Analytical Methods Committee AMCTB No. 92

Received
25th November 2019
, Accepted 25th November 2019

First published on 18th December 2019

Although the general concept of a detection limit as a lower practical operating limit for an analytical method is widely understood, the precise interpretation of results near the IUPAC limit of detection is rarely as well understood or implemented. It is important to distinguish between statements about the analytical result, which is known, and inferences about the true value, which are not known. It is also crucial to understand that there are at least two quite different limits involved and that the limit most commonly referred to as the ‘limit of detection’ is, surprisingly, not intended as the criterion for detection of an analyte. This Technical Brief describes the principal internationally recognised approach to decision and detection limits.

In conformity with modern thought, these questions would best be answered by looking at the uncertainty associated with a result at low levels.^{1} Historically, however, the issue has been addressed by constructing limits for decisions and detectability. These have been defined differently over time, and are consequently often misused or misinterpreted. This Technical Brief describes the approach used by IUPAC^{2,3} and ISO^{4} for the construction and interpretation of decision and detection limits.

(i) At what value is an observed analytical result significantly different from the result for a true ‘blank’ test material (that is, a material free from the analyte sought)?

(ii) At what true analyte concentration will the analytical result reliably exceed the level defined in (i)?

Question (i) defines the criterion for detection; any result above this value is to be declared ‘positive’. For that reason, together with its basis in hypothesis testing, IUPAC’s recommendation^{3} refers to this first level as a ‘critical value’, designated L_{C}. Question (ii) is essentially asking the question ‘what is the lowest level that will reliably be detected in practice?’ Importantly, it is the second question that corresponds to the IUPAC definition of ‘limit of detection’ (LOD). IUPAC designate this second value as L_{D}.

Notice also why these questions are described as “fundamentally” different. The first asks about an observable signal – the analytical result leading to a declaration of presence. The second is asking about a true value – something we cannot see directly and can only make inferences about.

L_{C} = _{0} + s_{0}t_{0.95,ν0} | (1) |

The rationale for this calculation is in statistical hypothesis testing. For the statistically minded, L_{C} is the critical value for a test of the null hypothesis H_{0}: μ = 0 with alternative hypothesis H_{1}: μ > 0. If we assume approximate normality, this implies a one-tailed Student t-test, conventionally at the α = 0.05 significance level (95% confidence). Fig. 1, step 1 illustrates the rationale, assuming results in the units of calculated concentration.

Fig. 1 The basis of decision and detection limits (adapted from ref. 3). |

Given L_{C}, which answers question (i) above, we can move to the second question: How much analyte needs to be present to give results that are reliably above L_{C}? Fig. 1, step 2 illustrates the basis for this new estimate, with the area to the right of L_{C} showing the proportion of results that fall above L_{C} given a true analyte concentration equal to L_{D}. Usually, ‘reliably’ is taken to mean ‘95% of the time’, so we adjust L_{D} to make the shaded portion equal to 95% of the distribution. This just needs another one-sided critical value, so we have

(2) |

Notice that we have distinguished, here, between the standard deviation for the blank material and the standard deviation for a material with a true level of analyte L_{D}. The standard deviation at L_{D} is hard to determine, as L_{D} is not known. Most texts therefore simplify by assuming that the standard deviation is approximately constant near zero, leading to

L_{D} = L_{C} + s_{0}t_{0.95,ν0} = _{0} + 2s_{0}t_{0.95,ν0} | (3) |

As before, the Student’s t value is often replaced by the large-sample value, leading to the well-known approximation L_{D} ≈ _{0} + 3.3s_{0}.

It is worth noting, here, that several approximations have been made. Normality has been assumed, and we have assumed that the standard deviation is constant at least from zero up to L_{D}. Perhaps even more importantly, we have also used an observed standard deviation without specifying the conditions of measurement. In practice, most texts use a repeatability standard deviation, though at least one regulation^{6} uses a rough estimate of the within-laboratory reproducibility (intermediate precision) standard deviation. If day to day (or some other) variation dominates, the calculated value for L_{D} could be a severe underestimate.

Table 1 overleaf summarises what can be said about the true concentration when the analytical result x falls in each of these ranges.

Result in region | Description | Detection | Inference about the true value^{a} |
Quantitative inference (all regions) |
---|---|---|---|---|

a Assuming α and β are set at 5% for 95% confidence. | ||||

A |
x ≤ L_{C} |
Not detected | Less than L_{D} with at least 95% confidence |
The best estimate of the true value is x. The standard uncertainty associated with this value is at least s_{0} [see text] |

B |
L
_{C} < x ≤ L_{D} |
Detected | Greater than zero with at least 95% confidence | |

C |
x > L_{D} |
Detected | Greater than L_{C} with at least 95% confidence |

A result in region C can, of course, also be interpreted as evidence that the analyte is present with at least 95% confidence.

Notice that results in region B – between the critical value and the detection limit – are positive for the presence of analyte. It is the critical value that is the decision criterion for presence of analyte – not the LOD.

The quantitative inferences that can be made should also be noted. All observed results are valid estimates; it is, however, important to allow for the associated uncertainty in interpretation. Near zero, the relative uncertainty is usually large. For a single observation, the standard uncertainty cannot be less than the standard deviation used for calculating the detection limit; it is usually much larger because an estimate of uncertainty includes all of the factors that might affect the result.

Finally, results below L_{D}, or even the critical value L_{C}, are not meaningless; they just have a larger relative uncertainty than results above these limits. While it is clearly not harmful to report results as ‘less than’ L_{D} where this meets client requirements, it is important that laboratories remain able and willing to provide the raw results if requested.

The AMC recommends that if statements such as ‘less than LOD’ are made in relation to results, the basis for the calculation and interpretation should be made clear to the client, either explicitly in the report or by reference to a documented standard or procedure.

The AMC also recommends that raw data be made available to the client irrespective of whether it is above or below detection or reporting limits wherever it is important for the client’s needs – for example, for trend analysis, averaging or other summaries, or (for example) in proficiency testing. Where possible, information on the uncertainty of results should be given with any numerical values to prevent over-interpretation.

- M. Thompson and S. L. R. Ellison, Towards an uncertainty paradigm of detection capability, Anal. Methods, 2013, 5, 5857–5861 RSC.
- L. A. Currie, Limits for Qualitative Detection and Quantitative Determination: Application to Radiochemistry, Anal. Chem., 1968, 40, 586–593 CrossRef CAS.
- L. A. Currie, Nomenclature in Evaluation of Analytical Methods Including Detection and Quantification Capabilities, Pure Appl. Chem., 1995, 67, 1699–1723 CAS.
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- Commission Decision 2002/657/EC, Official Journal of the EC, L 221/8, 2001, https://publications.europa.eu/en/publication-detail/-/publication/ed928116-a955-4a84-b10a-cf7a82bad858 Search PubMed.

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