[Rasch] Rasch analysis of interval data

Andrew Kyngdon akyngdon at lexile.com
Wed Jul 23 04:13:40 EST 2008

How can an ordinal structure be measured with an absolute scale?



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



From: Michael Lamport Commons [mailto:commons at tiac.net] 
Sent: Tuesday, July 22, 2008 2:04 PM
To: Andrew Kyngdon
Cc: rasch at acer.edu.au
Subject: Re: [Rasch] Rasch analysis of interval data


The orders of hierarchical complexity are ordinal, universal, context,
content, and participant free.  The are an analytic measure of the
hierarchical complexity of tasks.  We know of 15 orders.  You can see a
description on Wikipedia.  There is a non-arbitrary zero.  Rasch
measures performance.  Hierarchical complexity measures tasks
properties.  This is psychophysics.  The y-axis is Rasch Score, the x
axis is order of hierarhical complexity.  The r's are mostly in the .9
-.99 range.


Andrew Kyngdon wrote: 

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


	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
	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


	*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
	Thank you for your explanations and for your presentation 10
	ago, which are very helpful to me. You raise a fundamental
	still controversial, still worth visiting in my opinion. You
make the 
	case that Rasch measures are equal-interval representations of
	(ratios of counts), and that is all, and that they do not
	capture a fundamental unit of measurement in the underlying
	I think your point is amplified by a simple desktop experiment.
Lay a 
	ruler on a sheet of paper. Draw dots (representing persons, say)
	various distributions on the paper. For each centimeter
increment on 
	the ruler (representing items), count the dots above and below
	increment and calculate their log ratio. One finds that the
	spacings of the ruler increments may be highly unequal (unlike
	centimeters), depending on how one distributes the dots. If one 
	distributes the dots equally up and down the ruler, the logit
	between increments appear to get fatter at the extremes. If one


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


	relationship with the centimeters, (I don't have a proof for why
	would be true, but presumably it has something to do with the 
	relationship between the logistic and normal distributions and
	account for the similarity between independently calibrated
	So, I agree that Rasch logits do not capture fundamental units
	measurement, and are sample-dependent in this sense (and in
	other senses, too). My question is: What does this do to Rasch
	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
	at all. The disappearance of the item parameter when calculating
	person parameter, and vice versa, has the same force and
	that it always did. And due to how Rasch conjointly calculates


	and items, whatever distortions may occur affect the persons and


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


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


	and readers are forced into a common space anchored on the
	properties represented by word frequency and sentence length (or
	transformations thereof). This was facilitated by the fact that 
	MetaMetrics discovered and exploited a linear relationship
	textual empirical variables and item difficulties. But even
	that relationship, the two types of variables could have been
	(by a method MetaMetrics did not use, or need to use) into a
	space through an anchoring procedure. This is still not a
	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>
<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
<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>
<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
	concatenation-based objective measurement? Suppose we make a
	scale with 6 points:


	*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
<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>
<mailto:rasch at acer.edu.au> 
	*Subject:* RE: [Rasch] Rasch analysis of interval data
	I think Paul Barrett did something like this once...


	Yep - 10 years ago to be exact!
	Sorry I haven't replied until now ...
	The presentation about the simulation can be downloaded at:
	From my web-page abstract ...
	**Beyond Psychometrics: the recovery of a standard unit of
	This 50-slide presentation was given at the British
	Society's Division of Occupational Psychology conference:
	in the Millennium: Beyond Psychometrics, November 1998, at
	(University of London). The theme of this presentation was about


	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
	did was to construct an equal-interval latent variable of
	lengths! This simulation was heavily criticised Ben Wright and


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


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


	perspective, these scaling methods are frankly of little
utility, but 
	they are ingenious from a psychometric perspective and do have
	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
	and conclusions of the presentation still make sense - but now I


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


	the numbers applied to its algorithms, without regard to whether


	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
	the variable of interest - latent variables are simply
	ad-hoc to possess linear properties of measurement. That is not
	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
	while invoking a latent variable, is the work done by
Metametrics. It 
	is no accident that the initial exploratory work was empirical
	based upon much cognitive psychological experimentation, PRIOR
to the 
	scaling/modeling exercises. Andrew has already provided
	explanations of the history of this work, along with another 
	exposition recently in his peer response to Michell's target
	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


	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
	IRT/latent variable model magically produces "fundamental


	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
	small and almost stupid simulation, the Rasch model cannot
	"uncover/discover" the true metric for a "statistically
	latent variable". It just does what it does given the data with
	it is presented. Whether or not that data is an accurate 
	representation/set of observations of the phenomenon of interest
	"bad ruler"), the Rasch scaling will simple create a latent
	anyway - given sufficient stochastic error in the observations
	with Wood's coin-tosses). Which is why I think the Metametrics 
	exemplar is so very important, the scaling is constructed around
	wealth of empirical phenomena and magnitude relationships - and
	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
	Kyngdon, A. (2008) Treating the Pathology of Psychometrics: An


	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,
	Wood, R. (1978) Fitting the rasch model - a heady tale.
	Journal of Mathematical and Statistical Psychology//, 31, ,
	**An aside**
	The journal "Measurement: Interdisciplinary Research and 
	Perspective"published issues two issues simultaneously - three
	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.

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