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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