**Sample Size Requirements for Stratified Random Sampling of
Agricultural Run Off Pollutants in Pond Water with Cost Considerations
Using**

**a Bayesian Methodology**

**A.A. Bartolucci**

**Department of Biostatistics, University of Alabama at Birmingham,
Birmingham, Alabama 35294-0022 USA**

**S.J. Bae and K.P. Singh**

**Department of Biostatistics, School of Public Health, University of
Texas Health Science Center at Forth Worth, Forth Worth, Texas
76107-2699 USA**

**ABSTRACT**

Estimating average environmental pollution concentrations from
fertilization components and their variance is a fairly straight forward
task in stratified random sampling. A more challenging concept is the
introduction of the cost factor into this environmental model.
Traditional statistical techniques have incorporated costs from sampling
within a stratum as well as stratum weights to determine the stratum
size and overall required sample size. Information in the form of
informative prior distributions to determine a more coherent variance in
the system yield a more precise Bayesian approach to the sample size and
cost calculations. This approach results in a more efficient sampling
strategy in terms of cost when considering a pre specified margin of
error for the sampling mean as well as the more complicated situation of
correlation among the strata samples.

**K*eywords:*** *Stratified; random sampling; cost; Bayesian; margin of
error; optimum*

**1.0 INTRODUCTION**

The traditional statistical approaches to calculating overall and
stratum sample sizes in a stratified random sample are fairly straight
forward. The procedure is somewhat complicated with the incorporation of
cost as well as the possibility of correlation among the stratum
samples. Applications of such approaches employing several monitoring
strategies are well known as in Thornton et. al (1982), Nelson and Ward
(1981), Reckhow and Chapra ( 1983), and Gilbert ( 1987). Our focus here
is to consider a pond water environment in which the strata are
basically depth levels. Weighting of the strata as well as the overall
variance of the sample mean are the main components in our derived
statistics to determine sample size within the stratum. The three
situations considered are that of pre specified margin of error, pre
specified fixed cost and correlation among the strata samples. Cost
efficiency is seen for most ations with the introduction of Bayesian
methodology developed by Dayal and Dickey ( 1976), Bartolucci and Dickey
(1977), Birch and Bartolucci (1983), Baldi and Long (2001) and
Bartolucci et. al. (1998). The thrust of the Bayesian approach is
through the derivation of the posterior estimate of the variance derived
from coherent inference on a normal variance in the Behren=s Fisher
context of Dayal and Dickey (1976), Bartolucci et. al. (1998).
Comparisons of the traditional or classical and Bayesian methodologies
are presented using summary data from determining the phosphorous
concentration in a pond water sampling environment.

The motivation is to assure that we have a design that conforms to cost
effectiveness guidelines recommended by the National Academy of Sciences
(1977,2004) and Bartram and Balance (2001). These chosen designs
incorporating a cost analysis will either achieve a specified level of
effectiveness at minimal cost or a specified effectiveness at a
specified cost. The incorporation of the Bayesian analysis as modeling
the strata variance allows a further cost savings in the overall
approach. The approach can be applied to sampling contaminants from well
water or pond water with special attention to agricultural runoff as
seen in Atzeni. Casey and Skerman (2001). Also Gilbert et. al. (1975)
weighed in on the importance of this approach when cost considerations
demanded attention when sampling radioactive pollutants from desert
sites in Nevada. Our proposed technique can be applied to the sampling
plans of Ward, R.C., Loftis, J.C. and McBride, B.G. (1990) as well as
others. Thus historically there are many applications requiring the cost
considerations as well as can be refined by cost considerations when
sampling from the environment.

In section 2.0 below we derive the traditional set up of the sampling
providing the basic statistics such as the sampling mean, variance,
depth stratum size and weights as well as the overall population size.
In section 3.0 we incorporate into our formulation the methodology for
computing the optimum sample size under the assumptions of the pre
specified margin of sampling error (PMOE). We then introduce cost
consideration into the approach at a pre specified fixed cost per
stratum for independent stratum as well as correlated stratum.

In section 4.0 we introduce the Bayesian considerations in our
methodology, especially as applied to the stratum variance which impacts
on the overall final cost. In section 5.0 we apply the method to an
example when sampling phosphorous concentration in pond water at 5 depth
strata and demonstrate the conditions of cost reduction with the
Bayesian methodology.

