# RE: st: RE: Econometrics Theory Questions on Dummies and Correlation Analysis

 From "Nick Cox" To Subject RE: st: RE: Econometrics Theory Questions on Dummies and Correlation Analysis Date Mon, 18 Apr 2005 19:20:43 +0100

```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
n.j.cox@durham.ac.uk

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
>
> ----- Original Message -----
> From: Nick Cox <n.j.cox@durham.ac.uk>
> Date: Monday, April 18, 2005 10:05 am
> Subject: st: RE: Econometrics Theory Questions on Dummies and
> Correlation Analysis
>
> > Please note various points about
> > Statalist procedure:
> >
> > 1. This message is just a repeat of
> > one sent yesterday.
> >
> > 2. Please don't send email junk
> > like vcards with your postings.
> >
> > As for your question, I don't think
> > there is anything to prohibit you
> > doing this. The results won't necessarily
> > be very helpful or meaningful, except
> > in the extreme cases in which variables
> > are identical, or nearly so, which will
> > produce correlations that are +1, or nearly
> > so.
> >
> > Nick
> > n.j.cox@durham.ac.uk
> >
> > Dr. Stephen Owusu-Ansah
> >
> > > I have econometric/statistical theory questions about dummy
> > > variables 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?
> > >
> > > 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?
> > >
> > > Thanks for your usual cooperation.

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