Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 241

X

H

X

H

X k2s h d k y k ¼ X k2s g k , POST d k y k ;

N h
N h

t POST , y ¼

N h y s h ¼ ð 10 : 11 Þ

h ¼1 h ¼1

X k2s h ^ k =N h and N h ¼ X k2s h 1

where y s h ¼ =π k . The characteristic of the adjusted

weights w k , POST ¼ ¼1, ... , H and,

therefore, will respect the frequencies of the known types defined by the codes of
the auxiliary variables X.
The variance of Eq. ( 10.11 ) in the simple SRS case can be estimated using
(S¨rndal et al. 1992 , p. 266)

g k , POST d k is to have sums equal to N h for every h

X

H

S y , h
n h ;

¼

V SRS t POST , y

N h ð 1 f Þ ð 10 : 12 Þ

h ¼ 1

where S y;h is the variance of the sample from the h -th group. Note that, given the
same partition, this approach is generally less efficient than a stratified sampling
(see Sect. 6.5 ) because there is additional uncertainty caused by the random sample
size of each stratum.
A strong incentive to post-stratify (rather than introducing the groups in a
stratified design) may arise from the multipurpose nature of a survey. The stratified
design fixes the strata size once for all the survey variables y, so it will rarely be
efficient for each variable. Post-stratification can change when estimating different
survey variables. Moreover, the auxiliaries X may not be available when the sample
has been selected. This case is typical of remotely sensed data that are known before
the estimation process but after sample selection, because the most discriminating
satellite data is acquired between flowering and harvesting.
The R survey package provides post-stratification through the
postStratify function. It modifies an existing design object into a post-
stratified object. In addition to adjusting the sampling weights, it adds information
that allows the standard errors to be adjusted (Lumley 2010 ). In this example, we
use the artificial population that we introduced in Chap. 6 . Note that we selected the
sample using a SRS. The output provides the poststratified estimate of the target
variable.

> library(sampling)
> library(survey)
> n < - 100;
> N < - 1000
> set.seed(160964)
> framepop < - data.frame(id ¼ 1:N, xc ¼ runif(N), yc ¼ runif(N))
> yobs < - (exp((framepop $ xc-0.5)^2)+exp((framepop $ yc-0.5)^2))
> yobs < - 100-((yobs-min(yobs))/(max(yobs-
+ min(yobs))))*100+(rnorm(N)+5)*5
> q1obs < - sample(1:3,N,replace ¼ T)