Sample Size and Sample Allocation - Sampling Spatial Units for Agricultural Surveys - page 200

q

E t HT t

p

Var t H ðÞ

2

ð Þ

CV t H ðÞ ¼

E t H ðÞ ¼

; ð 8 : 8 Þ

t

is the variance of the HT unbiased estimator of

N 2 S y , U 2
n

where Var t H ðÞ ¼

n
N

1

the population total t .
Then, using the estimator S y , it is possible to define a maximum limit for the
coefficient of variation, CV t H ðÞ ¼

p

p

NSy

n
N

1

c , where c is a fixed precision level.

t

Solving for n , we have

N 2 S y
c 2 t 2

n

þ NS y :

ð 8 : 9 Þ

In the following example of computing the sample size, we have used the same
artificial data set that was introduced in Chap. 6 (when describing basic common
sampling designs). In this case, we have included some additional survey variables
to the data set, which will be useful for the applications in this chapter. For an SRS
design, we can use the following R code.

> N < - 1000
> set.seed(160964)
> V1 < - c(2,5,8)
> V2 < - c(30,20,10)
> V3 < - c(30,50,70)
> framepop < - data.frame(id ¼ 1:N, xc ¼ runif(N), yc ¼ runif(N))
> q1obs < - sample(1:3,N,replace ¼ T)
> 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)*V1[q1obs]
> yobs2 < - 3*framepop$xc+2*framepop$yc+(rnorm(N)+5)*V2[q1obs]
> yobs3 < - 30*framepop$xc+20*framepop$yc+(rnorm(N)+5)*V3[q1obs]
> framepop < - cbind(framepop,yobs,yobs2,yobs3,q1obs)

If we set the maximum limit of the coefficient variation (ccv) to 0.015, we
obtain the following sample size. Note that the sample size is always rounded up to
an integer, and we apply a lower bound (typically fixed at n ¼5). This operational
choice is motivated by the need to prevent problems arising from nonresponses,
while integer sample size values are obviously required for practical purposes.

> ccv < - 0.015
> sampsize < - (nrow(framepop)^2*var(framepop[,4]))/
+ (ccv^2*sum(framepop[,4])^2+nrow(framepop)*var(framepop[,4]))