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st: positive interaction - negative covariance

From   andrea pedrazzani <>
Subject   st: positive interaction - negative covariance
Date   Fri, 22 Feb 2013 18:32:12 +0100


I have a simple regression model with an interaction: Y = b0 + (b1)X +
(b2)Z + (b3)XZ.
Z is a dummy (0 or 1).

b1 = .0021067  (SE= .0008513 and p=0.013)
b2 = -.3692713  (SE= .2329837 and p=0.113)
b3 = -.0010758  (SE= .000926 and p=0.245)

Hence, the combined coefficient (i.e., the coefficient on X when Z=1)
is positive:
b1+b3 =  .0021067 + -.0010758 = .0010309

with SE = sqrt( var(b1) + var(b3)*(Z^2) + 2Z*cov(b1,b3)  )
            = sqrt( .0000007246 + .0000008574*1 + -.0000007079*2 )
            = .00040768

To get the p-value for the combinet coefficient, I did
.0010309/.00040768 = 2.528699.  The corresponding p = 0.0114.

Summing up, X has a positive impact on Y when the condition Z is
present (.0010309), and a positive impact also when the condition Z is
not present (.0021067).
So, what can I say about the interaction? What kind of interaction is
it when the impact of X is positive both when the condition is present
and when it is absent? Moreover, the coefficients b1 and (b1+b3) are
very similar to each other.
Also, both b1 and the combined coefficient (b1+b3) are positive, but
the covariance between b1 and b3 is negative. It sounds strange to

Sorry, these are probably trivial questions. I would be really
grateful if someone can help me.


Andrea Pedrazzani
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