R help - Nov 2010 - How to do bootstrap for the complex sample design?

Hello;
 
Our survey is structured as : To be investigated area is divided into 6 regions,
within each region, one urban community and one rural community are randomly
selected,
then samples are randomly drawn from each selected uran and rural community.  
 
The problems is that in urban/rural stratum, we only have one sample. 
In this case, how to do bootstrap?
 
Any comments or hints are greatly appreciated!
 
Faye    		 	   		  
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Faye wrote:
>Our survey is structured as : To be investigated area is divided into
>6 regions, within each region, one urban community and one rural
>community are randomly selected, then samples are randomly drawn from
>each selected uran and rural community.
>
>The problems is that in urban/rural stratum, we only have one sample.
>In this case, how to do bootstrap?
You are lucky that your sample size is 1.  If it were 2 you would
probably have proceeded without realizing that the answers were wrong.

Suppose you had two samples in each stratum.  If you proceed naturally,
drawing bootstrap samples of size 2 from each stratum, this would
underestimate variability by a factor of 2.

In general the ordinary nonparametric bootstrap estimates of variability
are biased downward by a factor of (n-1)/n -- exactly for the mean, 
approximately for other statistics.  In multiple-sample and stratified
situations, the bias depends on the stratum sizes.

Three remedies are:
* draw bootstrap samples of size n-1
* "bootknife" sampling - omit one observation (a jackknife sample),
then
  draw a bootstrap sample of size n from that
* bootstrap from a kernel density estimate, with kernel covariance equal
  to empirical covariance (with divisor n-1) / n.
The latter two are described in 
Hesterberg, Tim C. (2004), Unbiasing the Bootstrap-Bootknife Sampling vs.
Smoothing, Proceedings of the Section on Statistics and the Environment,
American Statistical Association, 2924-2930.
http://home.comcast.net/~timhesterberg/articles/JSM04-bootknife.pdf

All three are undefined for samples of size 1.  You need to go to some
other bootstrap, e.g. a parametric bootstrap with variability estimated
from other data.

Tim Hesterberg

At 01:38 AM 11/4/2010, Fei xu wrote:
>Hello;
>
>Our survey is structured as : To be investigated area is divided 
>into 6 regions,
>within each region, one urban community and one rural community are 
>randomly selected,
>then samples are randomly drawn from each selected uran and rural community.
>
>The problems is that in urban/rural stratum, we only have one sample.
>In this case, how to do bootstrap?
>
>Any comments or hints are greatly appreciated!
>
>Faye
Just make a table of your data, with each row corresponding to a 
measurement. You columns will be Region, UrbanCommunity, 
RuralCommunity and your response variables.

Bootstrap resampling is just generating random row indices into this 
table, with replacement. I.e.,

index<- sample(1:N, N, replace=TRUE)

Then your resample is myTable[index,].

Because you chose UrbanCommunity and RuralCommunity randomly, this 
shouldn't be a problem. The fact that you choose a subsample size of 
1 means you won't be able to estimate within-region variances unless 
you make some serious assumptions (e.g., UrbanCommunity effect 
independent of Region effect).

===============================================================Robert A.
LaBudde, PhD, PAS, Dpl. ACAFS  e-mail: ral at lcfltd.com
Least Cost Formulations, Ltd.            URL: http://lcfltd.com/
824 Timberlake Drive                     Tel: 757-467-0954
Virginia Beach, VA 23464-3239            Fax: 757-467-2947

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