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# Re: st: positive interaction - negative covariance

 From andrea pedrazzani To statalist@hsphsun2.harvard.edu Subject Re: st: positive interaction - negative covariance Date Fri, 22 Feb 2013 23:55:09 +0100

```Thank you very much Nick, Jay and David.

My -x- is -dur-, whose range is from 1 to 1764.
I plotted the two curves as you suggested me:

local b0 = 1.663478
local b1 = .0021067
local b2 = -.3692713
local b3 = -.0010758

twoway function `b0' + `b1'*dur, ra(1 1764) || ///
function `b0' + (`b1' + `b3')*dur + `b2', ra(1 1764)

The two functions have the same shape and are very close to each
other. Both tend to slightly increase as -x- increases (the functions
are really jagged, because -dur- has many values). The first one
(where the condition Z is absent) is always a bit higher than the
second (where the condition is present). If I am not wrong, this
indicates that the impact of both on my dependent variable goes in the
same (positive) direction.

To Jay: actually my dependent variable is a proportion (I am using flogit)

To David:
> 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.

Thank you for clarifying the point. Indeed, if I compare the
confidence interval of b1 to the confidence interval of (b1+b3), they
are not statistically different. Is it the same? In a book I read, the
authors make this comparison.

Thanks again.

Best,

Andrea Pedrazzani

2013/2/22 David Hoaglin <dchoaglin@gmail.com>:
> 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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```