## 4 Discussion  
### Contributions and Practical Implications  
The increasing use of NIR spectroscopy and PLS calibration models in the agri-food industry is undeniable. The necessity of establishing analytical sensitivity and some permitted limits with specific assurance probabilities is also evident. However, little effort has been devoted to this issue, as concluded based on the bibliographic review made. In most of the few publications that deal with this matter, a limit of detection (LOD) or a limit of quantification (LOQ) is calculated. Usually, a confidence interval is calculated for the signal at a zero concentration (PL = 0) by calculating k times the standard deviation of the signal in the blank (k = 3 for LOD and k = 10 for LOQ). The interval is then used to decide that a sample does not contain the analyte when its signal is within the interval and that it contains the analyte when it is outside of it. Regarding this, the mistake is to use a confidence interval instead of a hypothesis test, because with the confidence interval, only the probability of a false positive (\(\alpha\)) can be estimated, but the probability of a false negative (\(\beta\)) is not evaluated, which, as it has already been shown, is 0.5 for the LOD or LOQ (as can be seen in Figure 2 with a PL = 0). What has not been found in any publication is the use of the terms of false non-compliance and false compliance for permitted limits. Using the concept of a hypothesis test and the accuracy line, the unified methodology presented in this work is a versatile and easy-to-apply tool. The hypothesis tests proposed in Section 2.2 are one-tailed (Equations (1) and (6)) or two-tailed (Equation (9)) for the mean y\({}_{0}\), which is the accuracy line for PL = x\({}_{0}\). Precisely,  
Figure 8: Capability of detection operative curves for difufenican for a PL = 0 mg kg\({}^{-1}\) at \(\alpha\) = 0.05.  
the structure of the data allows for a calculation of the density function of any value y that is outside the acceptance region of the test, that is, outside the intervals [y\({}_{c}\), y\({}_{0}\)], [y\({}_{0}\), y\({}_{c}\)] or [y\({}_{c}\)], [y\({}_{c}\)] for a minimum permitted or established limit, for a maximum permitted or established limit, or for the capability of discrimination, respectively. This density function includes the changes in the standard deviation \(\vartheta\) when increasing the distance \(\mid\)y \(-\)y\({}_{0}\) \(\mid\), which is calculated using the factor \(w_{x_{0}}\) (Equation (3)). The corresponding CD\(\beta\) (Equations (2), (7), (9), and (10)) is obtained using the accuracy line. In this way, all the key elements of each problem are being considered. Knowing the parameters of this distribution makes it possible to compute the probability of false compliance, \(\beta\), for each x different from PL = x\({}_{0}\). This information is important and is the differential contribution when considering the decision of accepting or rejecting the PL as a hypothesis test (instead of using an interval of confidence for PL). The probability of false non-compliance, \(\alpha\), manages the length of the intervals [y\({}_{c}\), y\({}_{0}\)], [y\({}_{0}\), y\({}_{c}\)], or [y\({}_{c}\), y\({}_{c}\)]. The smaller the value of \(\alpha\), the greater their length, and therefore, the greater the distance between the PL and CD\(\alpha\).