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Re: st: Definition of strata and PSUs when svysetting


From   Steven Samuels <sjhsamuels@earthlink.net>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Definition of strata and PSUs when svysetting
Date   Thu, 3 Apr 2008 16:44:13 -0400

Angel,

Let me back up a bit. You were correct in your original conception, but a bit imprecise in phrasing it. I jumped to assumed you were speaking about a no-stratum setup.

Let me correct your misconceptions and answer your original question.

You are dealing with sampling with replacement

Let Mi be the advance measure of size for a PSU. You stated that " Census tracts were randomly selected with probabilities proportional to the number of dwellings in them: (#PSUs x #dwellings in PSUi)/ dwellings in all PSUs. I write these as K = Mi/Mo."

This was imprecise or incorrect on three counts.

1. The quantities K and M0 should be stratum specific. So the description 'all PSU's' should be 'all PSU's in the stratum.'

2. K should not be the number of distinct PSU's in a stratum but the number of random number draws. As you are sampled with replacement, a PSU may appear more than one time in the sample. Whenever this happens, the standard procedure is to draw a different sample of 12 dwellings.

3. K x Mi/Mo is not a probability, but it may be close to one.

Zi = Mi/Mo is the draw-by-draw probability of selecting PSU i into the sample. K x Zi is the expected number of appearances of the PSU i in the sample. You may have been thinking of K x Zi as the 'inclusion probability', which is the probability that the PSU would appear in the sample. The inclusion probability is 1 - (1 - Zi)^K, and will be close to K x Zi for small Zi and K.

However your basic idea was correct. In sampling with replacement, Wi = 1/(K x Zi) does function as a proper sampling weight The hallmark of a sampling weight is that the sample sum of weigh.ts should estimate the number of population PSU's. and Wi does that.

You will need different K's for different strata, so K will not cancel out in numerator and denominator for stratified designs. And, of course, you need the Zi, not only the Mi.

-Steven


-Steven







On Apr 3, 2008, at 7:22 AM, Angel Rodriguez Laso wrote:


Steven,

Sorry for insisting on this weights matter but, if you need only relative
weights, why Z (number of dwellings in all census tracts that is constant
for all individuals in the stratum) is included in the design weights
formula?

Ángel

-----Mensaje original-----
De: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] En nombre de Steven Samuels
Enviado el: miércoles, 02 de abril de 2008 23:29
Para: statalist@hsphsun2.harvard.edu
Asunto: Re: st: Definition of strata and PSUs when svysetting


Angel, for sampling with replacement, the probability of selection
is zi = Mi/M0, where Mi the measure of size for PSU i and M0 is the
total of the M's over PSU's. . The hallmark of probabilities is
that they add to 1 over the population, and this is true of the
Zi's. You need to multiply zi by K= the number of PSU's (in a
stratum) only in the formula for estimating sample totals. See WG
Cochran, Sampling Techniques 3rd ED, Wiley Books, 1977, p. 252. For
estimating means, proportions, correlations, regression coefficients,
only relative weights are needed and K is not needed.


-Steven
On Apr 2, 2008, at 4:25 AM, Angel Rodriguez Laso wrote:


Steven,

In the formula you give for the current sample weight for an
interviewed
person, shouldn't the number of PSUs chosen in the sample design be
included
in the denominator?

I say so because the selection probability of a PSUi is:

#PSUs in the sample design  x  (#dwellings in PSUi)/ dwellings in
all PSUs)

And being the weight the inverse of the selection probabilities,
#PSUs would
go to the denominator.


PS The list of dwellings per census tract was very up-to-date and
only very
minor changes in the actual measure of size were expected.


Ángel


-----Mensaje original-----
De: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] En nombre de Steven
Samuels
Enviado el: martes, 01 de abril de 2008 23:47
Para: statalist@hsphsun2.harvard.edu
Asunto: Re: st: Definition of strata and PSUs when svysetting


Angel,

As as only one person was taken per household. you were quite right
to exclude the dwelling stage in your -svyset- command.

