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It is well known, that there are diffrent types of scales (see Wikipedia Level of measurement).

In psycholinguistics there is Semantic Differential technique. It uses a scale (usually 5 or 7 grades) which type as ordinal or interval is being debated. As Wikipedia says, the majority think it is ordinal. But at the same time processing researchers use arithmetic mean and Factor Analysis: and it uses + and - math operations. Is it critical error? May be, nevertheless, in fact the intervals between closest grades are equal - how to check it? What is the cause of scale's type?

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You don't really "check" the type of scales; you have to use logic and reasonableness to figure out what is sensible to do. Stevens' set of scales is not set in stone and has problems.

Likert scales are, in fact, in between ordinal and interval.

Technically, an ordinal scale is one in which any transformation can be applied that retains order and the meaning will stay the same. So, you could code a Likert scale 1, 2, 3, 4, 5. Or 0, 1, 2, 3, 4. Or 0, 0.00001, 2, 17, 19101821.2. But the last one isn't reasonable.

For factor analysis, much has been written. Search on "ordinal factor analysis" and you'll find some things. There are articles by e.g. Joreskog and Sorbom. Different people have different opinions.

Can you add Likert items? Technically, no. But people do so all the time and it gives reasonable results.

On the other hand, some additions make little sense.

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    $\begingroup$ I believe the scaling device you choose reflect practical matters and an underlying measurement model. Likert scales were proposed as a way to scale persons (not stimuli, in contrast with Thurstone's approach, or both, as in Guttman's approach), and adding individual response scores altogether would make sense in this context, see, e.g., Unidimensional Scaling by McIver and Carmines. $\endgroup$ – chl Oct 6 '13 at 10:10
  • $\begingroup$ As usual @chl raises good points $\endgroup$ – Peter Flom - Reinstate Monica Oct 6 '13 at 10:15
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@Peter has given a good answer. I just want to add one point: it is important how the scale is presented or formatted.

For most people, the less the scale's notches are subscribed the more the scale is interval rather than ordinal. Compare

(disagree)| --- | --- | --- |(agree)
(disagree)1 --- 2 --- 3 --- 4(agree)
totally disagree --- rather disagree --- rather agree --- totally agree

where the 1st scale is just a grating to measure while the 3rd one is clearly categorical, ordinal. Labels involves verbal semantics what isolates the points away from being landmarks and towards being islands.

Osgood's rating scale (used in semantic differential) is like the 1st or 2nd above; in addition, it is bipolar - that is, two equally fair epithets (or objects) symmetrize the scale, which measures proximity to either of them. Such bipolar proximity measuring device seems to be further closer to interval away from ordinal, in comparison to a unipolar intensity measuring device (such as shown above), because symmetric opposition de-granulates the "landscape" between the opposites.

So, the way scale is typically presented in a semantic differential makes one think it is fairly interval.

With ordinal scale it is of course incorrect to do arithmetics (such as computing mean or summing to a total score) or check whether the data distribution is normal. The distinction between interval and ordinal implies the notion of underlying feature which is measured to produce an observed value. If the relation between the underlying and the observed is assumed to be linear, we speak of interval (equiinterval) scale. If the relation is assumed monotonic and is somehow known (e.g. postulated), then the scale is non-equiinterval; such a scale can be easily transformed into equiinterval.

If the relation between the underlying and the observed is assumed monotonic and unknown, there comes ordinal scale. Ordinal scale can be transformed into interval if the transformation rule is worked out. We may draw such rules from our pragmatic desire of maximizing some quantity in the analysis we conceive. For example, one might want linear correlations between items to be as strong as possible. Then the transformation which maximizes the correlations can be solved for. This process of quantifying categorical data is often referred to as optimal scaling.

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    $\begingroup$ I would add that some researchers consider that the way we treat data (e.g., 'ordinal'--that is, a set of ordered response categories--, or 'interval'--leaving alone the controversy that data are truly continuous in very rare cases--) depends on the underlying measurement model. Beside Borsboom, De Boeck and coworkers have a nice paper on distinguishing between categories and dimensions. A 'practical' aspect of that distinction is the way we conceptualize several mental constructs, e.g. mental disorders. $\endgroup$ – chl Oct 6 '13 at 10:00
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The question about the `true' nature of a scale is indeed a tricky one. But a quick pragmatic answer is that, depending on the analysis that you are interested in running, it might be fairly straight forward to deal with the scale as ordinal. I would suggest doing so because that would be the more conservative position, as the interval nature of the scale would be a stronger assumption.

Now, regarding how to deal with the data... in general you could review the literature in item response models and the myriad of models offered there. However, if you are in a literature that relies mostly in factor analysis, you can easily deal with ordinal data by using polychoric correlations (See the polycor in R for instance) instead of the traditional Pearson product moment, and then running the factor analysis on that correlation matrix.

Even easier is to use a software like Mplus, which will allow you to declare your variables as ordinal, and it will run the a generalized version of traditional factor analysis automatically.

Again, this is a very interesting and contentious issue, but if your interest is mainly practical, you could simply treat the data as ordinal, and if you are in the mood of exploring, you could also run it under traditional FA and see if/how the results vary.

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  • $\begingroup$ Just a side note it might be fairly straight forward to deal with the scale as ordinal, also simply treat the data as ordinal. These are burdening advices. Analysis of ordinal data is more difficult and is less developed than that of scale or nominal data. No "simplicity" there. $\endgroup$ – ttnphns Oct 14 '13 at 0:26
  • $\begingroup$ Hi, I appreciate that you seem to think that the field of item response theory is underdeveloped, and that indeed it might be more complicated than assuming continuity of the observed variables. However, I do think that considerably improvements have been achieved in software like R, Stata, Mplus and LatentGold that make such analysis considerable more accessible and I believe it is an option worth considering. $\endgroup$ – David Oct 29 '13 at 19:31

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