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INTRODUCTION
In recent years, statistical methods have played a major role in environmental monitoring programs. With the development of a modern statistical approach to environmental regulatory statistics (Davis, 1994; Davis and McNichols, 1987; Gibbons, 1987a, b, 1994, 1996; Gilbert, 1987) there has been a major evolution in the way in which environmental impact decisions are made. This early work has focused largely on the earliest possible detection of a release, often termed environmental detection monitoring. As these new methods have become incorporated into state and federal regulation and guidance (U.S. EPA, 1987, 1988, 1989, 1992), the need for improved statistical approaches to related problems of assessment, compliance, and corrective-action monitoring has grown as well. Unfortunately, far less statistical work has been done in this area, and corresponding environmental impact decisions are still often based on a comparison of individual measurements to fixed standards, or at best, simple normal confidence bounds. Often, a facility or property is declared as environmentally impacted if a single measured concentration exceeds an environmental standard. First, we should be concerned about such a practice because we should be interested in comparison of the true concentration to the standard, not simply the measured concentration. Of course, without infinite sampling we can never really know the true concentration; however, statistical analysis provides a means of drawing inference to the true concentration distribution from a series of measured concentrations.Second, we should be concerned about such a practice because it treats all environmental problems as being equal. For example, exceeding an environmental standard in 1 of 5 samples is a very different problem than exceeding an environmental standard in 1 of 500 samples. Again, the statistical approach to this problem incorporates our uncertainty in the true concentration distribution rather than simply assuming that the measurements are made without error and represent truth in and of themselves. In fact, nothing could be further from the truth.
Since much of the threat of pollution comes from human beings and manufactured products, the constituents of concern are often anthropogenic and do not exist naturally in the environment. Here, the variability in the system is based in large part on the practice of the laboratory and the preparation and analysis of individual samples. In almost all cases, analytical measurements are accepted as true concentrations without regard to their uncertainty. Confidence bounds on measured concentrations are rarely reported, despite the availability of historical data that could routinely be used for this purpose. In the absence of such uncertainty estimates, laboratories most often rely on limits of detection and quantification to screen analytical measurements. The limit of detection, which we denote as LD following the pioneering work of Currie (1968), allows us to make the binary decision of "detected" with specified levels of confidence for errors of both the first (false positive) and second (false negative) kinds. The limit of quantification, LQ (Currie, 1968), is the concentration at which the true concentration can reliably be measured. In many ways, these two types of limits are simply points along a continuum that describes the relationship between true concentration and uncertainty. As we will see, the statistical modeling of such a relationship is complicated by the fact that uncertainty is rarely, if ever, constant, even for small intervals on this continuum. Of course, the role of this uncertainty must be incorporated in making environmental monitoring decisions, although in practice, it rarely is.
The purpose of this book is to describe the statistical theory that underlies the detection and quantification of environmental pollution in both the laboratory and the field. In the laboratory, we present the foundation of relating measured concentrations to true concentrations and the development of intervals of uncertainty for true concentrations given a new measured concentration. Related to this problem is the problem of estimating thresholds on this curve that define concentrations at which detection and quantification decisions can reliably be made. In this book we present a comprehensive review of this topic with directions for future research. In the field, we discuss how analytical measurements can be used in making environmental impact decisions and more broadly, how environmental data can be compared to regulatory standards, naturally occurring background concentrations, or both. Again, we present a comprehensive review of this problem and directions for future research.
Overview of the Book
In most chapters a general introduction to the problem is presented, followed by increasingly complex solutions. In some cases, statistical theory is presented that may not be accessible to all readers; however, it is included for completeness and the hope that this book may provide a foundation for further statistical research in this area. Despite complexity, for each solution or statistical approach to a particular problem, a relevant example is provided with computational details and/or tables that can be used for routine application of the statistical results. Attention is paid to statistical properties of alternative approaches, including false positive and false negative rates associated with each test and factors related to these error rates where possible. Recommendations are provided for specific problems based on characteristics such as number of monitoring wells, number of constituents, distributional form of measurements, and detection frequency. The reader may use the book to help craft an assessment, compliance, or corrective-action monitoring program for most environmental media that have been or potentially have been affected by one or more pollutants. Although discussed to some degree for completeness, the reader interested in routine environmental detection monitoring is referred to the previous book by Gibbons (1994), where this topic is dealt with in considerable detail. Similarly, the reader interested in statistical aspects of environmental sampling is referred to the excellent book by Gilbert (1987).
Part I contains 10 chapters in which conceptual and statistical issues of detection and quantification in the laboratory are discussed. Although much of the work and illustrations involve problems in the environmental sciences, the problems and solutions apply to all aspects of analytical chemistry and to calibration problems in other fields as well. For example, much of the work done in analysis of the contents of the foods that we eat are directly amenable to solution using the methods described in Part I. Chapter 2 begins with a discussion of the conceptual foundations underlying the chemical measurement process, calibration function, sensitivity, precision, and accuracy. In addition, in Chapter 2 we discuss the conceptual basis for detection and quantification decisions both within and across laboratories.
In Chapter 3 we review the statistical foundations that underlie the methods described in the book. They are divided into the areas of hypothesis testing and interval estimation. Under hypothesis testing, we describe issues of confidence and statistical power. Under interval estimation we discuss confidence limits, prediction limits, and tolerance limits.
