## 2 Materials and Methods
### Statistical Method
#### 2.2.2 Capability of Discrimination or Multivariate Sensitivity  
The sensitivity of an analytical method can be defined as the smallest amount of substance or analyte that can be accurately measured. So, an increased sensitivity means a higher increment of the response for the same amount or concentration of an analyte. To fix the notation, consider the interval of concentrations (x\({}_{0}-\epsilon,\)x\({}_{0}+\epsilon\)) around a value x\({}_{0}\). The sensitivity (the increase in the response of the method in that interval) should be defined in terms of what probability that a value outside the interval is significantly different from x\({}_{0}\) or not is obtained. If only the quotient of the increment in the response regarding the increment of the concentration is considered (e.g., the slope of an univariate calibration), the inherent uncertainty in any determination in the decision is omitted, and it also depends on the units in which the response is expressed. Given that the hypothesis tests have already been formalized so that, given a predetermined value (x\({}_{0}\)), it is possible to decide whether it is significantly greater or less than x\({}_{0}\), it is reasonable to generalize the situation to the case of deciding whether the concentration is significantly different from x\({}_{0}\). Consequently, given a nominal concentration, PL = x\({}_{0}\), in order to know the performance of an analytical procedure in samples with a similar concentration, the minimum discriminable concentration (or multivariate sensitivity) is defined as the smallest concentration of the analyte in a sample that can be distinguished with a probability 1 - \(\beta\) from x\({}_{0}\). Some extended details about the multivariate capability of discrimination can be consulted in reference [33]. In process quality control, univariate charts are used to control a single parameter or several parameters at the same time with multivariate charts. Both cases are formalized as a hypothesis test with the evaluation of probabilities \(\alpha\) and \(\beta\). Unlike control charts, the multivariate sensitivity approach proposed in this work does not imply taking the decision based on the parameter measurement (e.g., the fat % of a milk sample). Instead, an NIR spectrum of the sample is recorded, PLS calibration is then applied, and the result is used to determine whether the sample fulfills the quality standards using multivariate analytical sensitivity. That is, the decision is not made with a direct measurement of the controlled parameter but rather indirectly through its NIR spectrum. Figure 3: Schematic representation of CD\(\alpha\) and CD\(\beta\) linked to the accuracy line (upper right) and Operative Characteristic Curve (bottom left) for a maximum limit (x\({}_{0}\) = PL) according to the hypothesis test in Equation (6), where x is the true concentration of a sample, y is the obtained concentration based on the PLS model, and where y\({}_{c}\) depends on the significance level of the test (\(\alpha\)). Formally, the compliance of an analytical procedure corresponds to x = x0, while, if x \(\
eq\) x0, the procedure is not compliant. Therefore, a two-tailed hypothesis test can be outlined as in Equation (9). \[ \begin{array}{l}\text{H}_{0}\text{: x = x}_{0}\text{ (the parameter is equal to x}_{0}\text{, compliant sample)}\\ \text{H}_{\text{a}}\text{: x \
eq x}_{0}\text{ (the parameter is smaller or greater than x}_{0}\text{, non-compliant sample)}\end{array}  \tag{9}\]  
This text formalizes the definition of the minimum discriminable concentration considering that a given concentration will be different from x0 if it is greater than x0 + \(\varepsilon\) or if it is less than x0 - \(\varepsilon\). In terms of the previous section, x0 + \(\varepsilon\) is a maximum PL and x0 - \(\varepsilon\) is a minimum PL. Nevertheless, both \(\alpha\) and \(\beta\) must be distributed within both limits.