# Presentation: 540015

## 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. Bae and K.P. SinghDepartment of Biostatistics, School of Public Health, University of  North Texas Health Science Center at Forth Worth, Forth Worth, Texas  76107-2699  USA

**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. Bae and K.P. Singh****Department of Biostatistics, School of Public Health, University of  North Texas Health Science Center at Forth Worth, Forth Worth, Texas  76107-2699  USA**

## GOAL:

**USING A BAYESIAN APPROACH WE WISH TO DETERMINE THE OPTIMUM SAMPLE SIZE , n,  AND SAMPLE SIZE,  n****h**** , FOR SAMPLING  WITHIN STRATUM, h, WHERE h=1,2,......L  AND n=n****1****+n****2****+........+n****L**** .**
**THE STRATA ARE BASICALLY DEPTH LEVELS IN A POND.  SAMPLING IS TO DETERMINE THE AMOUNT OF POLLUTION IN THE POND .                                                                               **

## Three Approaches

**Pre Specified Margin of Error (PMOE)**
**Pre Specified Fixed Cost**
**Correlation Structure Among the Strata**

## 	TRADITIONAL SETUP

**	****TRADITIONAL SETUP**
**N=Total number of population units in the target population**.
**For  L  strata,  h=1,........L ,**

**n** **= total number of sampling units in the target sample.**
**n = n****1**** +n****2**** +......n****L**** =   **

- .

## Weight of stratum h,

**Weight of stratum h,**
**W****h ****=N****h ****/ N.**
**The mean, μ, of the population of n units:**
**    **

**Estimate** **u****h**** by :**
**    **

**where x****hi** **=ith observation in stratum h.**

## An unbiased estimator of μ is :

** **

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

## Variance:

**Variance:**
**	****Var(m****st**)** = **

**Estimate the stratum variance**, **σ****2****h**** , by**

**It can be shown that for large N,**

## Optimum n: Using Prespecified Margin of Error

**	**
**PMOE**
**Let d= pre specified margin of error, i.e.**
**	**
**d=|m****st**** -μ|**
**that can be tolerated and a small probability, α,    **
**of exceeding that error.  i.e. **
**	****P( |m****st**** -μ|********d) = α.**
**Then by Cochran an optimum n is****:**

**	**

**For N********,**
**	**

**Thus the optimal n****h**** for each h is:**

- .
**For our example we let d=0.2 and α =0.10.****	**

**For our example we let d=0.2 and α =0.10.****	**

## Optimum n:   Using Prespecified Fixed Cost

**	****                          **

** **
**where** **c****h**** is the cost per population unit in the hth stratum and c****0**** is the fixed overhead cost.**
**Thus the optimum n is:**

## As above,  the optimum nh  per stratum is:

|  |
| --- |

** **
**Our examples will reflect both conditions of prespecified margin of error and prespecified fixed cost.**

## Correlation Among the Depth Strata

**Let ρ****c**** = the average correlation among the depth strata, i.e. average of all possible pairwise correlations. **
**Let n****s****  = number of strata.  ns=L.**
**Let n****h**** = the number to be sampled in each of the L strata or n****h**** =stratum size.  Thus:****	****	****	****	****	**

## 	Bayesian Considerations

**Derivation of the posterior variance using the Bayesian approach to the solution of the Behren’s Fisher problem for inference on mean (μ) and variance (σ)****  of the normal distribution when both paramters are unknown.**
**Likelihood function for n observations:**
** **

**υ=n-1,**
**nm=x****1**** + x****2**** +......x****n**** , υs****2 ****= (x****1****-m)****2**** +(x****2****-m)****2****+.....+**
** ****(x****n****-m)****2**  .

## Consider the t-density:

**Consider the t-density:**
**φ(x;s****2****) = s****-1****[υ****1/2****Beta(υ/2,1/2)]****-1****(1+υ****-1****(x/s)****2****)****-(υ+1)/2**
**where **

**The prior for the mean, μ, is: **

-  **normal for υ****o************.**

## Prior:

**p(********2****) ******** ********g****2****/ ************2**
**  **

** **
**Let  B=υ+τ.**
**The posterior variance is:**
** **
**ε****2**** = (υs****2****+τg****2****)/B.**
**Thus substitute ε****2****h**** for s****2****h**** in the above computations of n and n****h****.**

## Example:

**Example:**
**Estimate the average phosphorous concentration**
**(μg/100ml) in Pond water.**
**The phosphorous concentration of a 100-ml aliquot from each 1-Liter  sample will be measured.**
**N=total number of 100-ml water samples in the pond.**
**N****h**** =number of aliquots in stratum h.**
**There are five strata of depth levels, h=1,2,....5**

## 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                     ****	****	****(For cost per strata please see next slide)**

| 	Strata | 	Nh | 	Wh | 	nh | 	mh | 	s2h |
| --- | --- | --- | --- | --- | --- |
| 	1    | 	4.25M3 | 	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 | 	- | 	- |

## The unit cost to sample each depth level is:

| Level (h) | 	1 | 	2 | 	3 | 	4 | 	5 |
| --- | --- | --- | --- | --- | --- |
| Cost | 	1 | 	1 | 	2 | 	2 | 	3 |

**The assumption being that the cost is higher at greater depths.**

## Table 2.  Bayesian Results (PMOE)

| 	(υ,τ,g) | 	n1 | 	n2 | 	n3 | 	n4 | 	n5 | 	Total | 	s2(mst) | 	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-c0 | 	υ | 	τ | 	g | 	n | 	n1 | 	n2 | 	n3 | 	n4 | 	n5 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 	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,  ρc.

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

## Slide 22

## Slide 23

## Slide 24

## 		     Conclusions:

**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:**
**1.  Reduction in required number of samples thus    lowering the cost , especially when realistic (empirical) prior hyperparameters are utilized.**
**2. No serious adverse impact on standard error of the estimates of the mean concentration.**

## -  There were no real differences between classical  and  Bayesian approaches in the pre specified fixed cost analysis

**-  Given current computational tools the Bayesian calculations proved to be fairly straight forward.**
**- Given the current availability of databases, future Bayesian approaches to environmental sampling should be given serious consideration.**