# [Rasch] Rasch analysis of interval data

Andrew Kyngdon akyngdon at lexile.com
Wed Jul 23 03:14:12 EST 2008

```Michael,

I'm not familiar with your work at all, but I take it by "absolute"
scale that you mean you have a continuous, quantitative attribute that
you can measure with a scale possessing a non-arbitrary zero point? If
so, that is quite a feat in the behavioural sciences, given the lack of
natural concatenation operations.

But you state that you can transform Rasch scores into this supposedly
"absolute" scale of "Stage scores". Now, correct me if I am wrong, but
Rasch logits are usually advanced as interval scale measurements
(leaving aside the obvious problem of a lack of a defined unit,
something Steve Humphry has been at pains to point out). Interval scale
measurements cannot be meaningfully transformed into ratio or absolute
scales, unless your substantive theory is sufficiently understood to
enable this, such as in temperature with converting Celsius and
Fahrenheit measurements into the "absolute" Kelvin scale. But if you can
measure something with an absolute scale in the first place, why would
you bother with an interval scale, unless there are historical reasons
(as in temperature) for so doing?

Andrew Kyngdon, PhD
Senior Research Scientist
MetaMetrics, Inc.
1000 Park Forty Plaza Drive
Durham NC 27713 USA
Tel. 1 919 354 3473
Fax. 1 919 547 3401

-----Original Message-----
From: Michael Lamport Commons [mailto:commons at tiac.net]
Sent: Tuesday, July 22, 2008 12:38 PM
To: Andrew Kyngdon
Cc: Mark Moulton; Paul Barrett; rasch at acer.edu.au
Subject: Re: [Rasch] Rasch analysis of interval data

We have an absolute scale in order of hierarchical complexity. We
transform the Rasch scores into Stage scores which are based on the
absolute scale.

Michael Lamport Commons

Andrew Kyngdon wrote:
>
> This is still not a guarantee that the lexile unit captures the "true"

> spacing, but fortunately it does not seem to matter.
>
> The Lexile unit is best defined unit I have come across in the
> behavioural sciences, so by what criteria one judges to be a "true
> spacing" (whatever that is) seems to be a mystery...
>
> Andrew Kyngdon, PhD
>
> Senior Research Scientist
>
> MetaMetrics, Inc.
>
> 1000 Park Forty Plaza Drive
>
> Durham NC 27713 USA
>
> Tel. 1 919 354 3473
>
> Fax. 1 919 547 3401
>
>
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>
> *From:* rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au]
> *On Behalf Of *Mark Moulton
> *Sent:* Monday, July 21, 2008 6:09 PM
> *To:* Paul Barrett
> *Cc:* rasch at acer.edu.au
> *Subject:* Re: [Rasch] Rasch analysis of interval data
>
> Paul,
>
> Thank you for your explanations and for your presentation 10 years
> ago, which are very helpful to me. You raise a fundamental issue,
> still controversial, still worth visiting in my opinion. You make the
> case that Rasch measures are equal-interval representations of counts
> (ratios of counts), and that is all, and that they do not necessarily
> capture a fundamental unit of measurement in the underlying construct.
>
> I think your point is amplified by a simple desktop experiment. Lay a
> ruler on a sheet of paper. Draw dots (representing persons, say) in
> various distributions on the paper. For each centimeter increment on
> the ruler (representing items), count the dots above and below that
> increment and calculate their log ratio. One finds that the logit
> spacings of the ruler increments may be highly unequal (unlike the
> centimeters), depending on how one distributes the dots. If one
> distributes the dots equally up and down the ruler, the logit lengths
> between increments appear to get fatter at the extremes. If one clumps

> the dots in multiple modes, the logit lengths can be distorted in all
> sorts of cool ways. Interestingly, as the distribution approaches
> normal, the logit lengths between increments seem to approach a linear

> relationship with the centimeters, (I don't have a proof for why this
> would be true, but presumably it has something to do with the
> relationship between the logistic and normal distributions and may
> account for the similarity between independently calibrated scales).
