Two practical tools: sample weights and the design effect
Use of sample weights: We have learned about many important topics
in designing sample surveys, such as stratification, clustering, and the use of
auxillary information with ratio estimation. These ideas can be combined
in many ways to obtain very complex multi-stage sampling designs, and our
chapter on two-stage cluster sampling gave us a glimpse of the complexity
that can be involved in more sophisticated designs. As we saw in the chap-
ter on two-stage cluster sampling, the usual estimators (particularly variance
estimators) can become very complicated. In actual sampling studies, re-
searchers often simplify calculations by using sampling weights and compu-
tational approximations for variance calculations. Here we will introduce
sampling weights by rewriting the expression for the estimator of the mean
in stratified random sampling. Recall in the chapter on stratified random
sampling that the estimator of µis:
bµ=yst =1
N
L
X
i=1
Niyiwhich can be written as
=1
N
L
X
i=1
ni
X
j=1
Ni
ni
yij =
L
P
i=1
ni
P
j=1
wijyij
L
P
i=1
ni
P
j=1
wij
,
where wij =Ni/niis the weight for the jth observation in group i, and has
the interpretation that each observation in the sample represents wij =Ni/ni
members of the population. Thus if a population of N= 1000 elements are
divided into four strata each equal to Ni= 250,and if equal sample sizes
of ni= 100 are used for each stratum, then each observation in the sample
represents Ni/ni= 2.5 elements from the population. The general idea is
that a sampling weight is a reciprocal of a selection probability, so for the
StRS example above, δij = 1/wij =ni/Niis the probability of being sampled
for a member of the ith stratum. In many sampling studies, the sampling
weights are calculated as the sampling design is developed. Once the sam-
pling weights are calculated, any quantity of interest can be calculated as
a weighted sum as exemplified in the StRS expression above, and computa-
tional methods such as Taylor series, jackknife, or bootstrap methods can be
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