Hello Dino,

I don't want to chase this rabbit very far, because folks on both sides tend to get a bit crusty in support of their views. However, please permit me to point out that your example of IQ-type points does not come from any Rasch or IRT based method.

The deviation-type IQ score (which comes from the early days of classical test score theory, a.k.a. the "zero-parameter model," which presumes items are interchangeable) simply means that,

if....

(a) the target trait of interest (call it what you like) exists and is normally distributed in the population;

(b) the sample from which normative data are gathered is representative of the population;

(c) the instrument is, in fact, a valid measure of the target trait; and

(d) the scores are normalized (e.g., convert percentile rank to deviates on the standard normal distribution to correct for sample perturbations/departures from the population distribution);

then...

the deviations (from the arbitrarily set mean of 100) are on an equal interval scale.

This does not (and I don't know of anyone who claims otherwise) mean that the score units represent equal and operationally describable differences in examinee behavior/performance/capability throughout the scale.

There have been a number of measures for which the developers tried to tie scores to specific, scalable, observable performance targets/sentinels on the trait of interest (e.g., the old Key Math Test), not unlike Behaviorally Anchored Rating Scales. Rasch and IRT models make this task more tractable, from a number of perspectives.

Good luck with your survey.

Hello Dino,

As David Morse indicated this is a delicate point in psychometrics. A lot rests on assumptions.

An important assumption is interchangeability or exchangeability. In Rasch models, persons and items are expected to be exchangeable or interchangeable. It is also an important assumption in multi-level analysis (see Bosker and Snijders, Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling, 1st ed [due to size of book it is not in the second edition]).

It relates to a basic mathematical axiom: A+B=B+A. In scale analysis it comes back as: score 3 for item A + score 4 for item B = score 4 for item A + score 3 for item B. It must be true for all scores for items A and B. This must hold to have additivity. Showing it to hold is hard, very hard and in many cases impossible.

IRT models (Rasch, Graded Response Models, Mokken) are used for items with a small number of categories (true/false or likert scales). Factoranalyses assumes items are on interval level. Using 'normal' factor analysis with ordinal or binary data can produce incorrect results, but there are special factor analytic models for ordinal or binary data.

Sometimes distributional assumptions need to be made to estimate intervals. In Rasch and other IRT models sometimes the assumption has to be made that the population is normally distributed.

In order to have extremely good scales, you have to perform a lot of test (uni-dimensionality, differential item or scale functioning) before you might be able to assume linearity for all the uses you want to have for a scale.

Good information David

Dear David Morse and Peter Moorer

thank you very much for your quick reply to my question!

Regarding the IQ, I used it as an example to show how the property of linearity cannot be tested in the same way in physical systems and in psychometrics. Sorry if I was not clear on that.

Regarding the exchangeability property (I am still digging into Bosker and Snijders), I do not see how the sum of two scores might actually depend on the order. I know I am not depicting it in the right way, but I would just sum the two scores (such as in Likert scales) and in that case, the order would not matter.

Is there an example in which it is easy to observe this dependancy?

Thank you again for your attention!

The sum score should not depend on the order of scores, but this is not a fact. In fact it could be true that the score 3 for item A + score 4 for item B score 4 for item A + score 3 for item B. A simple some score like in a Rasch or extended model can not be calculated. For Rasch-like models the sum score is a sufficient statistic. We do not need to know more about the items themselves, for instance we can assume the items to have (nearly) identical item trace lines. It means, you only need item difficulties, but do not need discrimination parameters and can assume these to be (nearly) identical (usually set at 1 or 1.7, but any identical value would be allowed).

Maybe you should read work by van Ark and Sijtsma on the relationship between IRT models to appreciate the differences. I have read the work by Ivo Molenaar on item steps, but it has never been submitted to a journal. Ivo became editor of Psychometrica and wanted to send it to this journal. Not to be involved in two roles, he never submitted it. It has been published as a Heymans Bulletin, but it is hard to find. Maybe other literature on item steps might help you.

Ok, I see.

If I understood correctly, the "linearity" is deduced by testing the assumptions of conjoint theory. Is that correct?