# Risk Assessment for Linear Regression Models in Metrology
## 2 Materials and Methods
### Model Parameters
#### 2.2.2 Measurement Uncertainty  
The standard uncertainty \(\mathrm{u}_{0}\) associated with the best estimate of a measurand \(\mathrm{y}_{0}\) was determined in two ways: by applying a simplified measurement model with the use of comprehensive data on measurement performance and by applying a statistical model, specifically the linear regression analysis (LRA), which was adapted to the measurement data. A simplified measurement model that describes the measurement uncertainty of probe calibration is expressed by following equation:  
\[\mathrm{u}_{\mathrm{c}}(\mathrm{L}_{\mathrm{P}})=\sqrt{\mathrm{u}^{2}( \mathrm{L}_{\mathrm{ULM}})+\mathrm{u}^{2}(\mathrm{L}_{\mathrm{LEI}})+\mathrm{ u}^{2}(\mathrm{L}_{\mathrm{LEII}})+\mathrm{u}^{2}(\mathrm{L}_{\mathrm{N}})}, \tag{10}\]  
where \(\mathrm{u}_{\mathrm{c}}(\mathrm{L}_{\mathrm{P}})\) represents combined standard uncertainty of the probe calibration. The value denoted as \(\mathrm{u}(\mathrm{L}_{\mathrm{ULM}})\) is the standard uncertainty due to the influence of the accuracy of the ULM device. The marks \(\mathrm{u}(\mathrm{L}_{\mathrm{LEI}})\) and \(\mathrm{u}(\mathrm{L}_{\mathrm{LEII}})\) stand for standard uncertainty of the linear error due to the error of the probe tilt angle and for standard uncertainty of the linear error due to the error of the probe position, respectively. From experience, it is known that the error of the probe tilt angle can amount to a maximum of \(5^{\circ}\) and that the error of the probe position can amount to a maximum of \(0.3~{}\mathrm{mm}\). The standard uncertainty due to the influence of the resolution of the measuring device is indicated with \(\mathrm{u}(\mathrm{L_{N}})\). The components that contribute to the uncertainty of measurement results and their contributions to the combined measurement uncertainty are given in Table 2.  
The input parameter \(\mathrm{u_{0}}\), calculated in the risk assessment procedure based on the expression for the combined measurement uncertainty [49] (p. 21), amounts to \(\mathrm{u_{0}^{GUM}}=0.1247~{}\mathrm{\upmu m}\). The mark \(\mathrm{u_{0}^{GUM}}\) stands for \(\mathrm{u_{c}(L_{P})}\) in further text. The measurement uncertainty, which is calculated in this way, is the same for all points on the moderate scale \(\mathbf{x}\). All components that enter the calculation of the combined measurement uncertainty of the probe have the sensitivity coefficient \(\mathrm{c}=1\). Another approach to evaluating measurement uncertainty \(\mathrm{u_{0}}\) involves utilizing a statistical model and a functional relationship derived from linear regression analysis (LRA) [50; 51]. The measurement uncertainty was calculated after the measurement was performed, and after the regression line was determined. This procedure encompasses the computation of the partial derivatives of the expression for the regression line and the incorporation of standard errors for the slope and intercept in the calculation of the measurement uncertainty [52; 53]. Measurement uncertainty calculated using LRA is carried out for the equation:  
\[\mathrm{Y_{f}}=\hat{\beta}_{0}+\hat{\beta}_{1}\mathrm{X}. \tag{11}\]  
The uncertainty to be evaluated in the regression model is the uncertainty of the random variable \(\mathrm{Y_{f}}\) evaluated for the given value \(\mathrm{x_{j}},\mathrm{j}=1,2,\ldots,n_{2}\) of the explanatory variable \(\mathrm{X}\).