# [Rasch] What measure is halfway?

Mike Linacre (RMT) rmt at rasch.org
Sun Nov 18 09:24:17 EST 2007

```Dear Rasch practitioners ....

Here's a question I've been asked - and your viewpoints would be appreciated.

Imagine a rating scale for an item from 0 to 7 (8 categories). So the
categories are 0, 1, 2, 3, 4, 5, 6, 7. The central categories are 3 and 4.
What person measure corresponds to half-way along the rating scale?

A. The person measure for which the expected score on the item is half-way
between 0 and 7, i.e., 3.5 rating points.
B. The person measure for which the probability of being observed in the
bottom half of the rating scale (categories 3 and below) is the same as the
probability of being observed in the top half of the rating scale
(categories 4 and above).
C. The person measure for which the probability of being observed in
central category 3 is the same as the probability of being observed in
central category 4.

Which do you think? Or do you have a fourth answer?

Let's make the rating scale from 0 to 6 (7 categories): 0,1,2,3,4,5,6, so
that the central category is 3.
What person measure corresponds to half-way along the rating scale?

D1. The person measure for which the expected score on the item is half-way
between 0 and 6, i.e., 3 rating points
D2. The person measure at which central category 3 has the highest
probability (which is the same as D1 for the usual Rasch models).
E. The person measure for which the probability of being observed in the
lower section of the rating scale (categories 0,1,2) is the same as the
probability of being observed in the upper section (categories 4,5,6 and
above).

Now which do you think? Or do you have another answer?

Mike Linacre

```