Paul:

Thank you for the Excerpts and Commentary about measurement. The math 
behind the Rasch model does not seem to be in doubt. The Rasch model can 
be mathematically derived from many different sets of axioms. It is like 
Pythagoras theorem. It cannot be disproved, but it can certainly be 

A mathematical formula, such as Pythagoras Theorem, does not construct 
anything, but it is a powerful aid and guide in the construction 
process. We know that a planar triangle is substantively right-angled 
when its underlying data (the lengths of the sides) fit Pythagoras 
theorem. Similarly, we know that measures are substantively linear when 
their underlying data (ordinal observations) fit the Rasch model. 
Pythagoras Theorem says nothing about the purpose, utility or duration 
of the triangle, similarly for the Rasch model. However, architects and 
builders apply Pythagoras Theorem in the design and construction of 
buildings (artificial artifacts). Similarly test designers and analysts 
can apply the Rasch model in the design and construction of latent 
variables (artificial artifacts).

For the Rasch model, critiques of the error distribution are critiques 
of the data, not of the model. That is why, in the application of the 
Rasch model, we pay so much attention to parameter-level and 
response-level fit statistics. Everyone agrees that lucky guesses skew 
the error distribution and so the Rasch measures (also the raw scores, 
etc.). When lucky guesses are detected, there are ways to eliminate or 
mitigate them. Pythagoras has the same problem. That is one reason for 
the carpenter's maxim: "Measure twice, cut once." Even in physical 
measurement the data can be bad.

Of course, the perfect Pythagorean triangle does not exist, nor does the 
perfect Rasch variable. However, "Empirical problems are frequently 
solved because, for problem solving purposes, we do not require an 
exact, but only an approximate, resemblance between theoretical results 
and experimental ones." (L. Laudan).

Paul, thank you for provoking us to think about what we are doing :-)

Mike Linacre

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