Sample Size and Sample Allocation - Sampling Spatial Units for Agricultural Surveys - page 206

the strata population variances for the survey variable are significantly different,
and it is more efficient. Finally, the cost-optimal allocation in Eq. ( 8.17 ) is more
efficient in terms of survey costs than the optimal allocation in terms of sample
sizes, if the survey costs for the selected sample units vary noticeably from stratum
to stratum.
Note that if we let n h ¼ w h n for chosen w h s, a general formula for n is

X

H

N h S y , Uh
w h

h ¼1

n ¼ ; ð 8 : 18 Þ

þ X

H

N h S y , Uh

c 2 t 2

h ¼ 1

where c is the usual threshold for the coefficient of variation of t HT , STR .

8.4 The Multipurpose Allocation Problem

Generally, surveys on spatial units are designed to obtain estimates for many
variables, each with its own fixed sample error (Cochran 1977 ; Kish 1988 ). These
characteristics may be in conflict. In fact, when a stratified sampling design is used,
an allocation that is optimal for one variable might not be optimal for others. This
problem is certainly not new in survey methodology (Folks and Antle 1965 ). It can
be solved using a multivariate procedure, which searches for a compromised
sample allocation. In other words, it is in some sense optimal for all the variables
of interest.
Khan et al. ( 1997 ) discussed the problem of finding the compromised allocation
by minimizing the total relative increase in the variances, as compared with the
optimal allocation, when the costs for measuring several variables are fixed in
advance. In a more recent paper (Khan et al. 2010 ), they also suggested a method
for constraining the optimum to an integer solution using dynamic programming
techniques.
To implement the procedure in stratified sampling, classic Neyman univariate
formulas have been extended to the multivariate case. The importance of this
extension was recognized by Neyman in his fundamental paper (Neyman 1934 ).
The best solution to multivariate sampling allocation for spatial frame surveys
was proposed by Bethel ( 1989 ). The advantages of this method outweigh the
drawbacks of the convex programming approach. The algorithm is as follows.
Consider a population U of size N , divided into H sub-populations (strata)
according to some code. Let { U 1 , U 2 ,

, U H } be the subsets of U that are
the partition of the population induced by the H strata. Let N 1 , N 2 ,

... , U h , ...

... , N h , ... , N H be

the cardinality of each subset, so that N ¼ X

H

N h .

h ¼1