[Rasch] Counting and measuring
Stephanou, Andrew
Andrew.Stephanou at acer.edu.au
Wed Jun 18 08:37:58 EST 2014
Currently there are 500 subscribers to the ACER Rasch listserv and 800 to the MBC list (https://mailman.wu.ac.at/mailman/listinfo/rasch). Only 10% of the subscribers to the ACER list are also subscribed to the MBC list. Discussions on one list that are of general interest are missed by subscribers to the other list who don't subscribe to both lists. A typical example is the recent discussion on counting and measuring which I am including in this email.
Andrew Stephanou
From: Rasch [mailto:rasch-bounces at wu.ac.at] On Behalf Of Fabio La Porta
Sent: Monday, 16 June 2014 3:45 AM
To: The Matilda Bay Club
Subject: Re: [MBC-Rasch] Log-transformation of Rasch person estimates
Hi Andrew and Stephen,
thank you for your email.
Your arguments are convincing!
best regards
Fabio
Il giorno 11/giu/2014, alle ore 02:30, Stephanou, Andrew <Andrew.Stephanou at acer.edu.au<mailto:Andrew.Stephanou at acer.edu.au>> ha scritto:
Hi Fabio,
Thanks for your email; it is good that you keep trying to understand what is going on. Maybe my previous email was not sufficiently convincing. A book by Mark Wilson, Constructing Measures, published in 2005 by Lawrence Erlbaum, may help.
First of all you need to focus on the construct rather than on a list of measurement properties.
a) How has "gross domestic product" been defined? Is it a measurable variable or is it some sort of an indicator? As far as I know it is a composite variable defined with bits and pieces expressed in some currency. It is certainly not what Rasch taught us about constructing measurement variables.
Similarly to the definition of "gross domestic product", you could define an educational or health indicator by combining scores on groups of items. You may get a ratio scale, as you say you get it in the gross domestic product, and the rest, but you would not have a measurement variable.
Points b) to d) are dependent on Point a.
e) Counting dollars incorrectly is due to mistakes and not to measurement error. The calculation of a total is prone to counting mistakes, also with mechanical counters as far as I know. In fact if you could eliminate all mistakes, there would be one and only one value because your indicator is not a continuous variable. This is not the case in the measurement of your height. Is there a true value of your height? The best we can do is to measure it many times and take the average value as the true value. You can never eliminate the error of a measurement of length. You could make it very small but you can never eliminate it. I have never seen the value of a gross domestic product expressed with its "measurement" error: ten billion euro plus or minus one million. Actually it could be expressed like this as an approximation but this is not measurement error.
Accepting a currency as a unit of something, be it "material wealth" or "gross domestic product", is practising measurement methodology in reverse. First we have a unit and from the unit we make some sort of a ratio variable. You expect this unit to be "unidimensional" with the indicator. You expect it to be invariant. However I thought the value of currencies changes from day to day. A measurement variable is not directly observable. To construct it we focus on the idea of this variable and we prepare indicators of it, items/tasks, with which we collect data. Rasch showed us how to analyse the data with a thorough analysis of fit of the data to the model and construct a measurement scale with a probabilistic unit. The unit comes after we constructed a variable. You can't have a unit if you don't already have a measurement variable. The unit is not the starting point of constructing a measurement variable. Has any analysis of fit been done in the definition of gross domestic product? I don't think so. You may define anything you like with dollars and euro. Please don't claim that you have constructed a measurement variable without any analysis of fit to the measurement model. You may satisfy as many of the requirements of measurement as you like, but without a proper construction of a measurement variable any arguments about satisfying requirements of measurement are irrelevant.
I hope we have made further progress.
Andrew Stephanou
From: Rasch [mailto:rasch-bounces at wu.ac.at] On Behalf Of Stephen Humphry
Sent: Tuesday, 10 June 2014 7:01 PM
To: The Matilda Bay Club
Subject: Re: [MBC-Rasch] Log-transformation of Rasch person estimates
Fabio, comments below yours
May I act as a devil's advocate?
The dollar (or whatever currency):
a) is in linear relationship with a latent variable that is gross domestic product
Do you mean a rough and variable relationship?
b) is unidimensional to this latent variable
I don't understand what that means.
c) has a unit of measurement that is invariant
According to what theory and empirical evidence?
d) has a multiplicative structure (ratio scale), thus supporting all the mathematic properties typical of this kind of measurement.