> **2.0 TRADITIONAL SETUP**

Let N=total number of population units in the target population. N~h~ is
the number of population units within each of the h stratum,
h=1,\...\...L. Clearly N=![](media/image1.wmf). With reference to the
sample, n=total number of sampling units in the target sample. Likewise
as above, n= n~1~ + n~2~ + \...\...\...+ n~L~ = ![](media/image2.wmf).
We define the weight of the stratum, h, as W~h~=N~h~/N. The mean, μ, of
the population of N units is:

![](media/image3.wmf) (1)

where μ~h~ is the mean of the h stratum and is estimated by

![](media/image4.wmf) (2)

where x~hi~= ith observation in stratum, h. An unbiased estimate of u is

![](media/image5.wmf). (3)

Let N~h~/N = n~h~/n in all strata, then

![](media/image6.wmf). (4)

We define Var(m~h~ )= ![](media/image7.wmf) where σ^2^ is the variance
of the h stratum. We estimate the stratum variance by
![](media/image8.wmf) . It can be shown that for large N,

![](media/image9.wmf). (5)

It will be important to note the robustness of this sample mean variance
in the Bayesian context.

**3.0 COMPUTING THE OPTIMUM n**

An important aspect of stratified random sampling is to determine how
many samples are to be collected within a stratum. Gilbert (1987) has
proposed a method for doing so that will minimize the variance
s^2^(m~st~) in equation (5) above for a pre specified fixed cost per
stratum or that will minimize the value of s^2^(m~st~) under the
condition of a pre specified margin of error (PMOE). The PMOE is the
value d such that d=\|m~st~-μ\| or the minimal absolute distance we wish
to tolerate between the sample mean and population mean with some
acceptable error which we define below. We also relate these two
conditions in our development of computing the optimum n.

We give a brief overview of three methods to compute the optimum n.

i\) Pre Specified Margin of Error (PMOE)

Letting d=\|m~st~-μ\|, we denote d as the pre specified margin of error
as in Gilbert (1987). The value d is such that

P(\|m~st~-μ\|d)=α (6)

for small α. The optimum n (Cochran, 1977) is thus

![](media/image10.wmf) (7)

where for N,

![](media/image11.wmf) (8)

and z~1-α/2~ is the usual100(1-α/2) critical value of the standard
normal distribution. Thus the optimum n~h~ for the hth stratum is
![](media/image12.wmf).

ii\) Pre Specified Fixed Cost

We define the overall cost of the sampling as

![](media/image13.wmf) (9)

where c~h~ is the cost per population unit in the hth stratum and c~o~
is the fixed overhead cost. This is a standard cost representation. Thus
the optimum n can be derived as in Aczel (1999),

![](media/image14.wmf) (10)

As above the optimum n~h~ per stratum is

![](media/image15.wmf) (11)

One can examine equation (10) in terms of its sensitivity to changes in
the PMOE. Let W~h=~n~h~/n.

Then (10) can be rewritten as

![](media/image16.wmf) (12)

If we assume unequal PMOE, d~h~ , for sampling within stratum then we
can write n~h~=(Z~1-α/2~s~h~/d~h~). See Cochran, (1977) and Aczel,
(1999). Thus equation (12) can now be examined with respect to
sensitivity to changes in d~h~.

iii\) Correlation among Depth Stratum

Let ρ~c~ = average correlation among all possible lags in the depth
sampling environment. For example if L is the number of strata or depths
and ρ~l~=the correlation of the lth lag, then

> ![](media/image17.wmf). (13)

If n~h~ is the number to be sampled in each of the L strata or n~h~ =
stratum size, then

![](media/image18.wmf) (14)

**4.0 BAYESIAN CONSIDERATIONS**

Examining equations (7), (10), (12) and (14) we see that they all
involve the expression for the stratum variance, s^2^~h~. We reevaluated
these expressions adding a prior structure to the variance of Dayal and
Dickey (1977), Bartolucci et. al. (1998) and then estimating the
posterior expression for the variance, normal σ^2^. We assumed an
underlying normal distribution with both mean, μ, and variance, σ^2^
unknown. In this context we define the likelihood function for n
observations:

![](media/image19.wmf) (15)

for v=n-1, nm=x~1~+x~2~+\...\.....+x~n~, vs^2^=(x~1~-m)^2^ + (x~2~-m)^2^
+\...\...\...\...(x~n~-m)^2^ and ![](media/image20.wmf) denotes a
proportional relationship. Consider the t-density,

![](media/image21.wmf) (16)

where ![](media/image22.wmf)

and υ, s\>0.

