Yes, X might have a zero correlation with Y despite being causally
related to Y. One way this can happen is when there is no direct effect but
two indirect effects, one indirect effect being positive, the other negative,
and their magnitudes being similar. Alternatively, with only one mediator, the
direct effect of X on Y and the indirect effect of X through M on Y might be of
similar magnitudes but opposite signs. And yes, this can also be conceived as
suppression.
See http://core.ecu.edu/psyc/wuenschk/SimData/XD-Mediate.htm .
For more on suppressor effects, see
http://core.ecu.edu/psyc/wuenschk/MV/multReg/Suppress.docx (with thanks to Jake
Cohen).
Cheers,
Karl L. Wuensch
-----Original Message-----
From: Mike Palij [mailto:[email protected]]
Sent: Friday, October 10, 2014 12:48 PM
To: Teaching in the Psychological Sciences (TIPS)
Cc: Michael Palij
Subject: RE: [tips] Spurious Correlations
On Fri, 10 Oct 2014 08:20:18 -0700, Jim Clark wrote:
>Hi
>
>A lot of the discussion of how to interpret correlations involves the
>presence of a simple correlation, as in the spurious correlation
>examples. It is equally important to emphasize to students that the
>absence of correlation is subject to all the same concerns. That is,
>absence of correlation does not imply absence of relationship between X
>and Y because of all the same mechanisms. For example, Z might be
>positively related to X and negatively related to Y, masking a direct
>positive association between X and Y.
I admit to not completely understanding everything that is said above. A few
points:
(1) In the simplest case, a correlation may not be statistically significant
for two reasons:
(a) The null hypothesis (population rho = 0) is true or
(b) There is insufficient power to achieve significance in the sample.
(2) The first idea that popped into my mind when I read the example above was
"Jim is talking about suppression effects"
but it did not quite sound right to me. I went over to David Howell's website
to look at his stat notes on suppression but could not find a situation
described by Jim; see:
https://www.uvm.edu/~dhowell/gradstat/psych341/lectures/MultipleRegression/multreg3.html
Howell describes three suppression situations (which he borrows from Cohen &
Cohen; I don't have the latest edition handy to check):
Since this is in the context of multiple regression, allow me to restate the
variables Y (criterion), X1, and X2 (predictors).
These are the situations (about half way down the webpage)
(a) Classical suppression: r(Y, X1) is significant but r(Y, X2) is not.
r(X1,X2) is significant which means that including it in a regression of Y on
X1 and X2 will provide the best model because the variance in X1 that is
related to X2 but not Y, will provide a stronger effect because what was error
variance in X1 is now removed because it is recognized as systematic variance
between X1 and X2. Howell provides an example.
(b) Net suppression: all r's are positive, that is r(Y,X1), r(Y,X2), and
r(X1,X2). As in (a) above, r(X1,X2) reduces the error variance in X1 but now
the "error" variance in Y is also reduced by partialing out the variance due to
r(Y,X2), assuming that the correlation of interest is r(Y,X1). One problem
with this is that the "Ballantine" or Venn-Euler diagrams are misleading if
there is variance that is common to Y, X1, and X2 (i.e., the intersection of Y,
X1, and X2 in set theory terms). I believe Darlington goes into more detail
about this in his textbook on regression.
(c) Cooperative suppression: the situation most similar to Jim's example above
is cooperative suppression where
r(X1,X2) < 0.00, that is. there is a negative correlation.between
X1 and X2.
There is no situation where X1 or X2 is negative related to Y.
Perhaps Jim is referring to something other than suppression?
Summarizing Howell on suppression effects from his website:
|To paraphrase Cohen and Cohen (1983), if Xi has a (near) zero
|correlation with Y, we are talking about possible classical
|suppression. If its bi is opposite in sign to its correlation with Y,
|we are looking at net suppression. And if its bi exceeds rYi and is of
|the same sign, we are looking at cooperative suppression.
NOTE: Post #3 for me today.
-Mike Palij
New York University
[email protected]
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