# [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
mis-applied.

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
```