The prior for μ is

![](media/image23.wmf) (17)

for υ~o~.

The prior for σ^2^ is

p(σ^2^) ![](media/image24.wmf)τg^2^/![](media/image25.wmf) , τ\>0, g\>0
(18)

Where![](media/image26.wmf) is chi square on τ degrees of freedom. Thus
considering expressions (15), (17), and (18) the posterior variance for
each stratum is

ε^2^=(υs~h~ ^2^ +τg^2^)/B (19)

where B=υ+τ.

Thus substituting![](media/image27.wmf) for![](media/image28.wmf) in
(7), (10), (12), and (14) yields the Bayesian estimates of n and n~h.~
Thus in the following section we apply the Bayesian analysis to these
expressions to demonstrate and determine the efficiency of these
expressions in terms of the sample size requirements and cost of
sampling.

**5.0 EXAMPLE**

We wish to estimate the average phosphorous concentration (μg/100 ml) in
pond water. The concentration of 100 ml aliquot from each 1 liter sample
will be measured. The statistics for a classical representation of the
data using the pre specified margin of error (PMOE) d=0.2are given in
Table 1. The PMOE=0.2 is a fairly reasonable choice in environmental
sampling (see Gilbert, 1987). There are 5 depth strata to the pond in
which N=total number of 100 ml water samples in the pond. N~h~ is the
number of aliqots in stratum h. Note that we have left N~h~ as a non
integer just for the sake of generalization as this could be a depth
measurement or volume or any convenient measure the sampler wishes to
use or is convenient or has some environmental application. The weights
are W~h~= N~h/~/N for each strata. The number samples from each strata,
mean and variance of each strata are given as well all derived from our
previous formulations above in section 3.0. We have assigned costs to
each strata. For the sake of simplicity and without loss of generality
we have reduced the costs to integer units. The cost for sampling
stratum 1 and 2 are each 1. The costs assigned to strata 3, 4, and 5 are
2, 2, and 3 respectively - the assumption being that costs increase as
the depth increases. Thus the overall cost of sampling is 74 units. For
example for the first strata we have n~1~ =10 or 10x1=10 as a cost for
the first strata and we have for the last strata n~5~=7 at a cost of 3
per sample in that stratum or 7x3=21 for the cost of sampling that
stratum. Thus doing likewise for the rest of the strata, we have a total
cost of 74. Using the PMOE approach in Table 2, setting d=0.2
demonstrates the Bayesian results using empirical prior sampling
information and incorporating that into the variance calculation
overall. See equation (19). One sees that for realistic prior
assignments of υ, τ and g in (19) and incorporating that variance into
the calculation for n~h~ in section 3.0 one realizes a reduction in
assigned number per strata overall as well as a cost reduction in Table
2. In Table 3 using pre specified overall cost (i.e. holding C constant
in (9) ) did not yield any savings using the classical (top row) vs. the
Bayesian approach (bottom row) this makes sense somewhat in that the
cost is already fixed. However, we did examine these results using (12)
in which we varied the PMOE , d~h~ , to determine the effect on cost
using sensitivity changes and the classical and Bayesian results
remained fairly equal (results not shown here). Table 4 summarizes the
data results introducing correlation among the strata as per (14). The
average correlation is in the first column. One can see that as you
increase the average correlation , (13), then the required number
sampled within each strata will increase, but at a slower rate in the
Bayesian context.

6.  **Discussion**

Overall it appears that: Compared to the classical sampling analysis for
the pre specified margin of error approach as well as the correlational
approach, the Bayesian analysis resulted in a reduction in required
samples thus lowering the cost, especially when realistic (empirical)
prior hyperparameters are utilized. Also there was no serious impact on
the posterior standard error of the estimates of the mean concentration.
However, there were no real differences between the classical and
Bayesian approaches in the pre specified fixed cost analysis. Given the
current computational tools the Bayesian calculations proved to be
fairly straight forward. Also given the current availability of
databases, future Bayesian approaches to environmental sampling should
be given serious consideration especially where costs are concerned.

