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Re: st: variance when using svy: mean


From   Steven Samuels <sjhsamuels@earthlink.net>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: variance when using svy: mean
Date   Mon, 3 Dec 2007 13:52:24 -0500

You are very welcome. One observation and 10 observations per psu will not give the -same- precision; increasing the number of observations will decrease the standard error, but there is a limit, depending on the relative variation within-and between- psu's. Theory can help determine the optimal balance between the number of psu's and the number of observations to take from each. The formulas take into account the relative cost of working in a new psu versus the cost of taking an additional observation once you are in the psu.

I'm still not sure that you took a simple random sample, as this requires a finite population to begin with. Still, it sounds like you are on the right track.

Good luck!
Steve

On Dec 3, 2007, at 1:26 PM, David Merriman wrote:


Thanks, I am understanding better.
Within each area I took a random (iid) sample from an infinite
population.  The sample sizes varied for reasons that are contextual
and difficult to explain without telling you all the details of the
project. (Can you help me w'ith the computer code here?  How do I tell
stata that I took a random sample within psus?)

So, if I understand you correctly the variance of my estimate of the
population mean is roughly independent of how many observations I
have.  In other words, if I had gone to 100 different areas and
collected a single sample I would estimate the population mean with
about the same precision as if I had collected 10 samples within each
area?  I still have trouble getting my mind around that but your
example with a Census of the two areas does help.  Thanks again.


On Dec 3, 2007 12:08 PM, Steven Joel Hirsch Samuels
<sjhsamuels@earthlink.net> wrote:
Thanks, David that is clearer.

Was there a sampling procedure within areas? If not, you can still
apply the -svy- commands, but without a sampling procedure of some
kind, the observations within a single area may not 'represent' that
area in any statistical sense.

In any case,  the major portion of the variance of sample estimates
will still arise from PSU-to-PSU variation. The effective sample size
is 100, the number of PSU's.  Thought experiment:  you take a sample
of n= 2 areas and collect information about everybody (a census).
Even if the number of observations is 200,000, your effective sample
size is still n=2 for making inferences about the original
population. As long as the two area means are different, the standard
error will be nonzero.

Your simulations are therefore correct in producing the same standard
error for the two PSU's.  If you had simulated sampling within the
PSU's, then the you would the SE's to change, as the sample
proportions would vary from simulation to simulation.

-Steven


4. The sampling weight for a PSU

On Dec 3, 2007, at 12:21 PM, David Merriman wrote:

Thanks. I do have a lot of trouble with the terminology.
I selected a weighted random sample of 100 geographic areas from 930
such areas. The weights were designed so that my weighted random
sample would be representative of the population (e.g. I oversampled
areas with a high number of people). I do not think I have any strata
(I did NOT for example oversample high poverty areas). My psus are
the 100 geographic areas.

I am afraid I still do not know what to do next. It sounds like you
are saying that the variance should not differ between my two cases
but this does not make intuitive sense to me. Any help you can
provide would be appreciated.

On Dec 3, 2007 11:11 AM, Steven Joel Hirsch Samuels
<sjhsamuels@earthlink.net> wrote:

David, it doesn't sound like your study is a probability sample; if
not, you don't need -svy- commands. Instead, use non-survey
commands and assign an -iweight- or other weight variable to
properly represent your population.

If your data do arise from a probability sample, your 'areas' appear
to be strata, not primary sampling units (psu's). Strata are units
which partition a population. A psu is the highest stage unit
selected by random numbers within a stratum. Standard errors for
survey data depend mainly on the number of psu's, not on the number
of observations within them.

-Steven

On Dec 3, 2007, at 11:19 AM, David Merriman wrote:


Dear Statalisters:
I am a long time Statauser but new to svy: commands and am quite
confused.
I apologize if this is long-winded I am trying to say it as
concisely
as possible.

I have collected primary data in several geographic areas. Each of
the geographic areas has a different weight so that my entire sample
should be representative of the population. In each geographic
area I
have collected a number of observations but the number of
observations
in the area tells me nothing about the density of the activit

area. I want to estimate the population mean (for all geographic
areas) and the variance of that estimate. The problem is that
while I
get sensible means the variances do not seem to be a function of the
number of observations I have. Intuitively I think that the
variance
ought to change (fall) as the number of observations increases.

