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

Stephen Humphry shumphry at cyllene.uwa.edu.au
Sun Jul 20 03:36:04 EST 2008

```Agreed perceptual magnitudes need not necessarily have a linear
relationship with physical magnitudes. Agreed also they could in some
cases, such as those relating to three-dimensional visual and auditory
location. I'm not sure in what sense Rasch 'used ratios of
probabilities': he used functions of ratios of magnitudes given in
terms of probabilities.

> 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").

Yes, I agree this is entirely possible. In principle, invariance
neither guarantees nor requires a linear relationship between
perceptual and physical magnitudes. I wouldn't refer to these as
modalities (in particular the physical) but agree with what I
understand your point to be. Measurements in perceptual units, if
actually possible, need not have a simple linear relationship with
measurements of the relevant physical quantity in a physical unit.

Steve

Quoting Mark Moulton <markhmoulton at gmail.com>:

> 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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>> Today's Topics:
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>>   1. Re: Rasch analysis of interval data (Stephen Humphry)
>>
>>
>> ----------------------------------------------------------------------
>>
>> 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>
>> Content-Type: text/plain;       charset=ISO-8859-1;     DelSp="Yes";
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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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