## Validation of liquid chromatography mass spectrometry (LC-MS) methods

# 9.2. Decision limit and Detection capability

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**Decision limit (CC_{α}) and detection capability (CC_{β})

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http://www.uttv.ee/naita?id=23306

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https://www.youtube.com/watch?v=posQ05DUCIc&t=1s

As mentioned in the previous section, by definition may or may not take into account both (results where the analyte is declared to be present although actually it is below LoD) and (results where analyte is declared to be below LoD although it is not) errors at the same time.

An example is given in Table 1 to better explain the concept of false positive and false negative results on the basis of a widely banned pesticide DDT (dichlorodiphenyltrichloroethane). The concentration of DDT is compared to the capability of the analytical method.

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Table 1. Explanation of false positive and false negative results.

To account for both of these errors, Currie [] suggested to use two different quantities: decision limit (CC_{α}) and detection capability (CC_{β}). For example, when validating the analysis method for measuring DDT, it was found that CC_{α} was at 0.005 μg/L. Therefore, if a higher result than 0.005 μg/L is received from the analysis there is ≤ 5% probability that the result is just noise (noise can be defined as signal received when no analyte is present or the analyte concenrtation is so low that the analysis method cannot detect it). If we are relatively certain that the results are not simply noise, then we can say that we have detected DDT. In other words, CC_{α} can be considered as the concentration at which we can decide that we are not measuring noise but we are receiving a signal from the analyte.

CC_{α} is defined as the concentration level, as determined by the method, at which there is probability α (usually defined as 0.05 or 5%) that a blank sample will give a signal at this level or higher. CC_{β} is defined as the concentration level of the analyte in sample at which there is a probability β (again usually defined as 0.05 or 5%) that the method will give a result lower than CC_{α}, meaning that the analyte will be declared as undetected (although the analyte content in the sample is in fact higher than CC_{α, }see Figure 1 for illustration of and definitions).

For example, in the example of DDT analysis method, CC_{β} was found to be Y. This means that if a result above Y is received then the probability that the measurement is just noise is ≤ 5%. Because of random variability of results, even if DDT is present in the sample at a concentration above CC_{α}, there is the possibility that we receive a result that is below CC_{α} and therefore interpret the result as negative (see the red curve in Figure 1). Therefore, we need to find CC_{β} where the probability for this kind of false negative probability is small, in our example ≤ 5%.

By these definitions, if the analyte is present at concentration of CC_{β} (or above) it can be said with good certainty that the measurements are not simply noise (which otherwise would lead to false positive results) and that the analysis method can measure this concentration with only small probability of giving results below CC_{α} (i.e. false negative results).

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Figure 1. Relation between permitted limit, CC_{α} and CC_{β}

It must be noted that in some cases the name “limit of detection” is used in place of CC_{β}. When dealing with LoD, the precise definition under question must be clarified. Although in most cases LoD can be considered equal to CC_{β} this is not always the case. The approaches to calculate LoD make many assumptions about the properties of the system – e.g. and . The approaches given by standards to estimate CC_{α} and CC_{β} are usually provide more complex and do not make the same assumptions. Therefore, in principle LoD and CC_{β} could be considered equal, but their values cannot always be compared in a meaningful way.

If the value has been set for the analyte in the particular sample, then the following should be considered: (a) the MRL value should be above the CC_{β} or LoD value (see Section 9.1), and (b) the analyte concentration should be estimated where it can be said with known confidence (taking into account both false positive and false negative probabilities) that the concentration of the analyte is above the MRL. For the second part, CC_{α} and CC_{β} must be evaluated not for 0 concentration but at the MRL level (see Figure 1 where C = 0 or **at MRL**). It should be noted that for example Council Directive 96/23/EC has defined CC_{α} and CC_{β} so that if a MRL is set, these values are not used to estimate the minimum amount that the method can detect, but for estimating when the result is over or under the MRL value. The same approaches to estimate CC_{α} and CC_{β} can be used.

In Figure 1, a normal distribution is used only to clearly and simply explain the concept of CC_{α} and CC_{β}. It has been shown that often results from blank and very low concentration samples do not have a normal distribution. Moreover, it is more correct to use a t-distribution rather than a normal distribution in case if only a small number of replicate measurements are made. In addition, it should be noted that the data is heteroscedastic (width of the distribution is wider at higher concentration) in Figure 1 and homoscedastic (width of the distribution is same at both concentrations) in the video “Decision limit (CC_{α}) and Detection capability (CC_{β})”.

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Interpreting results with CCα and CCβ

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http://www.uttv.ee/naita?id=23349

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https://www.youtube.com/watch?v=DP1wAdmXkIg

With CC_{α} and CC_{β, }the results must be interpreted in the following way: (a) in case the result is below CC_{α}, it must be reported that the analyte concentration in the sample is below CC_{β} (this is so because it is not known whether our result is falsely negative or truly under CC_{α}), (b) in case the result is over CC_{α}, but below CC_{β}, the sample can be said to contain the analyte with good probability, and (c) in case the value is over CC_{β}, then the same statement can be made as in the point (b).

However, only the CC_{β} value can be used for the characterization of the method as at CC_{β} we take into account the possibility of both false positive and false negative error.