Measurement doesn't have 'mathematical properties'. If you measure something properly, you can use mathematics involving real numbers and inferences will be coherent. If you don't you can't. Most statisticians struggle to grasp this. "Multiplicative structure" is also ambiguous, at best.
e) is indeed prone to error of measurement, otherwise there would be no need for mechanical currency counters.
I don't understand your reasoning.
Are we entirely sure that the dollar (or whatever currency) does not satisfy, both conceptually and practically, the scientific requirements for a ratio measure?
Are you sure that it does?
Steve
From: Rasch [mailto:rasch-bounces at wu.ac.at] On Behalf Of Fabio La Porta
Sent: Tuesday, 10 June 2014 6:07 PM
To: The Matilda Bay Club
Subject: Re: [MBC-Rasch] Log-transformation of Rasch person estimates
Thanks Andrew for your explanation.
May I act as a devil's advocate?
The dollar (or whatever currency):
a) is in linear relationship with a latent variable that is gross domestic product
b) is unidimensional to this latent variable
c) has a unit of measurement that is invariant
d) has a multiplicative structure (ratio scale), thus supporting all the mathematic properties typical of this kind of measurement.
e) is indeed prone to error of measurement, otherwise there would be no need for mechanical currency counters.
Are we entirely sure that the dollar (or whatever currency) does not satisfy, both conceptually and practically, the scientific requirements for a ratio measure?
Fabio
From: Rasch [mailto:rasch-bounces at wu.ac.at] On Behalf Of Andrew Kyngdon
Sent: Monday, 16 June 2014 2:51 PM
To: The Matilda Bay Club
Subject: Re: [MBC-Rasch] Log-transformation of Rasch person estimates
Nick,
Prices are complex, not only because of supply and demand, but because the one person will almost certainly have different prices for the selling and buying of the same good or service. Such systematic differences are consistent with utility (e.g., losses being more unpalatable than what gains of commensurate magnitude are pleasureable, thus causing the lower bound of a selling price to be higher than the upper bound of a buying price). Paul Samuelson (1938) hoped that his "pure" theory of consumer choice (involving only price and income) would lead to the demise of utility. That of course did not happen, because decision makers are not the strictly rational "homo-economicus" beings that Samuelson assumed.
I would argue that money has been thought of as discrete by most of its users for most of the time since its conception. That is, statements to the effect of "I will exchange X chickens for Y shekels/drachma/unciae/ducats/pounds/dollars" have been quite common in marketplaces from 3000BC onwards. Treating it as continuous, I would argue, is a relatively recent occurrence.
I don't believe that money needs to be in some kind of ontological contest with the attributes of physics. Only the most abstruse ideologue would argue that money is not real, as survival for most people depends upon access to it. Money doesn't need to be "as real" as a physical attribute for us to take it seriously, anymore than attitudes need to be "as real" as physical quantities. You could argue that attitudes (evaluative beliefs) aren't really "real" either, but those fortunate enough to escape the regimes of Hitler, Pol Pot and Stalin would likely disagree.
Cheers,
Andrew
Dr Andrew Kyngdon
Chief Psychometrician
Manager, Quality, Processing & Research
From: Rasch [mailto:rasch-bounces at wu.ac.at] On Behalf Of Nick Connolly
Sent: Tuesday, 10 June 2014 12:35 PM
To: The Matilda Bay Club
Subject: Re: [MBC-Rasch] Log-transformation of Rasch person estimates
I would think "price" is something about which there is a kind of variability that is akin to measurement error.
There are three issues I see with money as a kind of measure:
1. It is an intersubjective quantity rather than an objective physical quantity (i.e. it quantifies a kind of agreement among economic actors)
2. It has qualities that appear to be discrete and qualities that approximate continuity
3. It seems to be self-referential - the quantity it measures seems only to exist because of a common medium of exchange which is the very thing we are using as the unit of measurement.
The relation with utility (as Andrew K points out) suggests ways out of two of those issues.
Sent: Tuesday, June 10, 2014 at 4:02 AM
From: "Stephanou, Andrew" <Andrew.Stephanou at acer.edu.au<mailto:Andrew.Stephanou at acer.edu.au>>
To: "The Matilda Bay Club" <rasch at wu.ac.at<mailto:rasch at wu.ac.at>>
Subject: Re: [MBC-Rasch] Log-transformation of Rasch person estimates
Thanks for your email, Fabio.
