[Rasch] Re: Rasch Digest, Vol 36, Issue 18

Mark Moulton markhmoulton at gmail.com
Sat Jul 19 14:06:29 EST 2008


Steve,
Thank you for your explanation of the Rasch relationship to the
Weber-Fechner law.  Yes, it seems clear that "perceptive" spaces (a more
precise term, perhaps, than "probabilistic" spaces, though Rasch used ratios
of probabilities to erect non-physical spaces) have no necessary linear
relationship to physical spaces, and a probable non-linear relation.  I
would expect, however, based on a few perception experiments, that
perceptive spaces will sometimes have a  linear relation to physical spaces
(e.g., distance, height, weight, color intensity, etc.), depending on the
type of physical space.

What is of greater interest to me, here, is the role of invariance as it
relates to linearity.  A Rasch analysis of perceptions of height may fully
satisfy the requirements of invariance.  A physical analysis of height may
also satisfy the requirements of invariance.  Yet the perceptual measures
and physical measures may have a non-linear relationship to each other.  The
conclusion seems straightforward.  Satisfaction of the Rasch definition of
invariance does not guarantee a linear relationship between invariant
measures derived from different modalities (e.g., "perceptive" and
"physical").

This leads to the following questions:  What geometrical relationships
doesinvariance require?  Does it require
any?  Does it matter?

A thought experiment:  A set of humans and a set of robots are given the
task of comparing the same set of objects.  a) What is the relative spacing
of the objects when calibrated separately for the two types of perceivers?
 b) If the robots and humans space the objects differently (e.g.,
nonlinearly), what happens if we do a Rasch analysis of the humans and
robots together, rating the objects?

Will the data misfit the model?


Mark Moulton
Educational Data Systems







On Fri, Jul 18, 2008 at 7:00 PM, <rasch-request at acer.edu.au> wrote:

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>   1. Re: Rasch analysis of interval data (Stephen Humphry)
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> Message: 1
> Date: Sat, 19 Jul 2008 04:54:22 +0800
> From: Stephen Humphry <shumphry at cyllene.uwa.edu.au>
> Subject: Re: [Rasch] Rasch analysis of interval data
> To: rasch at acer.edu.au
> Message-ID: <20080719045422.zq03n8hlc8oc0c00 at webmail-5.ucs.uwa.edu.au>
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>
> Mark, Thurstone argued Weber's law gives Fechner's 'law' when the
> discriminal dispersions are equal for all stimuli, as in Case V of the
> law of comparative judgment. In this case, the Bradley-Terry-Luce
> model also holds, which has the form of the Rasch model with only one
> kind of parameter.
>
> To the extent the "Weber-Fechner" law holds, therefore, the Rasch
> model holds, and the relationship between physical and perceptual
> magnitudes of stimuli is not linear; rather it is logarithmic.
> Nevertheless, it is likely to appear linear within a relatively small
> range.
>
> > A question I would like to answer is:  "Under what
> > conditions must a probabilistic space correspond to a "physical" space,
> or
> > to any other space."  I think an effort to answer this question
> rigorously
> > would prove very fruitful.
>
> Clearly, there are good reasons to suspect some correspondence between
> the perception of certain physical quantities and actual
> three-dimensional physical space. Examples are the perception of the
> locations of objects and sounds in physical space. However, I think it
> is unfortunate that coexistent dimensions are invoked so commonly in
> the social sciences in situations in which there is no connection to
> three-dimensional physical space, and in the absence of any compelling
> theory with direct empircal evicence to support it.
>
> Although I'm not sure what you mean by a probabilistic space in this
> context, I agree that an effort to rigorously answer this kind of a
> question would prove fruitful.
>
> Steve
> ting Mark Moulton <markhmoulton at gmail.com>:
>
> > Anthony,
> > I did a number of experiments in this vein, almost exactly as you
> described.
> >  I did indeed find that my Rasch logit measures had a strong linear
> > relationship with centimeter measures.  While this was suggestive as a
> > demonstration, and spurred me to carry the analogy between Rasch
> measurement
> > and spatial geometry quite a bit further, it does not constitute proof
> that
> > Rasch measures must necessarily correspond to such "physical" measures in
> a
> > strictly linear way.  A question I would like to answer is:  "Under what
> > conditions must a probabilistic space correspond to a "physical" space,
> or
> > to any other space."  I think an effort to answer this question
> rigorously
> > would prove very fruitful.
> >
> > Mark Moulton
> > Educational Data Systems
> >
> >
> > 2008/7/16 Anthony James <luckyantonio2003 at yahoo.com>:
> >
> >> Hi all,
> >>
> >> Has anyone ever tried to Rasch analyse a variable for which there's
> >> concatenation-based objective measurement? Suppose we make a height
> scale
> >> with 6 points:
> >>
> >> Very short (1), short (2), average height (3), quite tall (4), rather
> all
> >> (5), Very Tall (6)
> >>
> >> We measure some people with this scale and then Rasch analyses it and
> >> obtain Rasch height measures. How do these measures compare with
> persons'
> >> heights in centimeters or inches? Does the Rasch measure difference
> between
> >> a person who is 170 cm and a person who is 175cm equal to the Rasch
> measure
> >> difference between a person who is 180cm and one who is 185cm?
> >>
> >> I'd love to see what happens. Pairing persons' heights in cm and their
> >> Rasch heights simultaneously on a vertical line and comparing the
> >> calibrations give a good test of Rasch's interval scaling, I guess.
> >>
> >> Cheers
> >>
> >> Anthony
> >>
> >>
> >>
> >>
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> >>
> >
>
>
>
>
>
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> End of Rasch Digest, Vol 36, Issue 18
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