>
> So, I agree that Rasch logits do not capture fundamental units of
> measurement, and are sample-dependent in this sense (and in several
> other senses, too). My question is: What does this do to Rasch claims
> of "invariance," aka "special objectivity," the notion that the
> relative logit spacings of persons will remain the same regardless of
> how the items are spaced? Strangely, I don't think it has any effect
> at all. The disappearance of the item parameter when calculating the
> person parameter, and vice versa, has the same force and implication
> that it always did. And due to how Rasch conjointly calculates persons

> and items, whatever distortions may occur affect the persons and items

> equally.
>
> I am left with a relativistic notion of psychometric spaces. Each
> Rasch analysis erects a unique space. That space bears no /necessary/
> relationship to any other Rasch space (except perhaps in some
> topographic one-to-one homeomorphic kind of way). However, objects
> within that space are distributed in a way that is reproducible, hence

> objective, with respect to other objects in that space. Two Rasch
> spaces can be reconciled only by "anchoring" one space to the other
> via common persons or items. This forces the two spaces to share the
> same "distortions," and thus to become one space, and to preserve
> invariance for all objects residing in that space.
>
> Your point about the MetaMetrics lexile scale is well-taken. All texts

> and readers are forced into a common space anchored on the physical
> properties represented by word frequency and sentence length (or log
> transformations thereof). This was facilitated by the fact that
> MetaMetrics discovered and exploited a linear relationship between
> textual empirical variables and item difficulties. But even without
> that relationship, the two types of variables could have been forced
> (by a method MetaMetrics did not use, or need to use) into a common
> space through an anchoring procedure. This is still not a guarantee
> that the lexile unit captures the "true" spacing, but fortunately it
> does not seem to matter.
>
> One space is as good as another, so long as they are internally
> consistent. Am I reading this right?
>
> Mark H. Moulton
>
> Educational Data Systems
>
> 2008/7/21 Paul Barrett <pbarrett at hoganassessments.com
> <mailto:pbarrett at hoganassessments.com>>:
>
> *From:* rasch-bounces at acer.edu.au <mailto:rasch-bounces at acer.edu.au>
> [mailto:rasch-bounces at acer.edu.au <mailto:rasch-bounces at acer.edu.au>]
> *On Behalf Of *Anthony James
> *Sent:* Wednesday, July 16, 2008 7:19 AM
> *To:* rasch at acer.edu.au <mailto:rasch at acer.edu.au>
> *Subject:* [Rasch] Rasch analysis of interval data
>
> 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:
>
>
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>
> *From:* rasch-bounces at acer.edu.au <mailto:rasch-bounces at acer.edu.au>
> [mailto:rasch-bounces at acer.edu.au <mailto:rasch-bounces at acer.edu.au>]
> *On Behalf Of *Andrew Kyngdon
> *Sent:* Wednesday, July 16, 2008 8:13 AM
>
>
> *To:* rasch at acer.edu.au <mailto:rasch at acer.edu.au>
>
> *Subject:* RE: [Rasch] Rasch analysis of interval data
>
> I think Paul Barrett did something like this once...
>
>
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>
> Yep - 10 years ago to be exact!
>
> Sorry I haven't replied until now ...
>
>
> http://www.pbarrett.net/presentations/BPS-rasch_98.pdf
>
> From my web-page abstract ...
>
> **Beyond Psychometrics: the recovery of a standard unit of length**:
> This 50-slide presentation was given at the British Psychological
> Society's Division of Occupational Psychology conference: Assessment
> in the Millennium: Beyond Psychometrics, November 1998, at Birkbeck
> (University of London). The theme of this presentation was about Rasch

> scaling, and its capacity to construct a standard unit from
> observational data. This presentation contained a data simulation that

> attempted to hide a true quantitatively structured latent variable of
> length behind some poor ordinal observations. All the Rasch scaling
> did was to construct an equal-interval latent variable of ordinal
> lengths! This simulation was heavily criticised Ben Wright and others,

> and I have included these criticisms as an addendum to the
> presentation - along with my reply. However, recent papers seem to
> have vindicated my conclusions in some respects.....The reality is
> that these methods simply construct linear latent variables in
> complete isolation of any empirical evidence that such variables might

> indeed be quantitatively structured.. In my opinion, from a scientific

> perspective, these scaling methods are frankly of little utility, but
> they are ingenious from a psychometric perspective and do have great
> utility in a more pragmatic sense. It all comes down to what the
> purpose is for using such scaling, science or number scaling.
