Sample Size and Sample Allocation - Sampling Spatial Units for Agricultural Surveys - page 202

survey results for the whole population. In this case, the stratification is essentially
used to improve the sampling efficiency of the results.

8.3.1 Proportional Allocation

Proportional allocation is realized when the number of sampled spatial units in each
stratum n h is proportional to the size of the stratum. The sampling fraction n h / N h is
the same for each stratum, which also implies that the overall sampling fraction n/N
is the fraction taken from each stratum. In other words n h ¼ N h ( n / N ).
Using proportional allocation, the estimated population total t HT , STR reduces to
the form

X X k2s h y k

n

H

h ¼1

t P HT , STR ¼ N

; ð 8 : 10 Þ

and the variance of the estimator reduces to

X

¼

H

N n
n

Var t P HT , STR N h S y , Uh : ð 8 : 11 Þ

h ¼1

where S y , Uh ¼ X U

2

ð y k y h Þ

. Using proportional allocation, each unit in the

N h 1

sample represents the same number of units in the population, and the sample is
denoted as self-weighting. Nonresponse, particularly when its rate differs from
stratum to stratum, can have dramatic effects on the validity of the estimate.
Proportional allocation ensures an equal share of the sample in all strata, but can
produce less efficient estimates than generally expected.
Since the sampling fraction is constant equal to n / N in each stratum, the first-order
inclusion probability of any population element is also constant. Therefore, the
scheme provides an equal probability sampling design, which appears equivalent
to that of SRS (the second-order inclusion probabilities are actually quite different).
In general, the variance of the estimator t P HT , STR , even if not fixed and not under
control, will be smaller than the variance of t HT , SRS . Thus, it can represent an upper
bound for the precision of the estimates.
In large surveys where a lot of information is collected for each unit of the
sample, the proportional allocation is often used because of its simplicity, even if it
is not the optimal design in terms of the precision of estimates.
Proportional allocation can be simply obtained without the use of any R package,
using the following commands.