I tried using
svyset psuedo_psu [pweight=obs_weight]
svy: mean psuedo_chicago_tax_paid

where psuedo_psu is the variable indicating the primary sampling
unit,
obs_weight is the psu_weight divided by the number of
observations in
that psu and psuedo_chicago_tax_paid is the (zero-one) variable for
which I want to estimate the mean and variance.

I created a simulated data set (the real one is more complex) with 2
psus. In the first trial, each psu had 50 observations. psu 1
had a
weight of 1 and a 50 percent chance of a 1. psu 2 had a weight of 5
and a 20 percent chance of a 1. I get a sensible mean of .25 and a
standard error of .0833333.

In the second trial, I also had two psu. Psu 1 has 900 observations
and psu 2 has 100 observations. psu 1 had a weight of 1 and a 50
percent chance of a 1. psu 2 had a weight of 5 and a 20 percent
chance of a 1. I get a sensible mean of .25 but the same standard
error of .0833333 as in case 1. This does not make sense to me. I
have more observations in case 2 so I think I should get a smaller
variance.

I imagine I am not using the correct design. Can anyone help?
Below,
I show the computer code for my simulation (fake data set) but you
don't need to read this if you understand the comments above.
Thanks
so much.


#delimit ;
****************************************************************
* created the simulated data
***********************************************************;
set obs 100;
****************************************************************
* generate psu
***********************************************************;
gen psuedo_psu=1 if _n<51;
replace psuedo_psu=2 if _n>=51;
****************************************************************
* generate chicago_tax_paid
***********************************************************;
gen psuedo_chicago_tax_paid=1 if _n<=25;
replace psuedo_chicago_tax_paid=0 if _n>25 & _n<=50;
replace psuedo_chicago_tax_paid=1 if _n>50 & _n<61;
replace psuedo_chicago_tax_paid=0 if _n>=61;
****************************************************************
* generate psu weights
***********************************************************;
gen sample_weight=1 if psuedo_psu==1;
replace sample_weight=5 if psuedo_psu==2;
summarize;
****************************************************************
* generate OBSERVATION weights
***********************************************************;
sort psuedo_psu;
by psuedo_psu: gen obs_weight= sample_weight/_N;
summarize;
svyset psuedo_psu [pweight=obs_weight];
**********************************************************
* psu1 has a mean of .5 and a weight of 1
* psu2 has a mean of .2 and a weight of 5
* (5*.2)+(1*.5)=1.5
* 1.5/6=.25
*
* so the mean estimate makes sense to me
*******************************************************;
svy : mean psuedo_chicago_tax_paid;
mean psuedo_chicago_tax_paid;
*********************************************************
* do a second round with unequal size groups
*****************************************************;
clear;
#delimit ;
****************************************************************
* created the simulated data
***********************************************************;
set obs 1000;
****************************************************************
* generate psu
***********************************************************;
gen psuedo_psu=1 if _n<901;
replace psuedo_psu=2 if _n>=901;
****************************************************************
* generate chicago_tax_paid
***********************************************************;
gen psuedo_chicago_tax_paid=1 if _n<=450;
replace psuedo_chicago_tax_paid=0 if _n>450 & _n<=900;
replace psuedo_chicago_tax_paid=1 if _n>900 & _n<921;
replace psuedo_chicago_tax_paid=0 if _n>=921;
****************************************************************
* generate PSU weights
***********************************************************;
gen sample_weight=1 if psuedo_psu==1;
replace sample_weight=5 if psuedo_psu==2;
****************************************************************
* generate OBSERVATION weights
***********************************************************;
sort psuedo_psu;
by psuedo_psu: gen obs_weight= sample_weight/_N;
summarize;
svyset psuedo_psu [pweight=obs_weight];
**********************************************************
* psu1 has a mean of .5 and a weight of 1
* psu2 has a mean of .2 and a weight of 5
*
* I get the same answer for the mean in case 1 and case 2
* which I think is correct but
* I also get the same answer for the variance which I think is not
correct
*
* I think I should have a lower variance in case 2
*******************************************************;
svy : mean psuedo_chicago_tax_paid;
mean psuedo_chicago_tax_paid;


--
David Merriman
dmerrim@gmail.com
*
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Steven  Samuels

sjhsamuels@earthlink.net
18 Cantine's Island
Saugerties, NY 12477
Phone: 845-246-0774
EFax: 208-498-7441





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--
David Merriman
dmerrim@gmail.com
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--
David Merriman
dmerrim@gmail.com
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