Stas - Thanks for pointing out the asymptotic joint normality property
for U-statistics. Your comments suggest that if I could get my hands on
the standard error covariance matrix for the original tau-s, I could try
using the delta method to get a standard error for tau_xy.z.
I think I can get this by three runs of -somersd-.
1.
somersd y x z
matrix Vyxz = e(V)
matrix list Vyxz
symmetric Vyxz[3,3]
y x z
y 0
x 0 .11
z 0 -.09777778 .09
So Var(tau_yx) = 0.11, Cov(tau_yx, tau_yz) = -.09777778, Var(tau_yz)
=0.09
2.
somersd x y z
matrix Vxyz = e(V)
....
gives Var(tau_xy), Cov(tau_xy, tau_xz), etc
3. somersd z x y
...
gives Cov(tau_zx, tau_zy),etc
So if this is correct, the problem can be approached by only counting to
three - not four!
Of course, the delta method on the original tau-s may not be as good as
doing jackknife on resampled tau_yx.z values - but the former is easy to
implemnet given -somersd- in its present form.
Al
It looks as thoiug I could do this by running somersd twice, once
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of Stas
Kolenikov
Sent: Wednesday, January 21, 2009 8:29 AM
To: [email protected]
Subject: Re: st: Partial Correlation Kendall's Tau (or Somers D)
Al,
there are three types of statisticians: those who can count to four and
those who cannot :)).
Since the original tau's are U-statistics, they will have a (joint
multivariate) asymptotic normal distribution, and hence the partialized
version you presented would also be asymptotically normal.
The necessary condition for the jackknife standard errors to be
consistent is that the statistic of interest has an asymptotically
normal distribution, and I would guess that other more subtle regularity
conditions would also be satisifed (although jackknife is not consistent
say for a median which is also asymptotically normal).
If you have a complex survey design then you would need to omit the
complete PSU when computing the standard errors, and Stata's -svy
jackknife- does that for you (although you would need to write a
wrapper of -eclass, properties(svyj)- and see that the conditions
outlined for those properties are satisfied).
You guys do have some powerful computers at NASA, I guess. I wouldn't
think of doing jackknife over -ktau- with the capacities I have :)).
On 1/21/09, Feiveson, Alan H. (JSC-SK311) <[email protected]>
wrote:
> Hi - I have been reading in Gibbons and Chakraborti (Nonparametric
> Statistical Inference) about a Kendall's Tau analog to partial
> correlation, where one would like to quantify the association between
> y and x after correcting for z. Specifically, the authors define
> tau_xy.z in terms of the three pairwise associations tau_xy, tau_yz,
> and tau_xz, and then give an expression that loooks exaclty like
> Pearson partial
> correlation:
>
>
> tau_xy.z = (tau_xy - tau_xz*tau_yz)/sqrt((1-tau_xz^2)*(1-tau_yz^2))
>
>
> My questions on this are
>
> 1. Can the jackknife method for standard errors be extended to
> tau_xy.z or its Somers D analog?
>
> 3. Are there extensions to defining tau_xy.z within or betwen strata
> and for obtaining standard errors with clusters as Roger Newson has
done?
>
> 4. Most importantly - Roger - do you have any plans for updating your
> Somers D program to include partial association?
>
> Thanks,
>
> Al Feiveson
>
--
Stas Kolenikov, also found at http://stas.kolenikov.name Small print: I
use this email account for mailing lists only.
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