# st: RE: Gini index with pweights

 From "Newson, Roger B" <[email protected]> To <[email protected]> Subject st: RE: Gini index with pweights Date Thu, 23 Oct 2008 17:09:02 +0100

```A possible solution might involve the -somersd- package, downloadable
from SSC. This can calculate confidence limits for an unweighted Gini
index, as demonstrated in Newson (2006) and in Newson (2008).

The Gini index is a special case of a between-scenario Somers' D of
wealth with respect to "scenario". It is defined by assuming that there
are 2 lotteries, in which the sample (or poulation) all participate. In
the first lottery, each individual buys one ticket. In the second
lottery, each individual buys a number of tickets proportional to his or
her wealth. The Gini index is the difference between 2 probabilities,
namely the probability that the winner of the second lottery is richer
than the winner of the first lottery and the probability that the winner
of the first lottery in richer than the winner of the second lottery.
This difference will always be non-negative, but will be higher if the
distribution of wealth is more unequal.

To calculate the unweighted Gini index, we can expand each observation
(corresponding to a person) to 2 observations (corresponding to the same
person in the 2 lotteries), to produce a dataset with 1 observation per
person per lottery. The Gini coefficient is then the Von Mises Somers' D
of wealth with respect to lottery sequence (1 or 2), with sampling
probability weights equal to 1 for each person in the first lottery and
to the amount of wealth owned by each person for that person in the
second lottery.

To calculate a sampling-probability-weighted Gini coefficient, we must
define a second set of sampling probability weights. We then expand the
dataset as before, to have one observation per person per lottery, and
multiply the second sampling probability weight by the sampling
probability weight used to calculate the unweighted Gini index, to
produce a third sampling probability weight variable. This third
sampling probability weight variable is then used with the -somersd-
command to calculate a confidence interval for the weighted Gini index.

I hope this helps. Let me know if you have any queries.

Best wishes

Roger

References

Newson R. Confidence intervals for rank statistics: Somers' D and
pre-publication draft from
http://www.imperial.ac.uk/nhli/r.newson/papers.htm

Newson R. Confidence intervals for rank order statistics and their
SSC archive site, and is a post-publication update of a Stata Technical
http://www.imperial.ac.uk/nhli/r.newson/papers.htm

Roger B Newson BSc MSc DPhil
Lecturer in Medical Statistics
Respiratory Epidemiology and Public Health Group
National Heart and Lung Institute
Imperial College London
Royal Brompton Campus
Room 33, Emmanuel Kaye Building
London SW3 6LR
UNITED KINGDOM
Tel: +44 (0)20 7352 8121 ext 3381
Fax: +44 (0)20 7351 8322
Email: [email protected]
Web page: www.imperial.ac.uk/nhli/r.newson/
Departmental Web page:
genetics/reph/

Opinions expressed are those of the author, not of the institution.

-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of
francesca.modena
Sent: 23 October 2008 16:08
To: [email protected]
Subject: st: Gini index with pweights

Deal all,
I would like to estimate the Gini index with sampling weights pweight.

fweights
and aweights.

How can I do it?
Thanks
Francesca Modena

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