I am not sure that your weights are correct.  You state that your
weighting computation simplifies, because the number of dwelling
units in a census tract cancels out in numerator and denominator.
Yet rarely does the advance measure of size for a PSU match the
actual measure of size. (L Kish, Survey Sampling, Wiley Books, 1965,
p. 239)

Let Z be your advance count of the number of dwellings in all census
tracts. If you anticipated 200 dwellings in a sampled census tract,
you selected the tract with probability equal to 200/Z.  Suppose when
you got to the census tract, you discovered the actual number of
dwellings was 210. Your target number of dwellings is 12. If you
maintain the intended probability of 12/200 (so that the 200 cancels
in the weight computation), the attained sample size will be random,
n= 12 or 13). (Kish, p. 239). If you select exactly 12 dwellings,
with probability 12/210, your current sampling weight for an
interviewed person (Z x (# hh members)/12 should be multiplied by
210/200.

This assumes that you obtained interviews in all 12 selected
dwellings. If you reached the quota of 7 younger and 3 older people
after interviewing in n = 10 or 11 dwellings, I suggest that you
change '12' in the weight computation to the value of n.

-Steven


On Apr 1, 2008, at 3:57 AM, Angel Rodriguez Laso wrote:

Steven,

1. Because only one person was interviewed in each dwelling, I
don't see the
need to include a third stage in the design (there is no
clustering of
individuals by dwelling, only by census tract).

2. I agree with dropping the age stratum.

3. I appreciate your advice on oversampling of the elderly. When
listing and
selecting separately younger and elderly people in each dwelling, I
see the
need to include the dwelling variable, because then you can have two
participants living in the same dwelling.

4. and 5. Census tracts were randomly selected with probabilities
proportional to the number of dwellings in them:

(#PSUs x #dwellings in PSUi)/ dwellings in all PSUs.

As probability of selection of each dwelling is:

12/#dwellings in PSUi,

#dwellings in PSUi cancels out and the result of these two
components of the
weight is constant for all individuals in the stratum and can be
dropped.
The only weights used were then: a) #people in the dwelling; b)
post-stratification weights to make age proportions match those of
the
census.

Many thanks for your help.

Ángel Rodríguez Laso
Institute of Public Health of the Region of Madrid


-----Mensaje original-----
De: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] En nombre de Steven
Samuels
Enviado el: lunes, 31 de marzo de 2008 19:30
Para: statalist@hsphsun2.harvard.edu
Asunto: Re: st: Definition of strata and PSUs when svysetting



Angel

"Gender" in point 2 should have been "age"-fixed below.  I apologize
for the confusion.

-Steven
On Mar 31, 2008, at 9:32 AM, Steven Samuels wrote:


--

Angel, you had a three-stage, not a two stage design

1. The proper -svyset-  should include the stage of selecting
dwellings.

-svyset censustract [pweight=???], strata(area) || dwelling || _n

For the proper pweight, see point 4 below.

2. You did not really stratify on AGE, so drop all reference to an
AGE stratum.

3. Your design, selecting one person at random, and hoping to get
enough elderly people, is not one I recommend. There are standard
approaches for oversampling sub-populations in household surveys.
At the least, one can list older and younger people in each
dwelling and select separately from each list.

4. The design makes it very difficult to calculate the sampling
weights.  You appear to be saying that you stopped interviewing
when you had enough elderly and younger people ( or when you ran
out of dwellings).  This is a version of 'sequential
sampling' (Sharon Lohr, Sampling: Design and Analysis, Duxbury, p.
403)

Here are my best guesses at sample weights.

4a. person weight  =
 1/(prob sel tract)   x (no. dwellings in tract)/(no. of dwellings
where you obtained interviews) x   (no. of people in the person's
dwelling)

4b. If you listed the ages of all people in the 12 selected
dwellings, not just those where you did interviewed, you can do
more:

weight for younger person  =
1/(prob sel tract)   x (no. dwellings in tract)/12 x  (no. younger
people in the 12 sampled dwellings)/(no. of younger people
interviewed)


weight for older person =
1/(prob sel tract)   x (no. dwellings in tract)/12 x  (no. older
people in the 12 sampled dwellings)/(no. of older people
interviewed)

4c. If you have  ages of all people in the sampled dwellings,
substitute  'no. of dwellings where you obtained interviews' for
'12 sampled dwellings' in the formulas in 4b.   These weights may
slightly over-estimate the proportion of elderly people.

5. If there are census figures available for your target
population, apply a  post-stratification weighting to make the
ratio of 'elderly' and 'younger' people match that in the census.
See Lohr, Chapter 8.

