Download An Expert Sample Allocation Program

AN EXPERT SAMPLE ALLOCATION PROGRAM
James W. Mer&erson
INTRODUcrION
A univariate optimum allocation
where
formula
for
a
L
stratified sampling design with simple random
sampling within strata may produce Dh > Nb in
some strata.
CochraD (1977) gives a procedure
for determining the optimum allocation for this
situation
when
equal
costs
among
- number of strata
- popUlation size of stratum h
strata are
- standard deviation of data
in stratum h
assumed and a fixed desired precision level is
specified.
When costs among strata differ this
procedure is not applicable.
An
explicit
procedure for this case does
available in sampling textbooks.
- average data collection cost
in stratum h
not seem to be
L
N
When costs differ greatly among strat8$ and
especially when low costs are associated with
strata having high variances and high costs are
associated with strata having low variances. the
total sample size may exceed the sum of the
number of population units.
An example is
presented involving such a situation.
Also. a
general procedure for determining allocations
when some initial n h are greater than the
corresponding N is presented. The procedure is
applicable for aetermining optimum allocations
for a stratified sampling design. with simple
random sampling within strata. for a fixed
desired level of precision when costs among
strata are unequal.
The procedure has been
implemented
using
Pascal.
Input standard
deviations are computed using SAS.
-
~
N
h=1 h
D
- desired size of s
s
- standard error of the estimate
of the population mean
- mean of data in stratum h
Applying (2.1) n = 307. The allocations to each
stratum is computed using the formula:
(2.2) •
FXAMPLE
Given a desired coefficient of variation (CV)
equal to 0.02. consider determining the optimum
allocation using the following data:
ttl
h
y
100
~.!l~
25
3.54
20.0
= 17.
and n3
=
282.
,
___
1___1.!:1.L_~.Q,.!l_
The formula for computing the total sample
is as follows:
8. tt2
This allocation is impossible since n 3 ) N •
3
Also. n > N implies that attempts to follow,a
procedure similar to the procedure in Cochran
(1977) is fruitless.
Cochran's procedure is
outlined in Appendix A.
l--iiio---wo--"3-:33-1o.02
=
Since we have n3 > N3 • set ng = N~.
The formula for computing the varia~ce of the estimate
of the population mean is as follows:
size
772
REFERENCES
Clearly, the contribution to the standard error
of the estimate of the population mean from
stratum 3 will be zero. Strata 1 and 2 will
account for all of the sample variance-. Having
set ttg = N3 • let's compute a sub-total sample
size for strata 1 and 2. Since strata 1 and 2
will ac~ount for all of the variance. the quantity N2D2 in (2.1) remains the same. The formula for the sub-total sample size for strata 1
and 2 is:
Cochran. William (1977). Sampling
New York: John Wiley and Sons.
Teohniques.
Hansen. Morris. Hurwitz. William and Madow. William (1953), Sample Survey Methods and Theory.
New York: John Wiley and Sons.
Jessen. Raymond (1978). Statistical Survey Techniques. New York: 70hn Wiley and Sons.
Raj. Des (1968).
McGraw-Hill.
n
Applying (2.4) n': 43. The
allocation
to
Sampling
Theory.
New
York:
Sukhatme.
Pandurang,
Sukhatme.
Balkrishna.
Sukhatme. Shashikala and Asot. C. (1984). Sampling Theory of Surveys with Applications. Ames.
Iowa: Iowa State University Press.
each
stratum is computed using the formula:
"h
,
1/2
2
1/2
In ~sh/(ch)
1/1 ~ ~sh/(ch)
1
h=l
n
14 and n
n
1
,
8110e8 tion
29. The revised
2
APPENDIX A : Cochran's Procedure
Cochran's procedure for allocations requiring
more than 100 per cent sampling (equal costs
among strata) is outlined:
is
,
14, tt2 = 29 and n3 = 100.
1
1. I f
GENERAL PROCEDURE
When computing the allocation for
sampling
design
with
simple
> Nh
If n h
2.
Compute
set
a
,
"h
stratified
random
sampling
for
nh
= (n -
all
n
8
N s (C )1/2)( l
N s /(Ch)1/2)
h
d h h h ____!u.~h_____
_
where I is the index set {1.2 ••.•• L} and Q is a
set containing each h such that ~ > Nh
The
set d is the difference of sets I and Q.
3.
Compute
~=
,
1/2
In Nhsh/(ch)
1/1
had.
n , tt • tt3'
1
2
The
~ Nhsh/(c h )
h • d
optimum
1/2
allocation
for
1
is
then
~.
773
where
and
such that n h
> Nh •
Q
is
I
is
the
index
set
a set containing each h
The set d is the
difference
The optimum allocation is then
,
n l • n 2 • na'
2
N D2 + l NhS~
h • d
all
h • d
of sets I and Q.
l
h
n h = Nh
1 Nk)(Nhsh)/( 1 Nhs h ) ,
kaQ
h e d
{1.2 ••••• L}
,
(
,
set
2. Compute
within strata, one or more nh may initially be
greater than the corresponning Nh • When this
sit_uation occurs the following procedure may be
applied:
1.
> Nh
nh
•••• nL"