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# Re: st: positive interaction - negative covariance

 From David Hoaglin To statalist@hsphsun2.harvard.edu Subject Re: st: positive interaction - negative covariance Date Fri, 22 Feb 2013 15:17:50 -0500

```Dear Andrea,

The basis for a statement about the interaction is the estimate of b3
and its standard error: After taking into account the contributions of
X and Z, the interaction is not significant (p = .245).

Thus, X has a positive impact on Y when Z is present and when Z is
absent, but those contributions are not significantly different.  That
is, the interaction is essentially absent.

A negative covariance between b1 and b3 is to be expected.

You may want to remove XZ from the model.

Regards,

David Hoaglin

On Fri, Feb 22, 2013 at 12:32 PM, andrea pedrazzani
<andrea.pedrazzani.piter@gmail.com> wrote:
> Hello,
>
> 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
> me...
>
> Sorry, these are probably trivial questions. I would be really
> grateful if someone can help me.
>
> Best,
>
> Andrea Pedrazzani
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