CHAPTER 1
BACKGROUND

1.1 INTRODUCTION

For some purposes, qualitative detection of a substance of interest may be sufficient; for example, is there melamine adulterant in milk [1] or 210Po in an ex-spy [2]? In many cases, quantitation of the substance of interest, generally referred to as the analyte, is either desired or required. Three simple examples are as follows:

• What is the total organic content (TOC) in a drinking water specimen?
• What is the Cr3+ number density in a ruby laser rod?
• What is the pinene concentration in an air specimen collected in a pine forest?

In each of these cases, what matters is quantitative, since it is already known that all drinking water contains some organic content, every ruby has (and gets its color from) its Cr3+ content, and pinene contributes to the fragrance of a pine forest. What matters are the specific numerical concentrations, quantities, or amounts, that is, the quantitative analyte contents. In the drinking water example, unacceptable TOC levels should, politics and costs aside, trigger subsequent decisions and corrective actions.

In science fiction, it is common for instruments to be capable of scanning entire alien planets, and perfectly inventorying everything in them, perhaps in preparation for errorlessly teleporting up everything valuable. The real world is different: the laws of nature are always obeyed and experimental measurements are afflicted with measurement uncertainties. The immediate consequence of the latter is that only estimates of underlying true values may be obtained and these estimates, except by generally unknowable coincidence, do not equal the true values. These limits may be reduced further only if the relevant measurement uncertainties are reduced.

The elementary theory of detection limits in chemical measurement systems evolved over a century, through the dedicated efforts of many individuals. Based on their collective work, the next several chapters are devoted to the methodic and rigorous development of the concepts of decision level and limit of detection, with emphasis placed on understanding what they are and the purposes they serve. In Chapter 19, the concept of limit of quantitation is finally introduced, thereby completing the classic detection triptych.

1.2 A SHORT LIST OF DETECTION LIMIT REFERENCES

The refereed scientific literature contains hundreds of publications dealing with limits of detection. These attest to the fact that there was not a consensus understanding of chemical analysis detection power for many years. Although the large majority of early papers dealing with detection limits have few further insights to yield, those wishing to judge this for themselves have never had it easier, thanks to Internet availability of many publications and translation software for use with papers in some otherwise inaccessible languages. Accordingly, a lengthy list of early publications will not be presented, but references are provided to a few papers that contain such lists.

The review paper by Belter et al. [3] focuses on an historical overview of analytical detection and quantification capabilities and cites work back to 1911. Currie's 1987 book chapter [4] is also a valuable source of early references, as is his classic 1968 publication [5] and his paper in 1999 [6]. Additional papers containing especially relevant references include those of Gabriels [7], Kaiser [8-10], Boumans [11], Linnet and Kondratovich [12], Eksperiandova et al. [13], Mocak et al. [14], EPA document EPA-821-B-04-005 [15], and the Eurachem/CITAC Quantifying Uncertainty in Analytical Measurement (QUAM) Guide CG 4 (3rd Ed.) [16]. As well, there are maintained websites where useful lists of relevant publications may be found [17]. Additional references are provided throughout this text and in the Bibliography.

1.3 AN EXTREMELY BRIEF HISTORY OF LIMITS OF DETECTION

In 1947, Kaiser published what might be considered the first paper to deal explicitly with detection limit concepts as they apply to chemical analysis methods [18]. Others followed, including Altshuler and Pasternak [19], but it was the landmark 1968 publication by Currie [5] that marked the true beginning of the modern era of analytical chemistry detection limit theory. Subsequently, Currie tirelessly advocated for the basic precepts articulated in his heavily cited paper. His detection limit schema was based on classical Neyman-Pearson hypothesis testing principles, using standard frequentist statistical methodology. As a consequence, both false positives and false negatives must be taken into consideration. Neither Currie nor Kaiser was the first to recognize the value of considering both types of error: the prior development of receiver operating characteristics, briefly discussed in Chapter 6, clearly proves this. But Currie was the first highly regarded analytical chemist to clearly identify and discuss the issue, and bring it to the attention of practicing analysts.

Implicit in Currie's paper was his belief that a chemical measurement system possessed an underlying true limit of detection, temporarily denoted as LD, and that this was the fundamental figure of merit that needed to be estimated with minimal, or even negligible, uncertainty. As shown in Chapter 15, it is easy to construct a simple experimental chemical measurement system that possesses an LD, though this does not prove that every such system must possess an LD. It does, however, definitively rule out general nonexistence of an LD [20].

1.4 AN OBSTRUCTION

It might have been expected that Currie, or one of his contemporaries, would have rather quickly arrived at the results to be presented in subsequent chapters, for example, Chapters 7-14. Unfortunately, as every scientist knows only too well, scientific progress is far from a clean, linear progression. It actually evolves by creeps and jerks, many mistakes are made, and there is often no obvious way forward. Worse still, correcting mistakes may be a lengthy process even when the facts are incontrovertible. As it happened, a major problem arose only 2 years after Currie's paper: Hubaux and Vos [21] published an influential paper that effectively sidetracked Currie's program.

Hubaux and Vos' method obtained detection limit estimates by employing standard calibration curves, processed via ordinary least squares, with the customary prediction limit hyperbolas. The statistical methodology they employed was both familiar and entirely conventional. Not surprisingly, this led to a long period where very little progress was made because, with the notable exception of Currie, the experts at the time thought the matter was largely settled. A perfect example was provided in 1978 by Boumans, who confidently declared, as the lead sentence in his detection limit tutorial [22]: "Are there any problems left to be solved in defining, determining and interpreting detection limits?" His immediate answer was "Fundamentally most problems have been adequately discussed in the literature." It is now known that his answer was incorrect and that the Hubaux and Vos method, discussed in Appendix E, was an inadvertent obstruction. But this was not at all obvious at the time and there appeared to be no reason to doubt Boumans' expert opinion.

1.5 AN EVEN BIGGER OBSTRUCTION

Early in 1988, Currie published a lengthy book chapter [4] in which he came close to solving the problem of how to instantiate his 1968 detection limits schema. Tellingly, the Hubaux and Vos paper and method, which had absolutely no need for either true underlying detection limits or explicit hypothesis testing, was neither mentioned in Currie's book chapter nor listed in the references at its end. However, a more important event preceded Currie's book chapter by a few months: Clayton et al. published their highly influential detection limits paper [23]. Their work was based on the "statistical power" of the t test and used critical values of the noncentrality parameter, d, of the noncentral t distribution. These critical values are specific numerical values of the d parameter, and, for brevity below, they are generically denoted by dcritical.

Currie adopted the dcritical method and promoted it as the underlying basis of the analytical detection limits methodology currently sanctioned by various standards organizations, for example, the International Standards Organization (ISO) and the International Union of Pure and Applied Chemistry (IUPAC). Currie apparently felt that the dcritical method was the key to correctly bringing to fruition his 1968 detection limits schema, but an insurmountable problem eventually surfaced: in 2008, the dcritical method was proven to be irrelevant to detection limits [24, 25].

This quite unexpected finding is discussed in detail in Chapter 17, which, among other things, shows how to perform simple Mathematica® simulations to verify the result. Unlike German standard DIN 32645 [26], the ISO and IUPAC official detection limit protocols are directly based upon the dcritical method, thereby rendering them irrelevant. Accordingly, it is the primary purpose of this book to demonstrate the rigorous theory upon which Currie's detection limit schema is correctly founded and to demonstrate, via both properly designed experiments and comprehensive computer simulations, that it works as...