Detection limit hypothesis testing

Significance tests, however, also are subject to type 2 errors in which the null hypothesis is falsely retained. Consider, for example, the situation shown in Figure 4.12b, where S is exactly equal to (Sa)dl. In this case the probability of a type 2 error is 50% since half of the signals arising from the sample s population fall below the detection limit. Thus, there is only a 50 50 probability that an analyte at the lUPAC detection limit will be detected. As defined, the lUPAC definition for the detection limit only indicates the smallest signal for which we can say, at a significance level of a, that an analyte is present in the sample. Failing to detect the analyte, however, does not imply that it is not present. 

The beauty of this completely random approach to the analyte detection limit is the direct applicability of the statistical hypothesis testing formalism. Also, long-term trends in calibration slope or backgrounds have little influence. One important assumption is made that the form of the calibration curve [Equation 2c] is fixed. Also, a subtle change has occurred, the operation is no longer linear, with A in the denominator. Thus, the distribution of x is only asymptotically normal, as the relative standard deviation of becomes smaller. 

Space remains for only a brief glance at detection in higher dimensions. The basic concept of hypothesis testing and the central significance of measurement errors and certain model assumptions, however, can be carried over directly from the lower dimensional discussions. In the following text we first examine the nature of dimensionality (and its reduction to a scalar for detection decisions), and then address the critical issue of detection limit validation in complex measurement situations. 

Fig. 1. Hypothesis Testing and Detection Limits. The upper part of the figure Indicates the null [H ] and alternative [H ] hypotheses, with the corresponding decisions [D , D ] at the left. Two kinds of erroneous decisions may be made false positives [probability a] and false negatives [probability jS]. (S represents a signal level C, a decision point or "critical" level.) The lower section contrasts a number of "real world" H s and H. s where adequate detection limits for the have clear, practical consequences.

Hypothesis testing is applicable to all of the above factors. Detection decisions may be made, for example, using the critical level of Student s-t to test for bias, or the critical level of to test an assumed spectral shape or calibration model or error model. For a given measurement design and assumption test procedure, one can estimate the corresponding detection limit for the alternative hypothesis, e.g., the minimum detectable bias. As with analyte detection, the ability to detect erroneous assumptions rests heavily on the design of the experiment and the study of optimal designs is a field unto Itself. 

Note that the critical level of the appropriate test statistic c generally be used as a normalized alternative to Xp, S(-, etc. The "detection limit" for a test statistic, however, is meaningless, as Xj, Sj, etc. refer to the true underlying quantity. A corollary Is that the term "detection limit" Is also without meaning In the absence of an alternative hypothesis. (This Is perhaps an obvious philosophical matter, but In principle, the null hypothesis cannot be rejected, except by chance [a-error], if no alternative exists the 0-error Is then necessarily undefined. Of course an unexpected rejection can lead to an exciting search for the alternative.)... 

In reviewing the history of detection limits (in Analytical Chemistry) it is helpful to keep these several, often implicit, differences in mind. If it Is agreed that the concept of detection has meaning, then it is essential that the above questions be fully defined and explicitly addressed. In the view of this author a meaningful approach to analyte detection must be consistent with our approach to uncertainty components of measurement processes and experimental results the soundest approach is probably the last [hypothesis testing] tempered with an appropriate measure of the first [scientific intuition]. 

Table I has been prepared from this perspective. The authors selected are drawn primarily from those who have contributed basic statements on the issue of detection capabilities of chemical measurement processes ["detection limits"], as opposed to simply addressing detection decisions for observed results ["critical levels"]. In fairness to those not listed, it is important to note that a) a selection only, spanning the last several decades has been given, and that b) there also exist many excellent articles (15.16) and books (12.17.18 > which review the topic. It is immediately clear from Table I that the terminology has been wide ranging, even in those cases where the conceptual basis (hypothesis testing) has been Identical. Nomenclature, unlike scientific facts and concepts, can be approached, however, through consensus. The International Union of Pure and Applied Chemistry [lUPAC], which appears twice in Table I, is the international body of chemists charged with this responsibility. At this point it will be helpful to examine the position of lUPAC as well as the contributions of some of the other authors cited in Table I.

Impurity detection for the multivariable case may be treated as a direct extension of the single variable case. For two patterns, the Impurity detection limit (component A contaminating control component B) can be calculated from (4/0 )., , corresponding to P - 0.05, where x (a - 0.05) is used to test the null hypothesis [H Xg - 4g ]. For mixed impurities, a "worst case"... 

Ihe application of the concept of "detection limit" as a criterion by the analytical scientist for withholding the results of low-level measurement is not supported. This usage may arise from the mistaken belief that the one-tailed statistical t-test is a test of the quality of the result, rather than the extent to which the result indicates that analyte is present in the sample. It is also the result of confusing the limitations of detection and measurement with the limitations of the analytical process and the impact of sample matrix effects. A further argument against the use of this concept as the basis for not reporting results is the improper hypothesis that a low result Is necessarily derived from a population of results with mean zero. 

The process of statistical inference requires us to select an hypothesis (fancy VK>rd for assumption) about the result, and then prove that this hypothesis was incorrect. In the case of Method Detection Limit, we assume that the result belongs to a distribution whose mean is centered on "zero". Based on a one-tailed test, if the analytical result is far enough away from zero we then conclude that such a result must come from some other distribution, in which case there is some likelihood that the sample contains the target analyte. MDL is conventionally set at 3 SD. 

There is no logical basis for the initial hypothesis that a low result comes from a population with mean zero. A low result can be obtained from any of the billions of the result populations on the analog nanber line between zero and twice the detection limit. Before the statistical test is even applied, there is already an immeasurable risk of a "false negative" conclusion, aggravated hy the fact that conventional wisdom rejects the reporting of results below HDL. 

In their more reeent documents, ISO and lUPAC recommend that the detection limit (minimum detectable amounts) should be derived from the theory of hypothesis testing and take into account the probabilities of false positives (a) and false negatives (fi). Thus, the limit of detection is defined... 

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