Living microbes are discrete, not homogeneously distributed in environmental media, and the form of the distribution of their counts in drinking water has not been well established. However, this count may "scale" or range over orders of magnitude over time, in which case data representing the tail of the distribution, and governing the mean, would be represented only in impractically long data records. In the absence of such data, knowledge of the general form of the full distribution could be used to estimate the true mean accounting for low-probability, high-consequence count events and provide a basis for a general environmental dose-response function. In this article, a new theoretical discrete growth distribution (DGD) is proposed for discrete counts in environmental media and other discrete growth systems. The term growth refers not to microbial growth but to a general abiotic first-order growth/decay of outcome sizes in many complex systems. The emergence and stability of the DGD in such systems, defined in simultaneous work, are also described. The DGD is then initially verified versus 12 of 12 simulated long-term drinking water and short-term treated and untreated water microbial count data sets. The alternative Poisson lognormal (PLN) distribution was rejected for 2 (17%) of the 12 data sets with 95% confidence and, like other competitive distributions, was not found stable (in simultaneous work). Sample averages are compared with means assessed from the fitted DGD, with varying results. Broader validation of the DGD for discrete counts arising as outcomes of mathematical growth systems is suggested.

A New Theoretical Discrete Growth Distribution with Verification for Microbial Counts in Water

Frequently reported degradation of urban waters indicates that discharges of urban stormwater may cause a variety of impacts in receiving waters [1]. Such impacts can be characterised with respect to their nature, time scales, spatial scales, and the types of receiving waters. The nature of the impact is usually classified as physical, chemical, microbiological or combined. Examples of physical impacts include increased flows (and reduced recharge of groundwater aquifers), erosion and sediment transport/deposition, temperature rise, and densimetric stratification. Chemical impacts contribute to changes in water quality through dissolved oxygen depletion, nutrient enrichment and eutrophication, and toxicity (both acute and chronic). Microbiological impacts may affect both recreational waters and shellfish harvesting areas. Finally, it should be recognised that these types of impacts usually occur in various combinations, which may be referred to as combined impacts and further described by ecological impacts and impairments of beneficial water uses.

Overview of Urban Stormwater Impacts on Receiving Waters

Water pollution risk is concerned with assessing the impact of existing and future hazards that are associated with the beneficial uses of water. The focus of this article is on risks to human health due to exposure to water pollution. Some important statistical issues involved in the management of risk are discussed based on two pollution types: persistent toxic substances and microbiological contamination.


Keywords:

human health;
ecosystem health;
toxic substances;
microbiological contamination;
pollution indicators;
environmental standards and regulations

Water Pollution Risk

The EU, governments and agencies throughout the world seek to regulate and protect the state of the environment by setting environmental standards to control the level of certain pollutants present in various media of concern. It has become evident in recent years that many such standards are set without due consideration of uncertainty and variation and many are based on poorly defined principles. Barnett and O’Hagan (1997) conducted a study of current standards, concluding that most standards can be classified into one of two unsatisfactory categories; ideal and realisable standards. They proposed a new conjoint and more statistically meaningful concept; the statistically verifiable ideal standard(SVIS). We consider the implications of this approach with the aim of applying the results to standards formulation for practical problems of pollution in air, water and soil, working in co-operation with relevant bodies. We will discuss the case of a simple pollutant distribution, developing SVISs from various standpoints within a hypothesis-testing framework. We consider a best linear unbiased quantile estimator(BLUQE) for use in the compliance assessment of a SVIS and describe an approximate significance testing procedure using simulated critical values. Finally, we develop a SVIS for a practical example involving water quality and illustrate the use of the BLUQE.

Setting Environmental Standards: A Statistical Approach

The negative binomial distribution is widely used to analyze count data. In environmentrics, one is frequently interested in summarizing the changes in the abundance of particular organisms in the environment either for protection or impact assessment. For example, monitoring the safety of a drinking water source is based on the estimation of the density of an indicator bacteria from the number of organisms found in several water samples. Another example is the estimation of the abundance of threatened species and the impact of habitat changes on their numbers in ecological conservation.

Negative Binomial Distribution ‐ Applications

Microbiological water quality guidelines have been established in most countries to protect the users from the risk of waterborne diseases. A guideline requires that a minimum number of water samples be collected during a period of time and analysed for an indicator organism. One or more summary statistics are computed and compared to limits specified in the guidelines. When at least one of the limits is exceeded, then certain actions must be taken. Since uncertainty is involved in the implementation of each of these requirements, the use of the guideline is intended to control and not to eliminate the risk from the use of impaired waters. The sources of uncertainties are discussed using two Canadian guidelines for microbiological recreational water quality. However, the methods used are applicable in general to guidelines' specification and use. Criteria are proposed for the selection of an indicator organism and the specification of its limits. The probabilities that quality of recreational water meets the federal guideline are estimated by simulation from the lognormal distribution. These probabilities reveal that imposing a single limit on the maximum eliminates the need for also imposing a limit on the geometric mean. The distributions of the maximum in samples of various sizes are derived by simulation and used to estimate the number of samples needed to detect a predetermined increase in the mean bacterial level. These distributions can be also used to determine the increase in the mean that a sample of a specific size can detect. Finally, the Ontario provincial guideline is compared to its federal counterpart. When the number of samples analysed is less than 15, the Ontario guideline is more conservative, but the reverse is true for larger sample sizes. Copyright © 1999 John Wiley & Sons, Ltd.

Guidelines for indicator bacteria in waters: uncertainties in applications

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Guidelines for indicator bacteria in waters: uncertainties in applications