The importance of incorporating the correct elements into an
environmental study design has been emphasixed by many authors. For
example Smith (1984) discusses the efficiency of the design. In our case
efficiency not only means precision in terms of the pre specified margin
of error, but also on the cost considerations. Provost (1984) has also
touched upon several of the elements discussed in this paper. These
papers plus others examine the consequences of parameter estimation in
terms of efficiency as one varies both the type of design and the size
of the sampling effort. The stratified random sampling scheme discussed
above is a useful and flexible design for estimating environmental
concentrations, inventories and cost. They make use of prior information
in the classical statistical sense of dividing the population into
subgroups or strata that are basically internally homogeneous. We have
extended that prior knowledge to the Bayesian application of making use
of the distribution of the prior variation within the strata to
establish a more efficient design in terms of number sampled within the
strata as well as cost efficiency. See Reckhow and Chapra (1983) for a
further discussion of empirical modeling and data analysis in a
stratified setting. Thus, have extended the work of several authors by
using the Bayesian methodology to ensure not only an efficient design in
the sampling sense, but in the cost arena as well. A possible
opportunity for extension of this methodology is to consider multi stage
sampling designs and the consequences of incorporating prior information
into the variation components of primary as well as secondary units in
the two stage setting. For designs with more complicated staging an
extended multivariate model of the variation within the population can
be considered.

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**Table 1.** Data for stratified random sampling to estimate samples per
strata (PMOE)

Classical Approach (υ =1, τ =0, g =1) s^2^(m~st~) = 0.0140, Cost=74

  --------- ---------- ----------- ------------- -------------- -----------
  Strata    N~h~       W~h~        n~h~          m~h~           s^2^~h~

  1         4.25       0.266       10            1.67           0.4376

  2         3.96       0.248       9             2.83           0.4228

  3         3.23       0.202       8             3.59           0.5339

  4         2.85       0.178       9             4.23           0.7222

  5         1.70       0.106       7             5.31           1.3920

  Total     15.99      1.000       43            \-             \-
  --------- ---------- ----------- ------------- -------------- -----------

**Table 2.** Bayesian Results (PMOE)

  ----------- ------ ------- ------- ------- ------- ------- ------------- -------
  (υ,τ,g)     n~1~   n~2~    n~3~    n~4~    n~5~    Total   s^2^(m~st~)   Cost

  35,1,0.5    9      9       8       8       7       41      0.0140        71

  35,2,0.5    9      8       8       8       7       40      0.0141        70

  20,1,1.0    9      8       8       8       6       39      0.0138        67

  40,35,0.2   5      5       4       4       4       22      0.0143        38

  40,35,0.5   7      7       6       6       4       30      0.0195        42
  ----------- ------ ------- ------- ------- ------- ------- ------------- -------

**Table 3.** Pre specified fixed cost (Bayesian results in bottom row)

  -------- ------- ------ ------- ------- ------- ------- ------- ------- -------
  C-c~0~   υ       τ      g       n       n~1~    n~2~    n~3~    n~4~    n~5~

  50       \-      \-     \-      31      7       7       6       6       5+

  50       40      35     0.12    31      7       7       6       6       5
  -------- ------- ------ ------- ------- ------- ------- ------- ------- -------

**Table 4.** Example Using the Correlation Structure, ρ~c~.

  -------------- -------------- -------------- -------------- --------------
  prior (υ,τ,g)  Classical      (35,1,0.5)     (20,1,0.1)     (40,35,.2)

  ρ~c~           n~h~ Cost      n~h~ Cost      n~h~ Cost      n~h~ Cost

  0.05           10 90          10 90          10 90          05 45

  0.10           12 108         12 108         11 108         06 48

  0.15           14 126         13 117         13 117         07 63

  0.25           17 153         16 144         16 144         09 81

  0.35           21 189         20 180         19 171         11 99

  0.45           24 216         23 207         22 176         12 108

  0.55           28 252         26 234         25 225         14 126
  -------------- -------------- -------------- -------------- --------------