Sample Size and Sample Allocation - Sampling Spatial Units for Agricultural Surveys - page 203

> num_units¼tapply(framepop$yobs,framepop$q1obs,length)
> stratsize ¼ pmax(ceiling(sampsize*(num_units/N)),
+ rep(5,length(num_units)))
> c(stratsize,sum(stratsize))
123
77 92 86 255

The sample sizes for all the stratified sampling allocations should be rounded,
which should work well for large sample sizes. However, we must be careful. In
some situations the rounded allocations may violate the cost constraint and/or could
be sub-optimal, particularly if we are dealing with a very large number of strata in a
multipurpose allocation problem (Khan et al. 2010 ).

8.3.2 Optimal Allocation

There are two different practical possibilities for allocating sampling units across
the strata. In the first, the optimal allocation minimizes the variance of the sample
estimator of a certain population parameter, for a sample of fixed size over several
strata. In the second, the optimal allocation is obtained by respecting a required
accuracy (i.e., the coefficient of variation is fixed) using the smallest sample
possible.
When the total sample size has been decided, the optimal strata sample sizes
(i.e., those that minimize the variance) are

N h S y , Uh
X

n h ¼ n

N h S y , Uh :

ð 8 : 12 Þ

H

h ¼1

Equation ( 8.12 ) is called Neyman allocation (Neyman 1934 ; Cochran 1977 ; Levy
and Lemeshow 2008 ). If the variances ( S y , Uh ) are specified correctly, Neyman
allocation provides an estimator of the total that has a smaller variance than
proportional allocation.
In optimal allocation, a stratum that is large, or has a large within-stratum
variance, has more sampling units than a smaller or more homogeneous stratum.
This type of allocation provides the most efficient estimates under stratified
sampling.
If the variances ( S y , Uh ) of the strata are more or less equal, proportional allocation
is most likely the best allocation for increasing the precision. If there are large
variations in the S y , Uh , Neyman allocation can result in smaller costs. It also works
well for sampling units that are very different in size, and is effective when some
strata are much more expensive to sample than others.