Re: st: RES: generating a variable with pre-specified correlations with other two (given) variables

[Richard Williams <richardwilliams.ndu@gmail.com>]

fjc, I'll trust your math, but it seems awfully 
complicated to me. Why don't you just want to do something like
mat mCorr = (1, .6, .4\ .6, 1, .5 \ .4, .5, 1)
corr2data x y z, cstorage(full) corr(mCorr) n(100000) clear
corr
reg z x y

You can get the correlations (and also means & 
sds) for the real x and y from your data set and 
just plug them in, and then plug in the desired 
correlations for x and y with z.
As a sidelight, any random variable u you 
generate yourself will have a nonzero (albeit 
small) correlation with x and y, because of 
sampling variability. (Unless you generate u as 
part of the corr2data command, in which case you 
can force it to have 0 correlation with x and y.)
At 03:23 PM 8/31/2011, fjc wrote:
> Hi,
>
> Thank you all for the quick and useful responses.
>
> 1. I can do with covariances instead of correlations, so the methods
> proposed by Tirthankar and Richard work fine.
>
> 2. Still, if I wanted to stick to correlations, I think one can apply
> the same ideas (as suggested in the previous responses):
>
> Let z be given by
>
> (0) z = a * x + b * y + c * u,
>
> where x and y are the two variables in the dataset and u is a
> zero-mean random variable independent of x and y.
>
> From (0) one gets:
>
> (1)  Corr(x,z) = a * sd(x)/sd(z) + b * sd(y)/sd(z) * Corr(x,y)
>
> (2)  Corr(y,z) = b * sd(y)/sd(z) + a * sd(x)/sd(z) * Corr(x,y)
>
> (3)  Var(z) = a^2 * Var(x) + b^2 * Var(y) + c^2 
> * Var(u) + 2 * a * b * Cov(x,y)
> Once we have chosen Corr(x,z), Corr(y,z) and Var(z), we can solve the
> system above for a, b, and c. Actually, equations (1) and (2) can be
> solved for a and b to get:
>
> a = [sd(z)/sd(x)] * [Corr(x,z) - Corr(x,y)*Corr(y,z)] / (1 - Corr(x,y)^2)
>
> b = [sd(z)/sd(y)] * [Corr(y,z) - Corr(x,y)*Corr(x,z)] / (1 - Corr(x,y)^2)
>
> Then we can use (3) to obtain the value of c.
>
> Finally, we can use (0) to generate z.
>
> Thanks again,
>
> Francisco.
>
>
> On Wed, Aug 31, 2011 at 3:59 PM, Richard Williams
> <richardwilliams.ndu@gmail.com> wrote:
> > At 07:41 AM 8/31/2011, fjc wrote:
> >>
> >> Thanks, Tirthankar.
> >>
> >> This answers my question as originally posted.
> >>
> >> Now, something I didn't say in my earlier post (and I think I should
> >> have) is that after I generate the new variable (z) I would like tow
> >> run a regression of y on x and z. But if I generate z in the way you
> >> propose, I will get perfect collinearity. żIs there any other way to
> >> generate z without getting this collinearity?
> >
> > Slightly tweaking the earlier example, does this do what you want?
> >
> > mat mCorr = (1, .6, .4\ .6, 1, .5 \ .4, .5, 1)
> > corr2data x y z, cstorage(full) corr(mCorr) n(100000) clear
> > corr
> > reg z x y
> >
> > Again, mCorr is a combo of the given correlations for x and y with the
> > desired correlations for z. If you want, you can also specify standard
> > deviations and means, both observed (for x and y) and desired (for z). I am
> > faking all the data, although the 
> correlations etc. can come from real data.
> > If you want to do some combo of fake and real (e.g. generate a z using the
> > realx and realy) it can probably be done but would take a bit more work.
> >
> >
> > -------------------------------------------
> > Richard Williams, Notre Dame Dept of Sociology
> > OFFICE: (574)631-6668, (574)631-6463
> > HOME: Â  (574)289-5227
> > EMAIL: Â Richard.A.Williams.5@ND.Edu
> > WWW: Â  Â http://www.nd.edu/~rwilliam
> >
> >
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-------------------------------------------
Richard Williams, Notre Dame Dept of Sociology
OFFICE: (574)631-6668, (574)631-6463
HOME:   (574)289-5227
EMAIL:  Richard.A.Williams.5@ND.Edu
WWW:    http://www.nd.edu/~rwilliam


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