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Re: st: positive interaction - negative covariance


From   Nick Cox <njcoxstata@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: positive interaction - negative covariance
Date   Sat, 23 Feb 2013 00:47:00 +0000

Correction. The code should end

, ra(1 1764)

On Sat, Feb 23, 2013 at 12:45 AM, Nick Cox <njcoxstata@gmail.com> wrote:
> 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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