# Risk Assessment for Linear Regression Models in Metrology
## 1 Introduction  
Regression is the most used statistical method, applied for various purposes and in different scientific disciplines, including metrology. The basic purpose of regression is the prediction of a value that has not been measured, but which can be estimated based on the regression line or regression plane. So, for example, a regression can be used to predict the cutting temperature and surface roughness during the face milling process [1], or for predicting and monitoring vibrations on a lathe machine [2]. Regression can also be used to characterize various devices and sensors for atomic force microscope characterization [3] or magnetic field sensors characterization [4]. It is most often used in calibration, from calibrating pressure against ruby fluorescence shifts [5] to calibration that describes the pressure sensitivities of the optical fiber sensors [6]. Furthermore, regression is often used in interlaboratory comparison studies of calibration standards in different fields [7, 8], or for the analysis of different physical constants [9, 10]. The application of regression involves measuring the value of the response variable that corresponds to the given values of the explanatory variable [11]. There are numerous existing regression techniques that can be used to determine the relationship between these variables [12]. Among all of them, a straight-line relationship is the simplest one. This, themost common way of connecting only one independent (explanatory) variable and the dependent (response) variable, is called univariate linear regression. The subject of this paper concerns the risk assessment for linear univariate regression models in metrology. All recommendations and standards related to metrology are issued by the international metrology organization International Bureau of Weights and Measures (Bureau International des Poids et Mesures, BIPM). The basic procedure for risk assessment for an item of interest is given by the guide called Evaluation of measurement data--The role of measurement uncertainty in conformity assessment 106:2012 [13]. However, this method of risk assessment, well-known in metrology, has not been applied for regression so far. According to [14] (p. 15), and [15; 16] guidelines called JCGM 107, for the application of the least squares method, the most famous method for determining the coefficients of the regression line, is still in preparation. It is not yet known whether these guidelines will include a risk assessment. Risk assessment is carried out in the process of assessing the conformity of a product with specified requirements. Two types of risks may arise during conformity assessment. The producer's risk refers to situations where measurements or products meet the required specifications but are rejected as non-conforming. Consumer risk refers to situations when products or measurements are accepted as conformed but do not meet specifications. Whether a product conforms to the given specifications or not is determined based on measurements. The measured value of the item of interest must be within the given tolerance interval. The measurement uncertainty that occurs during measurement may lead to incorrect decisions regarding the acceptance of a non-compliant product or the rejection of a product that meets the specifications. This happens when the measurement is close to either the lower limit of the tolerance interval \(\mathrm{T_{L}}\) or the upper limit of the tolerance interval \(\mathrm{T_{U}}\), and the measurement uncertainty associated with that measured value goes over the tolerance interval \([\mathrm{T_{L}},\mathrm{T_{U}}]\)[17; 18; 19; 20; 21]. It is important to emphasize that the tolerance interval in the metrology domain is different from the regression tolerance interval in the statistical domain. The statistical tolerance regions, also known as simultaneous tolerance intervals, are constructed so that they, with a certain level of confidence, contain a specified proportion of the population in future sampling [22; 23].