-Steven


On Mar 31, 2008, at 6:27 AM, Angel Rodriguez Laso wrote:


Thank you, Steven, for your interest.

Answering to your questions, I didn’t go into more details on the
sampling
procedure because I didn’t think they were needed for the
definition of
strata and PSUs. There was intermediate sampling of dwellings.
There was a
list of all dwellings in census tracts and from this list 12
dwellings in
each selected census tract were chosen at random. From each
dwelling one
person was taken at random (and his/her weight calculated from the
number of
people living in the dwelling). People were interviewed until a
sample of 7
bellow 65 and 3 over 65 was obtained in each census tract. The
reason why 12
dwellings were selected initially is that it was expected that
taking only
10 would not yield the final 7/3 proportion desired. Nevertheless,
not in
all census tracts 7 and 3 individuals could be selected and that's
the
reason (more than the existence of missing items) why there are
census
tracts with only one individual over 65.

I'm trying to check if following your advice (merging strata in
single PSU
per stratum census tracts) or just dropping the second stage
specification,
would give very different results, but when I run a svy: prop
under the
first specification:

svyset censustract [pweight=pondef], strata(area) fpc
(#censustractsinarea)||
identificationvariable, strata(agegroupscorrected)

I get the message: 'Missing standard error due to stratum with
single
sampling unit; see help svydes.', but when I

svydes variable, single stage(2)

no single PSUs are displayed. Do you know why?


Ángel Rodríguez Laso
Institute of Public Health of the Region of Madrid

-----Mensaje original-----
De: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] En nombre de Steven
Samuels
Enviado el: viernes, 28 de marzo de 2008 22:25
Para: statalist@hsphsun2.harvard.edu
Asunto: Re: st: Definition of strata and PSUs when svysetting


Angel-
I'm sorry that I missed your initial post; I was on vacation and
canceled my Statalist subscription. I agree with Stas's suggestion
for the first specification.

I have some questions

1. Your description implies that you created a list of ALL
people in
each selected tract, stratified by age. Then selected by simple
random sampling: 7 from the below 65 list; 3 from the over 65 list.
Is that a correct description? Or, was there intermediate sampling
of dwellings?

2. Your PSU's are census tracts, not people. ("Primary" refers only
to the first stage.) You are saying that in some of the census
tracts, you had only one person either under or 'over' 65. Is that
correct?

For those tracts, I suggest that you go with option 1, but ignore
the stratification, but keep the sampling probabilities. That is,
create a single stratum for those tracts by recoding.


You may still analyze your outcomes by age. The analysis age
groups
need not match the stratum age-groups.

-Steven


On Mar 28, 2008, at 10:40 AM, Angel Rodriguez Laso wrote:



Thank you for your answer, Stas.

I´ve tried both specifications and the first surprise was that
Stata 9
ignores further stages when stage 1 is sampled with
replacement. It
was good
to come across this warning because in our survey sampling was
without
replacement and the sampling fraction of the census tracts was
quite high
(more than one third in some strata) what precludes assuming that
selection
was with replacement.

The problem with using age groups as second stage strata is that
being 3 the
number of people over 65 selected per census tract, whenever
there are
missing values in the variables some strata become single-PSU
(person)
strata, what prevents Stata from calculating standard errors. So,
the two
specifications I´ve tried are:

svyset censustract [pweight=pondef], strata(area) fpc
(#censustractsinarea)
svyset censustract [pweight=pondef], strata(area-by-age) fpc
(#censustractsin
area)

Not surprisingly standard errors with both specifications differ
only in
some hundreths. I believe this is mainly due to the fact that in
both cases
degrees of freedom are very large. This is something I want to
check with
you: From the reading of Korn and Graubard "Analysis of health
surveys" I´ve
understood that in complex surveys degrees of freedom are
calculated as
#PSUs - #strata (624 for the first specification and 1244 for the
second,
because Stata duplicates the number of census tracts because each
of them
belongs to two different strata). I do not follow you very well
when you
recommend doing a small simulation with census or simulated
data to
ascertain degrees of freedom or when you state that Taylor series
expansion
standard errors might be badly off with small samples. It´s usual
practice
to work with such low numbers of individuals per PSU (10 in my
case) and
I´ve never heard that there was a problem of a small sample size
then.

Unfortunately, I don´t have enough knowledge to go for option 3.

