Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 260

on all the subsamples to estimate the variance of the parameters. This is much easier
than writing code to linearize the variance, see Eq. ( 10.26 ). The
withReplicates function evaluates a user-supplied function or expression
within each replicate and computes the standard error estimates.

> bootrep < - withReplicates(dsrsb, quote(sum(.weights*yobs2)/
+ sum(.weights*yobs)),return.replicates ¼ T)
> hist(bootrep $ replicates, xlim ¼ c(0.70,0.80),main ¼ "")
> text(0.725,62,"Bootstrap",cex ¼ 1.5)
> jackrep < - withReplicates(dsrsjk1, quote(sum(.weights*yobs2)/
+ sum(.weights*yobs)),return.replicates ¼ T)
> hist(jackrep $ replicates,xlim ¼ c(0.749,0.755),main ¼ "")
> text(0.7525,15.5,"Jackknife",cex ¼ 1.5)

10.6 Multiple Frames

In an ideal sampling survey, we have a finite population U with N units, a random
sample s of size n is selected from the population, and a certain variable y k is
measured for any unit k

s . The HT estimator of the total is thus applied without any
concern, because it is unbiased. This is only true if the sampling frame includes
every unit in the target population, if all the selected units respond, and if there are
no measurement errors. These assumptions are clearly purely theoretical, and they
are rarely true in practical surveys. The number of nonresponses can be minimized
using organizational measures and weights adjusted by using models that mitigate
the effects of the nonresponse bias. Undercoverage (see Chap. 5 ) is a common
problem when dealing with frames based on farms or other legal body definitions of
statistical units. It implies a severe negative bias in the estimates, particularly
because it is generally not constant over the whole population and is concentrated
in some specific subpopulations. For example, it is known that undercoverage is a
function of the size of the statistical unit, and smaller sized units are less accurate.
This topic is becoming more important as it has had an increasing impact on the
quality of surveys over the last few decades. NSIs have put a lot of effort into
linking records for different frames and integrating census data with administrative
data sources, but this has not improved the situation. Typically, undercoverage
(similarly to nonresponses) is dealt with using weight adjustments, increasing the
weights of selected respondents in an attempt to reduce bias. Attempts are also
made to reduce the measurement error using careful survey and questionnaire
design, and modeling.
Multiple frame surveys can help to address some of these problems (for a review,
see Lohr 2009 , 2011 ). Multiple-frame sampling refers to surveys in which two or
more frames are used. Independent samples are taken from each of the frames. The
potential advantages of using a multiple frame survey include:

2