I believe many other subscribers to the list are asking the same question. The problem is that we are asking the wrong question. The question is not whether the dollar is an appropriate measurement unit and another currency like your euro is not. We should be asking what is the construct we are trying to measure using the dollar as a unit. The temptation to take "material wealth" as the construct and the dollar as its unit is very strong indeed. Calling "material wealth" a count of dollars is not the same as constructing a measurement variable of "material wealth" in the same way as calling mathematics ability the number of questions answered correctly in a mathematics test is not the same as constructing a Rasch variable for mathematics achievement.
If we were to construct a Rasch scale on which we could identify levels of understanding measurement, I would expect to find at least four levels. People in the lowest level think that counting anything is measurement. Higher up the scale people think that counting any identical objects constitutes measurement. Higher up people think that counting items answered correctly in a test consisting of identical difficulty items is measurement (CTT even provides a "measurement error" for test scores). Higher up people focus on a measurement construct and its unit, which is an agreed amount of the construct that can be used to express amounts of the construct.
Ben Wright used the example of counting oranges as opposed to measuring their weight to show the difference between counting and measuring. I found this useful but not without dangers. In fact this example invites to think that if all oranges had the same weight then counting oranges would be measuring. If you want to measure weight and you knew the weight of an orange in units of weight, then counting them would allow you to measure, but the actual counting is not measurement in itself. Counting measurement units of a particular construct is. The example of counting oranges also invites to think that if we had a test consisting of all questions of the same difficulty, test scores would be measurements with a question being the unit. This is wrong. In fact the score equivalence table for such a test does not show a linear relationship between test scores and logits.
Measurement error is a strong indicator of measurement. I don't know of any measurement that can be expressed without its associated error. What about when you count dollars? What is the error associated with a count of ten dollars? Mistakes are not the same as measurement errors: mistakes can be eliminated but measurement errors are always present.
I hope this helps.
Andrew Stephanou
From: Rasch [mailto:rasch-bounces at wu.ac.at] On Behalf Of Fabio La Porta
Sent: Monday, 9 June 2014 7:41 PM
To: The Matilda Bay Club
Subject: Re: [MBC-Rasch] Log-transformation of Rasch person estimates
Hi Andrew,
could you be more explicit? Why the dollar or any other currency is not a measurement unit?
thanks
Fabio
Il giorno 07/giu/2014, alle ore 17:35, Stephanou, Andrew <Andrew.Stephanou at acer.edu.au<x-msg://6/Andrew.Stephanou@acer.edu.au>> ha scritto:
Often in these discussions incorrect statements are made that are not explicitly addressed by the discussants for various reasons and followers of the discussion risk to believe that such statements are true. This time there is an implication that the dollar is a measurement unit. Well, it is not.
Andrew Stephanou
________________________________________
From: Rasch [rasch-bounces at wu.ac.at<x-msg://6/rasch-bounces@wu.ac.at>] on behalf of Stephen Humphry [stephen.humphry at uwa.edu.au<x-msg://6/stephen.humphry@uwa.edu.au>]
Sent: Saturday, 7 June 2014 5:10 PM
To: The Matilda Bay Club
Subject: Re: [MBC-Rasch] Log-transformation of Rasch person estimates
Hi Mike,
I'm glad you agree logarithms are of the numbers not units.
Contrary to the amazingly widespread misconception, E=mc^2 is far from being Einstein'shttp://galileowaswrong.com/wp-content/uploads/2013/06/E-equals-mc2-Not-Einsteins-Invention.pdf I'm not sure why anyone would want to statistically investigate this conversional relation.
Cheers,
Steve
-----Original Message-----
From: Rasch [mailto:rasch-bounces at wu.ac.at<x-msg://6/rasch-bounces@wu.ac.at>] On Behalf Of Mike Linacre
Sent: Saturday, 7 June 2014 2:57 PM
To: The Matilda Bay Club
Subject: Re: [MBC-Rasch] Log-transformation of Rasch person estimates
Stephen, yes, you are right: the logarithms are of the numbers, not of the substantive units.
Economists often use log(dollars) when tracking inflation, capital growth, etc.
How about a statistical investigation of Einstein's E=mc^2 based on experimental results? Surely it would be computationally easier to confirm this relationship in the logarithmic form:
ln(E) = ln(m) + 2*ln(c)
Mike L.
Stephen Humphry wrote: What about taking logarithms of measurements/numbers for statistical convenience?
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