Sample Size and Sample Allocation - Sampling Spatial Units for Agricultural Surveys - page 205

N h S y , Uh =

p

k h

n h ¼ K k 0 ð Þ

:

ð 8 : 16 Þ

X

H

N h S y , Uh

p

k h

h ¼1

In the second case, fixing the threshold ( c ) of the coefficient of variation, the
optimal allocation in the strata is (S¨rndal et al. 1992 , Result 3.7.4, p. 105)

!

X

H

N h S y , Uh

p

p

N h S y , Uh = k h k h

h ¼1

n h ¼ : ð 8 : 17 Þ

þ X

H

N h S y , Uh c 2 t 2

h ¼1

Note that this solution does not ensure that n h N h . This intuitive rule is often not
respected in practical applications, particularly when we are dealing with highly
skewed populations. In such situations, the stratum will be forced to be censused;
that is, n h ¼ N h .
The following R code calculates the sample allocation in Eq. ( 8.17 ).

> ccv < - 0.015
> numvar1 < - sqrt(var_yobs*(num_units)^2)
> numvar1 < - numvar1*sum(numvar1)
> numvar2 < - var_yobs*(num_units)
> div < - ccv^2*sum(framepop$yobs)^2+sum(numvar2)
> stratsize < - pmax(ceiling(numvar1/div),rep(5,length(num_units)))
> stratsize < - pmin(stratsize,num_units)
> c(stratsize,sum(stratsize))
123
64 74 71 209

Note that this sample size is much lower than in the previous examples, because
stratified sampling is more efficient than SRS (which was used to estimate the
previous size). The sample has been proportionally or optimally allocated in each
stratum. Moreover, note that if all variances and costs are equal, proportional
allocation is the same as optimal allocation. If the variances within each stratum
are known and different, the optimal allocation gives a smaller variance for the
estimate of t than proportional allocation.
Furthermore, the optimal allocation diverges from the proportional allocation if
the standard deviations ( S y , Uh s) vary between strata; the proportional allocation
approximates the optimum unless the strata population variances for the survey
variable are significantly different.
Stratified sampling is more efficient than SRS if the strata means for the survey
variable are different. Optimal allocation is different from proportional allocation if