Via:          UK.AC.LE;  3 SEP 92 12:54:46 BST
Site:         University of Leicester, Leicester, U.K.
X-Name:       David D Stretch
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Date:         Thu, 3 Sep 1992 12:49:20 BST
Reply-To:     dds@leicester.ac.uk
From:         David Stretch <dds@LE.AC.UK>
Subject:      ordinal, interval, cautions (fwd)

I would just like to add my support for what Ronan says here, and add a few
extra comments of my own. To recap...

> From: Ronan M Conroy <RCONROY@IE.UCD.IRLEARN>
> Subject:  ordinal, interval, cautions
> To: Multiple recipients of <STAT-L@EARN.MCGILL1>
>
>
> The classic 5-point scale rangine from "agree strongly" to
> "disagree strongly" does not meet the assumptions of equal-interval
> scales, since the scale steps do not correspond to increases in
> a defined unit of anything. More important, however, is that
> the use of these scales correlates with things like education
> and social class - two tendencies, in particular, are worth
> noting: one is for better-educated people to make more use of
> the extremes of the scale, and the other is for a response set
> to set in, so that responses to later items are patterened by
> responses to earlier ones (if you start with a few "don't know"
> items, you'll be more likely to use the "don't know" box in later
> items, for instance). For these reasons, I usually recommend
> rescaling bunches of 5-point items as 2-point, by grouping
> together all the responses that represent what you are interested
> in and scoring them as '1', and putting everything else as '0'.
> If you are intested in the presence of significant levels of
> depression, for example, you could score a 1 for any item agreeing
> with a 'depressed' statement or disagreeing with a 'cheerful' one.
>
> I am very much against adding responses on the 5-point scale, since,
> if it isn't an interval scale, you can't do addition on it. The 1/0
> system allows you to add items and explain what you've done.

I agree with this, the main point being whether you can attach any substantive
meaning to certain operations using this scale.

For example, on the Likert scale, the assignment of consecutive integers to
the consecutive categories may, in certain quarters, be usual and a
consensus act (if that is the right term), but it is an arbitrary assumption
that need not be justifiable --- For example, if we have the following set
up for a Likert-style scale:

Label              Score
----------------------------
Not at all          0
Sometimes           1
Moderately          2
Frequently          3
All the time        4

--- is it justifiable to assert (by actions of adding the scores together,
and so on), that the difference between 'Not at all' and 'Sometimes' is
the same as the difference between 'Frequently' and 'All the time'? And
yet the simple application of ANOVA assumes that this is the case.
(The labels may differ in instances of this kind of scaling, but the
general point remains valid).

Note that this is not the same thing as invoking the old dogma about
categorical, ordinal, interval, and ratio scales (as started by Stevens and
which was the subject of some debate on this, or a related list, a while
back). I prefer to concentrate of the *meaning* of the measurements in
each situation rather than appealing to dogma. For an account of a position
similar to mine, read Levy (1981) (refs given below).

Now, if the scale points are well-defined (a characteristic that I think
Ronan was questioning), it may be possible to transform the old 0,1,2,3,4
scoring to something more akin to an interval scale. There are two traditions
to my knowledge that can so this. One which is probably more familiar to
statisticians on this list are the set of procedures known as Box-Cox
transformations, and the other, which I know more about as it stems from
the Axiomatic Approach to Measurement Theory (developed in part by many
of my collegues in Mathematical Psychology), where the particular type of
measurement system is best represented by Conjoint Measurement Theory.
Both of these methods of attack have the following property:

If you are interested in the way independent variables interact together to
affect the dependent variable, then using these approaches can result in
the interactions being *removed*, or otherwise radically altered. In fact,
one of the aims of conjoint measurement theory is to determine whether an
apparent interaction can be removed by a suitable rescaling of the dependent
variable (at the same time, one obtains a scaling of the levels of the
independent variables). The developement of these techniques is rather
mathematical, and may be rather inaccessible to people. the main references
are Krantz et al. (1971), Luce et al. (1990), and Suppes et al. (1989), with
Falmagne (1985, I think) giving a slightly more accessible application in
Psychophysics (which can be seen as being allied to Psychometrics). I am
planning to finish a paper in November with a colleague dealing with these
measurements issues in (we hope) a non-technical way that will be accessible
to people who do not have a great expertise in mathematics, and we hope to
submit it to the journal Theory and Psychology. (If anyone would like an
advance copy of this, email me telling me how you would like it (hard copy,
emailed ascii, or uuencoded .dvi or postscript file - It definitely won't
be finished before the middle of November because of my other work
committments, though).