>
> 10 years on - with some better understanding of things (!) - the goal
> and conclusions of the presentation still make sense - but now I fully

> understand why. Rasch scaling cannot "uncover" a linear latent
> variable from ordinal measures. It simply scales counts and in effect,

> the numbers applied to its algorithms, without regard to whether those

> counts or numbers are drawn from an ordinal or linear scale.
>
> The mistake made by many psychologists is to forget that latent
> variable theory implies nothing about the measurement properties of
> the variable of interest - latent variables are simply constructed
> ad-hoc to possess linear properties of measurement. That is not how
> normal science proceeds, it is as Michell states a "pathology of
> science" (2000).
>
> I propose that a key exemplar which shows how to properly model data
> while invoking a latent variable, is the work done by Metametrics. It
> is no accident that the initial exploratory work was empirical and
> based upon much cognitive psychological experimentation, PRIOR to the
> scaling/modeling exercises. Andrew has already provided excellent
> explanations of the history of this work, along with another
> exposition recently in his peer response to Michell's target article
> in the journal Measurement (references below).
>
> However, if we view edumetrics-psychometrics as largely
> pragmatic/technical work, which is concerned with the efficiencies to
> be gained in standards-based testing/examination/cumulative risk-scale

> environments, then IRT models in general, and the Rasch model make a
> great deal of sense. I think it is an illusion that the Rasch or any
> IRT/latent variable model magically produces "fundamental measurement"

> in any sense of the word. Michell (2004, and now 2008) has put paid to

> this notion.
>
> I don't think this is a controversial point anymore - from the
> standpoint of simple logic, the work by Robert Wood, and from my own
> small and almost stupid simulation, the Rasch model cannot possibly
> "uncover/discover" the true metric for a "statistically constructed
> latent variable". It just does what it does given the data with which
> it is presented. Whether or not that data is an accurate
> representation/set of observations of the phenomenon of interest (my
> "bad ruler"), the Rasch scaling will simple create a latent variable
> anyway - given sufficient stochastic error in the observations (as
> with Wood's coin-tosses). Which is why I think the Metametrics
> exemplar is so very important, the scaling is constructed around a
> wealth of empirical phenomena and magnitude relationships - and not
> just banks of "item responses".
>
> Regards ... Paul
>
> __________________________________________________
> Paul Barrett 918.749-0632 x 326
> Chief Research Scientist Skype: pbar088
> Hogan Assessment Systems Inc.
> 2622 East 21st St., Tulsa, OK 74114
>
> **References**
>
> Kyngdon, A. (2008) Treating the Pathology of Psychometrics: An Example

> from the Comprehension of Continuous Prose Text. //Measurement:
> Interdisciplinary Research & Perspective//, 6, 1 & 2, 108-113.
>
> Michell. J. (2000) Normal science, pathological Science, and
> psychometrics. Theory and Psychology, 10, 5, 639-667.
>
> Michell, J. (2004) Item Response Models, pathological science, and the

> shape of error. //Theory and Psychology//, 14, 1, 121-129.
>
> Michell, J. (2008) Is psychometrics pathological science?
> //Measurement: Interdisciplinary Research & Perspective//, 6, 1, 7-24
>
> Wood, R. (1978) Fitting the rasch model - a heady tale. //British
> Journal of Mathematical and Statistical Psychology//, 31, , 27-32.
>
> **An aside**
>
> The journal "Measurement: Interdisciplinary Research and
> Perspective"published issues two issues simultaneously - three target
> articles and commenatries on the issue:
>
> //The Conceptual Foundations of Psychological Measurement//
>
> The target papers by Denny Borsboom and Keith Markus are also
> excellent expositions of their respective positions. Very nice
> position pieces.
>
> I've attached the journal link here so you can look at the paper
> titles etc.
>
> http://www.informaworld.com/smpp/title~content=g794512699~db=all
> <http://www.informaworld.com/smpp/title%7Econtent=g794512699%7Edb=all>
>
>
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