[Rasch] Re: Rasch-Poisson

Stephen Humphry shumphry at cyllene.uwa.edu.au
Sun May 24 21:25:43 EST 2009


Hi Anthony and others.

Mike said:
> Reply: assuming you mean a "scale of ratios", then e^1.5 = 4.5 is an
> "odds-units" ratio measure corresponding to 1.5 logits (log-odds-units)
> additive units.

You can obviously define 'unit' in whatever way you want. However, the  
definition of measurement that is the basis of the physical sciences  
is {Q} = Q:[Q] where Q is a magnitude of a quantity (not a number) and  
[Q] is a unit, also a magnitude of a quantity. Following Maxwell,  
magnitudes are expresed as

Q = {Q}[Q] (e.g. 5 cm, 3 newtons).

Odds are pure numbers and are no more units than logits, which are  
also pure numbers. There is no scale (of ratios) unless the definition  
of scale you use does not involve units of an actual quantity.

In general, pure numbers are not units: logits and odds are pure  
numbers. I am leaving aside so-called dimensionless units aside which  
still arise from ratios of quantities, and whose status as units is  
debatable). Defining "scale" in terms of pure numbers alone -- that is  
with no implicit or explicit reference to quantity -- is divorced from  
the emprical world and therefore not appropriate for science.

The Bureau International des Poids et Mesures provides docments  
providing definitions and nomenclature as a general reference for  
metrology. Measurement is clearly defined in these documents and  
various articles by Joel Michell urge social scientists to be  
cognisant of the classical/standard definition of measurement in the  
physical sciences.

Steve


Quoting "Mike Linacre (RMT)" <rmt at rasch.org>:

> Thank you for your further questions, Anthony:
>
> You asked: What's the equation that I wrote in my previous email then?
> P (X=1| b) = e^b/ (e^b+e^d)
>
> Reply: This is an alternative way of writing the Rasch dichotomous
> model. It is an algebraic transformation of:
> P (X=1| b) = e^(b-d)/ (1+e^(b-d))
>
> You asked: Is there any way to get ratio scale measures with winsteps
> (or by hand) according to that equation?
>
> Reply: there is some ambiguity about the term "ratio scale".
>
> If you mean a Steven's "ratio scale", then any additive scale is a
> ratio scale provided that a point on the additive scale is defined as
> the origin (zero point), e.g., absolute zero on a temperature scale, or
> the end of a rule on a length scale. So, if you can define a zero point
> on a Rasch logit scale, then it becomes a Stevens' ratio scale relative
> to that zero point,
>
> If you mean a "scale of ratios", so that, for example, 2 is twice 1, 3
> is twice 2, 4 is twice 3, etc. then numbers on an additive scale (such
> as Rasch measures) are converted to a "scale of ratios" by
> exponentiating the numbers. Georg Rasch used a "scale of ratios" in his
> original formulation of his models for measurement.
>
> You asked: How can one convert an interval scale logit measure of say,
> 1.5 to a ratio measure according to the equation in my email? Will  it
> be again in logits?
>
> Reply: assuming you mean a "scale of ratios", then e^1.5 = 4.5 is an
> "odds-units" ratio measure corresponding to 1.5 logits (log-odds-units)
> additive units.
>
> You asked: Is Poisson a mathematician?
> Thank you, Trevor, for answering this ....
>
> Mike L.
>
> Mike Linacre
> Editor, Rasch Measurement Transactions
> rmt at rasch.org www.rasch.org/rmt/ Latest RMT:  22:4 Spring 2009
>
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