Finally, I tend to agree with MacRae (1988) who states that the often thought
of equivalence between specific parametric tests and specific nonparametric
tests (e.g., the Wilcoxen Rank Sum Test is the nonparametric version of
the unmatched-pairs t-test) is, logically, not true. Because administration
of the two tests on the same set of observations will sometimes result in
different decisions, then, logically, they are testing different things. Now,
of course, the differences may be slight and not matter much in some cases,
but in others, it does matter. For example, if one uses a t test, this assumes
that:

"...the difference of interest between two groups is the one between the
means of the groups; the distributions are assumed to be normal and equal
in variance. The null hypothesis is that the groups of scores are drawn at
random from a single universe of normally distributed potential scores.

Fisher's two-sample randomization test for identical populations ... is
sensitive to any difference between the groups, but especially to a
difference in their means. Its null hypothesis is that both sets of scores
are drawn at random from the same universe but with no assumptions about
the characteristics of the universe.

The choice of a median test ... would declare that differences in distribution
of any kind may exist but are unimportant. The null hypothesis is that the
two groups have the same expected median; scores are drawn at random from
two universes of potential scores, with the probability of lyiong above or
below the common median being the same for every score.

The Mann-Whitney test [also the Wilcoxen Rank Sum test] ... focuses on
individual scores. The two groups of scores are assumed to be drawn at
random from two different universes whose characteristics are unspecified
except in the one respect stated in the null hypothesis. The null hypothesis
is that if two scores are selected at random, one from each universe, each
universe has the same probability of providing the larger score."
(MacRae, 1988, page 166).

He then goes on to state that rejection of one null hypothesis from the
above is not equivalent to rejection of another. He says that some do imply
others, but that the implication is never reciprocal.

What it needed is a justification of statistical test which focuses on which
null hypothesis of the possible tests is of most relevance to the research
question being investigated. This way, you should end up with a statistical
test (and, hence, a statistical model), that best fits together with your
psychological, economic, or whatever substantive model.

Once again, the focus should be on the *meaning* rather than applying a kind
of dogma about tests and levels of measurement. (I am currently finishing off
a chapter on experimental design for a psychology book on this matter, details
about this can be obtained by emailing me directly).

Sorry, its been a long posting, and, in places, full of self-advertisement,
but as I'm writing about these issues at the moment, they were fresh in my
mind and I thought sharing my thoughts with the members of list would
help. My points of view are not shared by all, so don't feel that I am
putting over "the party line" or anything. Furthermore, the courses of action
my views steer people towards may involve more hard work than is thought of
at first sight, and certainly more work than following a viewpoint that in
some people is just dogma, but good research isn't the product of computer
packages or dogma, but of *hard* *thinking*.

Thanks Ronan for posting a message that spurred me to write this, and I
hope its been of some use to the person who posed the original question.

David Stretch

References.
===========

Falmagne, J.-C. (?1985). Elements of Psychophysical Theory. Oxford: Blackwell.

Krantz, D. H., Luce, R. D., Suppes, P., and Tversky, A. (1971). Additive
   and polynomial representations. (first ed.) Volume 1 of "Foundations of
   Measurement" New York: Academic Press.

Levy, P. (1981). On the relation between method and substance in psychology.
   Bulletin of the British Psychological Society, 34, 265--270.

Luce, R. D., Krantz, D. H., Suppes, P., and Tversky, A. (1990). Representation,
   axiomatization, and invariance. (first ed.) Volume 3 of "Foundations of
   Measurement" San Diego: Academic Press.

MacRae, A. W. (1988) Measurement scales and statistics: what can significance
   tests tell us about the world? British Journal of Psychology, 79, 161--171.

Suppes, P., Krantz, D. H., Luce, D. H., and Tversky, A. (1989). Geometric,
   threshold, and probabilistic representations. (first ed.) Volume 2 of
   "Foundations of Measurement", San Diego: Academic Press.


  +-----------------------+---------------------------------------------+
  |  Dr. David D Stretch  | First get your facts;   then you can        |
  |  Dept. of Psychiatry  |     distort them at your leisure.           |
  |  dds@leicester.ac.uk  |                                  Mark Twain |
  +-----------------------+---------------------------------------------+