Sample Size and Sample Allocation - Sampling Spatial Units for Agricultural Surveys - page 209

set g can be chosen. In practice, a good starting value is the most expensive
univariate allocation.
2. For s 2, we determine an index u such that

t

ðÞ

, g .If a t u χ α

ðÞ

ð a u a v Þ χ α 0, v ¼ 1, ... 1, the algorithm stops,

ðÞ 2 1 must be calculated such that

otherwise t ðÞ 0 ;

h i G χ h i , 8t 2

ðÞ
ʴ vz þ

ðÞ ðÞ ðÞ

G χ

t 1 t α t ʴ vz þ 1 t ð Þ α ð : 0 ; ð 8 : 24 Þ

ʱ

ð sþ 1 Þ

¼ t ðÞ ʴ vz þ 1 t ðÞ

ð v .

3. Let ʱ

v

< ʵ

ð sþ 1 Þ ðÞ

v

4. If ʱ ʱ , v ¼ 1, ... , g , the algorithm stops, where ʵ is a

v

predetermined convergence parameter.

The solution obtained in Step 1 implies that g is very small, so the remaining
constraints ( v ¼2,3,

, g ) are not satisfied. In the next steps ( s 2), the sample
cardinality is increased ( v ¼2,3,

...

... , g ), and the objective function is such that

, until all constraints are satisfied. Bethel ( 1989 ) proved that
this algorithm converges. Therefore, the optimal values,

ðÞ sð Þ

G χ G χ

ˇ h and ʱ v , can be deter-

G χ ð .
Unfortunately, the Bethel algorithm is very computationally complex. It
becomes less practical if there are many strata and variables. As a consequence,
the Chromy algorithm is often preferred. The Chromy algorithm empirically seems
faster and more accurate than the Bethel algorithm. The only drawback is that it has
not yet been proven that the algorithm effectively converges.
The steps of the Chromy algorithms are as follows. First,

mined in such a way that 0 G χ

ðÞ

χ

ðÞ is calculated

ðÞ
v

according to Eq. ( 8.23 ) using 1/ g as starting value for 1 v g . If the

solution satisfies all the constraints of the problem, then the algorithm stops.
Otherwise, χ

ʱ , 8v ;

( s ) values

ðÞ is calculated using the updated

α

2

ð s 1 Þ

a t v χ α

ð s 1 Þ

ʱ

ðÞ
v

v

ʱ ¼

i 2 ,1 v g

; ð 8 : 25 Þ

h

X

g
v ¼1 ʱ

ð s 1 Þ

a t v χ α

sð Þ

v

is evaluated using Eq. ( 8.23 ) with

sð Þ .
The bethel package can be used to compute the multivariate allocation with
R. The reference manual can be downloaded at http://cran.r-project.org/web/pack
ages/bethel/bethel.pdf . The following is a simple code for allocating the sample.

ð s 1 Þ

ʱ ∗ ¼ ʱ

where χ α

> library(bethel)
> b1 < -as.data.frame(cbind(var_yobs¼tapply(framepop$yobs,
+ framepop$q1obs,var),

1 Bethel ( 1989 , p. 52) suggests how to quickly calculate this parameter.