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# Re: st: positive interaction - negative covariance

 From Nick Cox To statalist@hsphsun2.harvard.edu Subject Re: st: positive interaction - negative covariance Date Sat, 23 Feb 2013 00:45:22 +0000

```This is very confusing.

1. In terms of your previous statement of "a simple regression model"
you should have applied code like this, tailored to the special syntax
of -twoway function-.

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

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

after which the functions would appear as perfect straight lines; no
jaggedness is implied. The jaggedness is a consequence of using -dur-
when only -x- is allowed and needed. -x- is a generic x axis variable
and unrelated to any variable in the dataset.

2. But now you are telling us that it is a flogit model, a term I
don't recognise.

I think you won't get good help if you don't explain clearly and
consistently what you are doing.

Nick

On Fri, Feb 22, 2013 at 10:50 PM, andrea pedrazzani
<andrea.pedrazzani.piter@gmail.com> wrote:
> 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: 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.
>
>
> 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...
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