# 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 width of the tolerance interval of the M3 model is greater than the width of the tolerance interval of the M4 model, except in the middle of the scale, at the point x\({}_{7}=0\). At this point on the moderate scale, the widths of the tolerance intervals for both models are equal, amounting to 6 min \(\left(\text{u}_{0}^{\text{LRA}}\right)\) (Table 3). Consequently, the global risks of producers and consumers in the M3 model are lower than those in the M4 model (Figure 5a,b). In these two models, the risk values coincide at the point of minimum, x\({}_{\text{min}}\). The iso-risk curves of models M1 and M2, along the moderate scale, exhibit stable behavior with a narrower range of value changes. These models behave differently compared to the M3 and M4 models. In models M3 and M4, the range of variations in producer and consumer risk values is significantly greater. In the case of consumer risk, the iso-risk curves for models M3 and M4 intersect those for models M1 and M2 (Figure 5a). The same applies to iso-risk curves of producers (Figure 5b). All models are shown on the same scale for comparison. That is why the iso-risk curves for models M1 and M2 do not have such prominent parabola shapes as models M3 and M4. The maximal value for conformance probability is reached precisely at the point x\({}_{\text{min}}\), where the global risk of the producer R\({}_{\text{P}}\) and the global risk of the consumer R\({}_{\text{C}}\) have a minimal value. Higher risk values result in lower values for the conformance probability and vice versa. This can be observed when comparing models M1 and M2, i.e., M3 and M4, as well as all models (Figure 6). Models M3 and M4 have the same value for conformance probability at the point of moderate scale, x\({}_{\text{min}}\). Other points of the moderate scale show higher values for conformance probabilities for model M3 than for model M4. The lowest values for conformance probability are found on the left side of the moderate scale for all models. The values for p\({}_{\text{C}}\) drop to 72% for the M3 model, and even to 69% for the M4 model. Hence, it is possible to conclude that models M3 and M4 attach greater importance to risks at the edges of the scale. On the graphs of models M3 and M4, which show the behavior of the global risk of the producers, anomalies on the left edge of the scale can be observed (Figure 5b). Anomalies mean that the iso-risk curves deviate from the parabola graph. At the point of the moderate scale x\({}_{1}=-30\) um, the global risk of the producer for the M3 model is greater than the global risk of the producer for the M4 model. These anomalies occur when the tolerance interval is too narrow. If the range of the tolerance interval is too narrow, the regression line crosses the acceptance line, or both the acceptance and tolerance lines (or acceptance and tolerance curves). In that case, the risk graphs are no longer parabolas (Figure 7). Figure 7 illustrates the presence of anomalies, i.e., the deviations of the iso-risk curve from the parabola graph. Such deviations are obtained when, for example, with the M3 model, a too-narrow tolerance interval range of \(\Delta\mathrm{T}=3\mathbf{u_{0}}^{\mathrm{LRA}}\) is set. In that scenario, the regression line intersects with the lower acceptance curve \(\mathrm{A_{L}}\) and the lower tolerance curve \(\mathrm{T_{L}}\). Anomalies may manifest themselves for both the global consumer's risk and global producer's risk (Figure 7a,b). The graph for conformance probability clearly shows that the model behaves badly on the left side of the moderate scale, where the conformance probability falls below 20% (Figure 7c). The primary drawback of the described method for risk assessment in regression is the excessively narrow tolerance interval. It is necessary to expand its span to resolve the issue. To stay within the 6-sigma range used in statistics,  
Figure 6: Comparison of behavior of conformance probability along the moderate scale for all four models, \(\mathrm{r}=1\), \(\mathrm{u_{m}}=\mathrm{u_{0}}/2\).