# [Rasch] Rasch analysis of interval data

Michael Lamport Commons commons at tiac.net
Thu Jul 24 06:31:58 EST 2008

```By the way, hierarchical complexity goes beyond cognitive development.
It is a content and context free Mathematical theory.  For example,
molecules are made up of atoms.  An atom is made up of subatomic
particles.  Subatomic particles are made up of quarks.  Luce has been my
advisor on this project.  He says that there was no need to assume
anything beyond ordinality.  You need to read his reasoning on why with
ordinal scales, things converge to the rationals relatively fast.  I you
look at the graphs, you can see they are very very linear.

At each order, a task action is defined in terms of 2 or more different
lower order actions.  There can be an infinite number of such actions.
But in one sequence of such actions, there is only one such action.
This creates an upside down tree like structure.  At order 0, no actions
are organized.  At order 1, there are 2 simple actions.  At order 2, 2
order 1 actions are organized.  At order 3, 2 order 2 actions are
organized yielding 4 total actions.  At order 4, 2 order 3 actions are
organized yielding 8 actions.  As you can see, one can count the
actions.  Each action is discrete and different from all the other
actions.  An action that is defined in terms of other actions does not
equal any of those actions:  Set A does not equal the members of Set A.

Andrew Kyngdon wrote:
> Michael,
>
> >From the information on the wikipage and from the graph you have sent,
> it strongly appears to me that your model of hierarchical complexity is
> an ordinal theory of cognitive development. Ordinal things are not
> measurable, unless one subscribes to S.S. Stevens views and believes
> that ordinal "scales" produce measurements.
>
> I don't consider that you have an absolute scale here at all. You have a
> theory of cognitive development that is ordinal - you don't have a
> discrete quantity here (like an aggregate of some kind) which can be
> measured through counting. Your theory is not a mere collection of
> objects - it is much more complex than that.
>
> Don't get me wrong. Psychology needs explicit theories of cognitive
> processes of the kind you have developed. It desperately needs more
> Piagets than it does Fred Lords. The truly disturbing thing about the
> behavioural sciences is the attitude of whilst it is nice to have
> "substantive theory", it's not necessary for measurement. On the
> contrary, if we are genuinely measuring a natural system, then our
> "substantive" and "measurement" theories are one and the same thing,
> from a realist perspective. Instead, psychometricians are always hoping
> the practitioners will provide the substantive theory, whilst the
> practitioners seem to think that psychometric models actually provide
> this. This kind of thinking pervades and dominates the SEM and HLM
> lands, for example.
>
> By the way, the definition of an absolute scale given in the "Dictionary
> of Psychology" is, true to form, quite poor. In all measurement in
> physics, regardless of whether one is using the so called ratio,
> absolute or interval scales, the unit is always "fixed".
>
> Cheers,
>
> Andrew
>
>
> -----Original Message-----
> From: Michael Lamport Commons [mailto:commons at tiac.net]
> Sent: Tuesday, July 22, 2008 5:44 PM
> To: Andrew Kyngdon
> Cc: rasch at acer.edu.au
> Subject: Re: [Rasch] Rasch analysis of interval data
>
>
>   absolute scale
>
>
>     A Dictionary of Psychology | Date: 2001
>
> *absolute scale n.* In statistics and measurement theory, a ratio scale
> <http://www.encyclopedia.com/doc/1O87-ratioscale.html> in which the unit
>
> of measurement is fixed. In practice, values on an absolute scale are
> usually if not always obtained by counting.
>
> MLC:  What is counted is the number of times in a hierarchy that a
> higher order action coordinates two or more lower order actions in a
> non-arbitrary way.  It has a tree structure so one counts how high up in
>
> the tree one is.
>
> MLC
>
> Andrew Kyngdon wrote:
>
>> "It is absolute in the sense that it does not require norms, it does
>>
> not
>
>> need to be tested on people to know what the hierarchical complexity
>>
> of
>
>>
>> It does not logically follow from these premises that you can measure
>>
> an
>
>> ordinal structure with an "absolute" scale.
>>
>>
>>
>> -----Original Message-----
>> From: Michael Lamport Commons [mailto:commons at tiac.net]
>> Sent: Tuesday, July 22, 2008 4:47 PM
>> To: Andrew Kyngdon
>> Cc: rasch at acer.edu.au
>> Subject: Re: [Rasch] Rasch analysis of interval data
>>
>> Andrew Kyngdon wrote:
>>
>>
>>> How can an ordinal structure be measured with an absolute scale?
>>>
>>>
>>>
>> It is absolute in the sense that it does not require norms, it does
>>
> not
>
>> need to be tested on people to know what the hierarchical complexity
>>
> of
>
>>
>> MLC
>>
>>
>>>
>>>
>>>
>>>
>>> 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.
>>>
>>> MLC
>>>
>>> Andrew Kyngdon wrote:
>>>
>>> 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
>>>
>>>
>> <mailto: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
>>>>
>>>>
>>>>
>>>>
>>>>
> ------------------------------------------------------------------------
>
>>
>>
>>>
>>>
>>>
>>>> *From:* rasch-bounces at acer.edu.au <mailto: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 <mailto: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.
>>>>
>>>>
>>>>
>> 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>
>>
>>
>>>> <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>]
>
>>>>
>>>>
>>
>>
>>>> *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 there's
>>>> concatenation-based objective measurement? Suppose we make a height
>>>> 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>]
>
>>>>
>>>>
>>
>>
>>>> *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 ...
>>>>
>>>>
>>>> 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>
>>
>
>
>>
>> <http://www.informaworld.com/smpp/title%7Econtent=g794512699%7Edb=all>
>>
>>
>>>>
>>>>
>>>> _______________________________________________
>>>> Rasch mailing list
>>>> Rasch at acer.edu.au <mailto:Rasch at acer.edu.au>
>>>>
>>>>
>> <mailto:Rasch at acer.edu.au>
>>
>>
>>>> http://mailinglist.acer.edu.au/mailman/listinfo/rasch
>>>>
>>>>
>>>>
>>>>
>>>>
> ------------------------------------------------------------------------
>
>>
>>
>>>
>>>
>>>
>>>> _______________________________________________
>>>> Rasch mailing list
>>>> Rasch at acer.edu.au <mailto:Rasch at acer.edu.au>
>>>> http://mailinglist.acer.edu.au/mailman/listinfo/rasch
>>>>
>>>>
>>>>
>>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>
>>
>>
>
>
>
>

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