Chapter 4 provides a detailed overview of calibration-based regression models that relate measured concentrations or instrument responses to the underlying true concentration. Here we present two general types of estimation procedures: ordinary least squares (OLS) and weighted least squares (WLS). OLS assumes that the measurements are independent and have equal uncertainty or variability regardless of concentration, although the latter assumption is rarely true in practice. WLS assumes that the measurements are independent; however, their uncertainty or variability is nonconstant and often is some function of the true concentration in the sample. For both OLS and WLS estimators, we provide details for the computation of confidence, prediction, and tolerance intervals for the calibration function itself (i. e., a confidence interval) and for individual deviations from the calibration function (i. e., prediction and tolerance intervals).
In Chapter 5 we introduce the concept of the detection limit for single-concentration based designs. The detection limit is used of make the binary decision as to whether or not an analyte is present in the sample. The various approaches to this problem differ primarily as to whether and how false positive and false negative error rates are specified and at what concentration, if any, the samples are "spiked." Although the single-concentration design is the least statistically rigorous in that it provides no way in which to estimate the variance function (i.e., the relationship between variability and true concentration), it is by far the most widely used approach to estimating the detection limit and the easiest to compute. We present it here as a review and as a way of fixing ideas for the presentation of calibration-based (i.e., more than one concentration) detection limit estimators, which are the primary theme of this part of the book.
In Chapter 6 we introduce the concept of the quantification limit for single-concentration designs. The quantification limit describes the point on the calibration curve at which the signal-to-noise ratio is sufficiently large to have reasonable confidence in the true concentration based on measured value( s). As one might expect, this conceptual definition can lead to a multitude of operational definitions. Again, the single-concentration design estimators are presented here as a means of fixing ideas and preparing the reader for the more statistically advanced calibration-based estimators.
In Chapters 7 and 8 we extend the concepts of detection and quantification limits to calibration-based designs in which multiple concentration points are used to model simultaneously the relationships between measured and true concentration and variability and true concentration. A variety of approaches are described, based largely on the statistical foundation provided in Chapters 3 and 4. As in all cases, the methods are illustrated using a number of relevant examples.
In Chapter 9 we extend the methods developed in Chapters 4, 7, and 8 to the important problem of determining the number of significant digits for an estimated concentration. In this chapter we also present new guidelines for reporting measurements and corresponding uncertainty estimates.
In Chapter 10 we present experimental design issues that underlie detection and quantification limit studies. In this chapter we consider problems of variance component estimation, between-and within-laboratory studies, selecting calibration standards, and various traditional experimental designs that can be used to provide representative conditions.
In Chapter 11 we extend the previous developments to the case of multiple laboratories. When the same data can be analyzed by multiple laboratories, both within- and between-laboratory variance components must be incorporated into the interval estimates for the calibration function. This is often the case when split samples are obtained and analyzed by different laboratories for regulatory or quality control purposes. In this chapter we introduce new work on generalized mixed-effects models (i.e., the laboratory is random and the concentration is fixed) and corresponding interval estimates.
Part II deals with the impact that use of detection and quantification limits have on traditional environmental statistics. As described in these chapters, the presence of nondetected values have a profound effect on the usual complete data case for testing hypotheses and constructing interval estimates.
In Chapter 12 we discuss how laboratory uncertainty estimates can be used to determine if the underlying true concentration for an individual measured concentration exceeds a regulatory standard. In Chapters 13 to 16 we discuss methods for dealing with the resulting data, which comprise a mixture of detected and nondetected concentrations. In Chapter 13 we review the literature on methods for dealing with censored data. In this chapter we consider imputation methods, linear estimators, regression methods, methods based on normal order statistics, maximum likelihood estimators, restricted maximum likelihood estimators, and delta distributions. In Chapter 14 we present methods for testing for distributional form, including cases in which the data are censored. In Chapter 15 we provide a general overview of testing for outliers in environmental data. In Chapter 16 we provide a general overview of nonparametric methods for testing trend that are all suitable for data that comprise a mixture of quantifiable and nonquantifiable data.
Chapter 17 is a broad overview of statistical methods for environmental detection monitoring, including problems of statistical prediction, multiple comparisons, treatment of nondetects, nonparametric alternatives, intrawell comparisons, applied to problems in groundwater, surface water, and air monitoring.
In Chapters 18 to 21 we present a unified treatment of statistical methods for analysis of data collected as part of compliance, assessment, and corrective-action programs. Although statistical methods for detecting an environmental impact (Chapter 17) have been well studied, considerably less attention has been paid to the characterization of the rate and extent of a release and the effects of remediation. In Chapter 18 we provide a general overview of the problem and a sketch of the general approaches that are available. In Chapter 19 we discuss comparisons to regulatory standards and in Chapter 20, statistical methods for comparison to background concentrations. The methods are then illustrated in Chapter 21 using a series of three case studies. Much of the material presented in these four chapters will be new to practitioners in this area. We consider methods for normal and lognormal distributions, as well as nonparametric alternatives. For the comparison to regulatory standards, we present normal, lognormal, and nonparametric confidence bounds for the mean, median, and other percentiles of the concentration distribution. As in all cases in this book, methods for the treatment of censored laboratory data are emphasized.
The book concludes with a review of currently available software (Chapter 22), a summary (Chapter 23), and an annotated bibliography. The bibliography covers most papers published in the area of detection and quantification and a brief discussion of each, as a well as a list of references used in the text.