To conclude, although both specifications yield similar results, I
agree
with you that the second one implies linked selection of PSUs
while
the
first one is conceptually sounder.



Ángel Rodríguez Laso
Institute of Public Health of the Region of Madrid

-----Mensaje original-----
De: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] En nombre de Stas
Kolenikov
Enviado el: jueves, 27 de marzo de 2008 20:06
Para: statalist@hsphsun2.harvard.edu
Asunto: Re: st: Definition of strata and PSUs when svysetting


I would say your first specificaiton makes better sense, even
though
the design it produces is quite weird, and the degrees of
freedom in
that design are strange (and 7 initial strata won't get you very
far,
anyway). In Stata 10, that's doable with

svyset tract, strata(area) || person, strata(age_group)

if I am getting your design right.

In the second specification with region by age strata, you have
some
sort of coupled sampling when selecting a PSU in one stratum
implies
selecting a certain PSU in the another stratum linked by
geography.
You could still analyze that, but you would need to get accurate
pairwise probabilities of selection to compute Horwitz-Thompson
estimator, and Grundy-Yates-Sen estimator of its variance (which I
don't think is implemented anywhere commercially as those higher
order
probabilities of selection are rarely known; Jeff P, that might
produce a cutting edge addition to Stata's set of -svy- tools,
although I've no idea how to input and parse those :)). Any
reasonably
high level book would have it (Kish, Cochran, Mary Thompson's
books
spring to mind). For special cases, I think that can be
programmed in
Mata. Let's call that option 3. Note that the naive
implementation as

svyset tract, strata(area X age) || person

produces wrong probabilities of selection, and the variances are
likely to be understated, as there is more variability in this
specification than in your actual design.

If I were in your shoes, I would try both specifications you
described
and see whether they are producing comparable substantive results.
Keep in mind that either way you are getting asymptotic Taylor
series
expansion standard errors, and they might be badly
off with small samples like those you have. And I think you
need to
worry about your degrees of freedom, not your number of PSUs; I
would
do a small simulation to determine the approximate d.f.s for your
main
variables -- from census data if you have it, or from simulated
data
resembling the actual population. If I had infinite time to
work on
that project (meaning, a week or two of devoted programming), I
would
implement option 3 as the most proper.

On 3/25/08, Angel Rodriguez Laso
<angel.rodriguez@salud.madrid.org>
wrote:


Greetings to all members of the list,



 I have the following questions on svysetting for an analysis
of a
complex
 survey:


 We have carried out a regional health population survey. We
defined

strata

 initially as geographic areas in the region (n=7) and allocated
to each

of

 them a sample proportional to their population. But because we
wanted to
 over-represent the elderly, we set that the number of people
over 65

years

 sampled in all areas had to reach a minimum number. We didn't
change the
 sample size of people bellow 65 obtained through the
proportional
 allocation. Therefore the sampling fractions (and consequently
the

weights)

 are different for each area by age group (bellow/over 65)
category.

 Then we selected census tracts in each geographic area with
probabilities
 proportional to their total population, and randomly sampled 10

individuals

 in those selected, always keeping the proportion 7 bellow 65
years/3 over

65

years, which was the regional overall age distribution after the
oversampling explained above. My first question is if strata
should be
defined as geographic regions alone or as geographic area by age
groups
(bellow/ over 65 years) (n=14) when svysetting. The first
possibility

looks

more reasonable, because census tracts were selected within
geographic
areas, not within geographic-age groups areas. If this is
correct, then
probably the way to svyset would be declaring geographic
areas as
first
stage strata, census tracts as first stage PSUs and age
groups as
second
stage strata.

Alternatively, if the answer is that strata should be defined as
region

by

two age-groups categories, then the same census tract can belong
to two
different strata (for example area A bellow 65/ area A over 65)
depending

on

 the age of the individual considered. If I svyset: strata
(region
by age
 group categories) and PSU= census tracts, STATA interprets that
there are
 twice the number of PSUs than real census tracts are. Is that
correct?



 Many thanks.


 Ángel Rodríguez Laso
 Institute of Public Health of the Region of Madrid



--
Stas Kolenikov, also found at http://stas.kolenikov.name

Small print: Please do not reply to my Gmail address as I don't
check
it regularly.

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Steven Samuels
845-246-0774
18 Cantine's Island
Saugerties, NY 12477
EFax: 208-498-7441





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