# st: Confidence interval for a median with weighted, clustered data

 From Steven Samuels To statalist@hsphsun2.harvard.edu Subject st: Confidence interval for a median with weighted, clustered data Date Tue, 23 Dec 2008 12:49:28 -0500

In correspondence, Roger Newson (r.newson@imperial.ac.uk) described to me a method for computing a CI for a median with weighted, clustered, data. With his permission, I am posting his description and the do-file which accompanied it. (I have slightly edited both to delete some extraneous material.) The do file utilizes Roger's program -cendif- (part of -somersd- available from SSC) and his commands -keyby- ,-xcontract-, and -expgen- .
```
-Steve

From Roger Newson:

```
I would use the following method to make cendif calculate confidence intervals for a median (possibly with clusters and/or weights). This method involves the concept of between-scenario rank statistics, exemplified by the Gini coefficient and the population attributable risk. (See Subsection 5.3 of Newson (2006).)
```
```
I would use my own expgen package, downloadable from SSC, to duplicate every observation in the sample, replacing each observation with a pair of identical observations, and generating a copy identifier variable named scenario, equal to 1 for the first member of a pair and equal to 2 for the second member of a pair. I would then set the value of the variable whose median I wanted to know to zero in all the second members of all pairs (where scenario==2). I would then use cendif, with the options
```
by(scenario) funtype(vonmises)

```
and with a cluster() option putting both members of each pair in the same cluster, to calculate a median difference between the subsamples with scenario==1 and scenario==2. This should definitely produce a valid confidence interval, with or without weights, for the median difference between individuals sampled from the world as we know it (Scenario 1) and a hypothetical fantasy world in which the variable whose median we are estimating is always zero (Scenario 2). This median difference is (of course) simply the median of the variable whose median we wanted to know.
```
To illustrate my method I have run an example do-file, appended.

I have set options

transf(iden) tdist

```
which, based on my own simulations so far, are usually a good idea when estimating median differences. I also set options
```
by(scenario) funtype(vonmises)

```
and the appropriate cluster options. To produce an "unclustered" estimate, I have used the option cluster(make), implying 1 cluster for each car model. (Therefore, each observation in the old auto data corresponds to a cluster of 2 observations in the duplicated dataset.) To produce a "clustered" estimate, I have used the option cluster(firm), where firm is a derived variable, containing the first word of make. This cluster variable implies that we are sampling firms from a population of firms, instead of sampling car models from a population of car models. For both the clustered and the unclustered case, I have produced an unweighted estimate, and also a weighted estimate, using pweights to weight non-American cars higher, and therefore producing a higher median mpg.
```
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

**************************CODE BEGINS**************************
#delim ;
version 10.1;
*
Demo of confidence interval for median using cendif.
```
This program uses the SSC packages somersd, keyby, xcontract, and expgen.
```*;

clear;
set memory 1m;

sysuse auto, clear;
keyby foreign make;
gene firm=word(make,1);
lab var firm "Firm";
gene pwt=foreign + 0.25*!foreign;
lab var pwt "Probability weight";
desc;
xcontract foreign pwt, list(,);
xcontract firm, list(,);

preserve;
expgen =2, copy(scenario);
replace mpg=0 if scenario==2;
*
Unclustered analyses
*;
```
cendif mpg, by(scenario) tdist transf(iden) funtype(vonmises) cluster (make); cendif mpg [pwei=pwt], by(scenario) tdist transf(iden) funtype (vonmises) cluster(make);
```*
Analyses clustered by firm
*;
```
cendif mpg, by(scenario) tdist transf(iden) funtype(vonmises) cluster (firm); cendif mpg [pwei=pwt], by(scenario) tdist transf(iden) funtype (vonmises) cluster(firm);
```restore;

exit;
***************************CODE ENDS***************************;

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```