# Risk Assessment for Linear Regression Models in Metrology
## 2 Materials and Methods
### Model Parameters
#### 2.2.3 Tolerance and Acceptance Interval  
In the present study, four models were examined concerning the width of the tolerance interval: M1, M2, M3, and M4. The conventional tolerance interval for each point of the moderate scale was obtained based on the information provided by the manufacturer. The total span of error in both directions for the inductive contact probe in the standard range of \(\pm 30~{}\upmu\)m amounts to \(0.6~{}\upmu\)m. Based on this information, it was assumed that in the first observed model M1, the range of tolerance interval \(\Delta\mathrm{T}=\mathrm{T}_{\mathrm{U}}-\mathrm{T}_{\mathrm{L}}\) for each point of the moderate scale **x** was \(\Delta\mathrm{T}=0.6~{}\upmu\)m. Individual values of the upper and lower limits of the tolerance interval were determined for each reference value **x** of the measuring range. Through these points, a straight line could be drawn. These lines are parallel and symmetrically positioned relative to the line \(\mathrm{y}=\mathrm{x}\), i.e., symmetrically relative to the line  
Figure 3: The measurement uncertainty of the fitted regression line according to the law of propagation of uncertainties. obtained for a perfect measurement. Therefore, the upper and lower tolerance lines can be spoken about. In this sense, this model is linearized. If information about the tolerance interval does not exist, it can alternatively be taken that the range of the tolerance interval for each point of the reference scale \(\mathbf{x}\) of the moderate area is equal to \(\Delta\mathrm{T}=4\mathrm{u}_{0}^{\mathrm{CUM}}\). Model M2, where the range of the tolerance interval is \(\Delta\mathrm{T}=4\mathrm{u}_{0}^{\mathrm{CUM}}\), is also linearized. Model M3, where \(\Delta\mathrm{T}=6\mathrm{u}_{0}^{\mathrm{LRA}}\), was analyzed as well. In this model, the tolerance interval is placed symmetrically around the \(\mathrm{y}=\mathrm{x}\) line, but the range of the tolerance interval at the edges of the measurement area was wider than the range of the tolerance interval in the middle of the scale. This model is non-linearized and upper and lower tolerance curves can be talked about. These tolerance interval ranges \(\Delta\mathrm{T}=4\mathrm{u}_{0}^{\mathrm{CUM}}\) and \(\Delta\mathrm{T}=6\mathrm{u}_{0}^{\mathrm{LRA}}\) were chosen according to the 4-sigma and 6-sigma rules. The global consumer's and producer's risk was also estimated for model M4. This model favors the minimal value of measurement uncertainty calculated using LRA. For that model, \(\Delta\mathrm{T}=6\mathrm{min}(\mathbf{u}_{0}^{\mathrm{LRA}})\) is valid. Model M4 was artificially linearized based on model M3. Examples of a linearized and a non-linearized model are given in Figure 4.  
The primary distinction between the statistically defined simultaneous tolerance intervals and the tolerance intervals outlined in this paper lies in their construction. The tolerance intervals in this study are positioned around the \(\mathrm{y}=\mathrm{x}\) line, which describes the perfect measurement. Simultaneous tolerance intervals are constructed relative to the fitted regression line [22]. When constructing a two-sided simultaneous tolerance interval, according to [22], for the lower and upper limits of the tolerance interval, the following are valid:  
\[\mathrm{T}_{\mathrm{L}}=\hat{\mathrm{y}}_{\mathrm{j}}-\hat{\mathrm{o}}_{ \mathrm{y}}\cdot\mathrm{k}_{2\mathrm{j}},\mathrm{j}=1,2,\ldots,\mathrm{n}_{2}, \tag{15}\]  
and  
\[\mathrm{T}_{\mathrm{U}}=\hat{\mathrm{y}}_{\mathrm{j}}+\hat{\mathrm{o}}_{ \mathrm{y}}\cdot\mathrm{k}_{2\mathrm{j}},\mathrm{j}=1,2,\ldots,\mathrm{n}_{2}, \tag{16}\]  
where \(\mathrm{k}_{2}\) is a constant which can be calculated using the formula given in [54] or using tabulated values in [55].