# [Rasch] What measure is halfway?

David Andrich david.andrich at uwa.edu.au
Mon Nov 19 13:30:28 EST 2007

```Here is a go at replying though it is a bit of a strange question. It seems
to ask for a definition in terms of observed responses, not what will happen
if we define the half way point. See below in CAPS.
David

David Andrich, BSc., MEd. (UWA), PhD (Chic), FASSA
Chapple Chair, Graduate School of Education
The University of Western Australia
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-----Original Message-----
From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf
Of Mike Linacre (RMT)
Sent: Sunday, 18 November 2007 7:24 AM
To: rasch at acer.edu.au
Subject: [Rasch] What measure is halfway?

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?

FIRST - IT SEEMS WE ARE TALKING THEORETICALLY ABOUT JUST ONE ITEM.
SECOND -  AS A DEFINITION OF HALFWAY, ONE APPROACH IS TO DEFINE THE EXTREMES
OF THE MEASURES AND THEN HALF WAY IS OBVIOUS.
ONE CLEAR WAY IS TO DEFINE THE 6/7 AND THE 0/1 THRESHOLDS AS THE EXTREMES.
THEN HALF WAY ALONG IS THE ANSWER.
THIRD - THE ANSWER TO ANY OF THE OTHER TYPE OF DEFINITIONS INDICATES THAT
THE ANSWER WILL DEPEND ON THE LOCATIONS OF THE THRESHOLDS, SO IT IS UNLIKELY
THAT THERE WILL BE AN UNEQUIVOCAL ANSWER.
FOURTH -IN THE CASE THAT THE THRESHOLDS ARE SYMMETRICAL, THEN MORE THAN ONE
RELATIONSHIP WILL HOLD -E.G. MEAN, MEDIAN AND MODE, IN A SYMMETRICAL
DISTRIBUTION.
FIFTH - HOWEVER, THE CONDITION B BELOW IS OUTSIDE THE STRUCTURE AND FORM OF
THE RASCH MODEL AND SHOULD NOT BE CONSIDERED A DEFINTIION OR INTERPRETATION
OF ANYTHING WITHIN THE RASCH MODEL FRAMEWORK.

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

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