A Bayesian method for computing sample size and cost requirements for stratified random sampling of pond water

Abstract

Estimating average environmental pollution concentrations and its 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 stratum samples.

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 stratum samples. Cost efficiency is seen for most situations 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 Behrens 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, National Academy of Sciences, 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 et al. (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 et al. (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 below we derive the traditional setup 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 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 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 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.