Al,
I was picking on you posting items 1, 3 and 4 in the original email :))
Looks like you could do it both ways, then. You can get the covariance
matrix of the taus, provided that it accounts for the complex sample
design the way you need. Working with U-statistics in stratified
samples is a weird thing to do. I think the general asymptotic
normality in complex survey designs was only proven in the 1990s (I
cannot find a proper reference, but here's a partial older answer --
http://www.citeulike.org/user/ctacmo/article/1743121). Whether
-wstrata- or -bstrata- options of -somersd- should be used, or yet
something else that would need to involve all units in a proper
Horvitz-Thompson type estimator -- that is your call as the analyst.
Keep in mind that there is no single asymptotics in survey statistics,
too. The most popular setups corresponding to the most popular designs
with 2 PSU/stratum is to have # strata \to \infty with # units/stratum
bounded. Those are the more complex settings to analyze compared with
fixed # strata and growing # units per stratum asymptotics.
I wouldn't say that the delta method is always preferrable to the
jackknife, or the other way round. The general theory is that in most
survey settings, they are equivalent to the second order (i.e., agree
to the terms of $O(n^{-2})$) while the bootstrap (provided it is done
right) agrees with them to the order of $O(n^{-1})$, and BRR agrees to
the order of $O(n^{-1/2})$. What I would probably do is to code it
both ways, to make sure they produce answers that match to the fourth
decimal point or so.
On 1/21/09, Feiveson, Alan H. (JSC-SK311) <[email protected]> wrote:
> 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
--
Stas Kolenikov, also found at http://stas.kolenikov.name
Small print: I use this email account for mailing lists only.
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