# Risk Assessment for Linear Regression Models in Metrology
## 3 Results and Discussion
### Graphical Risk Analysis
#### 3.1.1 Behaviors of a Global Consumer and Producer Risk along the Moderate Scale  
The behavior of the global producer's risk \(\mathrm{R_{P}}\) and global consumer's risk \(\mathrm{R_{C}}\) along the moderate scale depends on the values of the slope \(\hat{\beta}_{1}\) and intercept \(\hat{\beta}_{0}\) of the regression line. The risk curves for \(\mathrm{R_{P}}\) and \(\mathrm{R_{C}}\) along the moderate scale are parabolas with upward openings. The minimum of these parabolas is located at the intersection of the regression line \(\mathrm{y_{f}}=\hat{\beta}_{0}+\hat{\beta}_{1}\mathrm{x}\) and the line \(\mathrm{y=x}\). The position of the minimum at the moderate scale is indicated by \(\mathrm{x_{min}}\) and is represented by the following expression:  
\[\mathrm{x_{min}}=\frac{\hat{\beta}_{0}}{1-\hat{\beta}_{1}}. \tag{33}\]  
It is easy to show how the minimal values of the global risk of producers and consumers along the moderate scale in all analyzed models are achieved for \(\mathrm{x_{min}}=16.12\) (Figure 5). Parabolas are translated to the right when \(\mathrm{x_{min}}>0\), i.e., if \(\hat{\beta}_{0}>0\) and \(\hat{\beta}_{1}<1\), or if, as is the case here, \(\hat{\beta}_{0}<0\) and \(\hat{\beta}_{1}>1\). That is the reason why the risk parabolas, i.e., the iso-risk curves along the moderate scale depicted in Figure 5, are asymmetric. In all models, due to translation, the values of global risk for producers and consumers on the left side of the moderate scale are higher compared to the risk values on the right side of the moderate scale. This disparity is readily apparent in Figure 4, as evidenced by the proportion of the length of the regression line located on the right side of the intersection of the regression line and the \(\mathrm{y=x}\) line, in contrast to the proportion located on the left side of the intersection of the regression line and the \(\mathrm{y=x}\) line. Therefore, the calculated global  
Figure 5: Comparison of behavior of consumer’s risk and producer’s risk along the moderate scale for all four models, \(\mathrm{r=1}\), \(\mathrm{u_{m}=u_{0}/2}\): (**a**) Consumer’s risk; (**b**) Producer’s risk. A violet circle indicates the point of minimum for the risks \(\mathrm{R_{C}}\) and \(\mathrm{R_{P}}\). risks of producers and consumers can be seen as risks of deviation of the regression line from the y = x line. In the case of a regression line where \(\hat{\beta}_{0}>0\) and \(\hat{\beta}_{1}>1\), or \(\hat{\beta}_{0}<0\) and \(\hat{\beta}_{1}<1\), the parabolas would be translated to the left. The minimal values for the parabolas of the global risk of producers and consumers would then be located at the point of the moderate scale x\({}_{\text{min}}<0\), and would be calculated from equation (33). The line marked with y\({}_{\text{s}}\) and given by the following equation:  
\[\text{y}_{\text{s}}=-\hat{\beta}_{0}+\left(2-\hat{\beta}_{1}\right)\cdot\text {x}, \tag{34}\]  
is axisymmetric to the regression line. These lines are axisymmetric concerning y = x. For the \(\hat{\beta}_{1}>1\) worth \(2-\hat{\beta}_{1}<1\), and conversely, for the \(\hat{\beta}_{1}<1\) worth \(2-\hat{\beta}_{1}>1\). The global risks of producers and consumers calculated for the line y\({}_{\text{s}}\) are equal to the risks calculated for the regression line from equation (9) up to the order of magnitude ranging from \(10^{-4}\) to \(10^{-15}\), depending on the observed model and assuming that the line y\({}_{\text{s}}\) is obtained under identical conditions as the regression line, with identical values for u\({}_{0}\) and u\({}_{\text{m}}\) and identical values for tolerance intervals and acceptance intervals. The same applies to conformance probability. Due to the narrower range of the tolerance interval \(\Delta\Gamma\), the M2 model exhibits higher values regarding the global risk of producers and consumers across the entire moderate scale compared to the M1 model.