This exchange is a reflection of a longstanding debate
about levels of measurement and their implications (if
any) for appropriate statistical techniques.
Depending on your discipline, you may have suffered
through much teaching or reading on this, or little or
In my experience, those who have the strongest views
on this come from psychology, sociology and parts
of biology, among a few other other disciplines.
In an interdisciplinary list, I am not sure that all
will be familiar with the terms nominal, ordinal,
interval and ratio. It strikes me as odd that these
worries exercise people from some disciplines and not
others. The correct logic, whatever it is, should cut
Paul makes a vigorous case for his point of view, but
he doesn't change mine, which just shows stubbornness
on my part, no doubt. I still think that where the
numbers came from doesn't much affect the interpretation
of correlations. Whether this is called technical or
non-technical is to me a question of labelling.
I am very happy with the idea that binary scales
blue and not-blue
male and female
define ordinal scales, just as much as do multiple-points
scales like socialist --- liberal --- conservative --- fascist.
There's certainly arbitrariness about the direction of any
coding, as there would be with the political example, and many
other ordinal scales, but the ordinality
is there if you want it to be. However, it is more
interesting to think of binary scales as literal
0s and 1s, because that permits most of the best things we can
do with them statistically.
I support simplicity too whenever possible. A 2 X 2
table, however, isn't necessarily a simpler beast
than a correlation. In context, particularly when
dummy variables are on the RHS of some regression-like
model, the 2 X 2 reduction doesn't help much.
More generally, the old nominal-ordinal-interval-ratio
scheme misses much that is really important about
different kinds of data. The special characteristics
of binary variables are just one example. Among
other things, they permit the usual tricks with
dummy variables as predictors. Other criticisms are given in
the discussion of, and references to,
Statistics and the Theory of Measurement
D. J. Hand
Journal of the Royal Statistical Society.
Series A (Statistics in Society),
Vol. 159, No. 3. (1996), pp. 445-492.
In that discussion, there is a fortuitous but
agreeable juxtaposition of D.R. Cox and myself.
We even make very similar remarks.
> I don't have any quarrel about whether correlational
> techniques *can* be applied here. Clearly there is no
> technical or mathematical reason why they shouldn't be.
> The question, as I read it, was rather are there any
> *non-technical* reasons for which there may be objections to
> using these techniques for binary data. Since I don't know
> the data, or what is being measured, I cannot determine the
> level of measurement of the variable solely from the fact
> that it is binary. One can assume that it is at least
> nominal, but it can, at least in theory, be any of the four
> possibilities (albeit at low precision). Certainly, binary
> data is not *always* ordinal. For example, blue eyes are not
> necessarily of higher rank than other eye colours, so a
> binary indicator for blue eyes would be nominal. One can
> assert other levels of measurement, but usually with some
> sort of justification. I don't recall any requirement of
> logistic regression with respect to level of measurement of
> the dependent variable, so long as it is binary. I stand by
> the general principle that, other factors held constant,
> simpler techniques are preferable.
> Anyway, I still claim that there are (non-technical) reasons
> that one might choose not to use the techniques described,
> even if one is being a bit nit-picky. One can still use them
> of course, it should just be accompanied with a comment on why.
> - Paul
> ----- Original Message -----
> From: Nick Cox <firstname.lastname@example.org>
> Date: Monday, April 18, 2005 12:20 pm
> Subject: RE: st: RE: Econometrics Theory Questions on Dummies
> and Correlation Analysis
> > There is much good advice here, but it still
> > is further than I would go, and bound up
> > with a more literal reading of the assertions
> > of Stanley Smith Stevens
> > http://www.nap.edu/openbook/0309022452/html/424.html
> > and others on nominal, ordinal, interval and ratio
> > scales, and what you can do with them, than seems defensible.
> > Also, arguments about what was designed to do what
> > don't help much here. The techniques work the
> > way they work because of the mathematics of what is
> > being done, not according to what was in the
> > inventor's mind at the time. Anyway, historically,
> > this is a most dangerous tack, as it was (Karl) Pearson
> > above all others who thought that correlations could
> > be pulled out of categorical data in all sorts of ways:
> > you just needed the right formula to do it.
> > Regression (correlation if anyone insists, but the logic
> > is the same) can't discern the categorical origins
> > of dummy variables. It just sees 0s and 1s.
> > At one extreme, suppose you have two identical
> > dummy variables (and some variation in each).
> > In terms of a scatter plot, you have two clusters,
> > one at the origin (0,0) and one at (1,1), like this
> > *
> > *
> > and a straight line is a perfect summary of such
> > data, and so the Pearson correlation is identically 1.
> > Also, this on the RHS of a model has implications
> > for the model. In practice, as Paul emphasises, you
> > would do well to count the numbers as well, but this
> > result holds irrespective of coding and it is perfectly
> > sensible statistically.
> > More generally, for paired dummies you have clusters of zero or
> > more data at (0,0), (0,1), (1,0) and (1,1)
> > and the correlation you get will depend on the
> > "votes cast" by each of those clusters. In many
> > cases, the results won't be especially easy
> > to interpret, but they are not crazy or stupid.
> > Mind you, almost no correlation is easy to
> > interpret without looking at the corresponding scatter plot,
> > so nothing has changed there.
> > I don't think the case of Spearman correlation
> > needs much extra discussion. Note that binary scales
> > are always ordinal. In correlating, the signs may
> > be arbitrary, but the magnitudes of Spearman
> > correlations won't be.
> > In fact, in many cases they
> > are counts too, in a perhaps strained sense (how
> > many women inside this person? answer: either 0 or 1).
> > Note that no one, to the best of my knowledge, argues
> > that logit regression is inapplicable to binary
> > responses because you can't (shouldn't) apply such techniques
> > to "nominal" data!
> > Nick
> > email@example.com
> > Paul Millar
> > > on Dummies and Correlation Analysis...
> > >
> > > 1. Is there any theory that prohibit one from undertaking a
> > > correlation analysis (i.e., correlation matrix) with either
> > > with Pearson or Spearman rank correlation test on variables,
> > > which are all dummies?
> > >
> > > Although technically there doesn't seem to be anything
> > > preventing the kind of analysis you propose, from a
> > > theoretical (or at least methodological) point of view you
> > > wouldn't normally use this method for at least two reasons.
> > > 1) The level of measurement of the variables does not
> > > coincide with the level of measurement of the techniques.
> > > Pearson correlations are designed for interval (or ratio)
> > > measures and Spearman for ordinal. You have nominal measures
> > > (or so it seems).
> > > 2) It is more complex than required, and potentially
> > > obscures, rather than helps, understanding of the
> > > relationships between the variables. A series of simple
> > > crosstabs might be more illuminating.
> > > From a methodological point of view, a compelling reason to
> > > overcome these objections would be advisable to make your
> > > choice of method more defensible.
> > >
> > > 2. If there is no prohibition, theory wise, can the bivariate
> > > correlation coeficients for the dummy variables be interpreted
> > > in the same way as one would do with continuous variables?
> > > As stated above, the interpretation would require that you
> > > treat your nominal measures as if they are interval or
> > > ordinal. You need to justify this treatment before
> > > interpretation, at least if you are picky picky picky.
> > >
> > > - Paul Millar
> > > Sociology
> > > University of Calgary
> > >
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