In assessing risk in regression models, the best estimates of a measurand $y_0$ can be obtained by calculating the values for the regression line $y_f$ in the points of the reference scale $\mathit{x}.$ In that case, the best estimate of the measured quantity is represented by the vector ${\mathit{y}}_0=\left(y_f\left(x_1\right),y_f\left(x_2\right),\dots ,y_f\left(x_{13}\right)\right),$ and the risk is estimated for each of the values of the regression line ${y_{f_j}=y}_f\left(x_j\right),j=1,2,\dots ,13.$ The coefficients ${\widehat{\beta }}_0$ (intercept) and ${\widehat{\beta }}_1$ (slope) of the regression line were obtained by the well-known least squares (LS) method from a simple linear regression model: where the intercept ${\beta }_0$ and slope ${\beta }_1$ are the model parameters and $n_2=13.$ For simplicity, in the subsequent text, where appropriate, the designation $n_1=3$ is utilized. Values ${\epsilon }_{i,j},i=1,2,3,j=1,2,\dots ,n_2$ are random errors [45,46]. Model parameters ${\beta }_0$ and ${\beta }_1$ were estimated for each pair of observations $\left(x_j,y_{i,j}\right),i=1,2,3,j=1,2,\dots ,n_2$ [47] (pp. 76-78). The best estimates, ${\widehat{\beta }}_0$ and ${\widehat{\beta }}_1$, of the parameters ${\beta }_0$ and ${\beta }_1$ are obtained by the least square method by minimizing the sum of the squared residuals:According to [45] and [48], it is not difficult to show that is: where is $n=n_1·n_2=39,$ and that is:The value of $\overline{\mathit{x}}$ in expressions (3) and (4) represents the arithmetic mean of the values of the moderate scale:It should be noticed that the values of the moderate scale are equidistant. Also, the values from the left side of zero on the moderate scale are of the opposite sign when compared to the values from the right side of the moderate scale. That’s why it’s always worth that is $\overline{\mathit{x}}=0.$ The value of $\overline{\mathit{y}}$ in equations (3) and (4) is calculated from:

To estimate the measurement uncertainty $u_0$ based on the law of propagation of uncertainty (LPU) applied to the functional relationship of the input data, whisch was obtained from the linear regression analysis, two additional quantities are crucial: the standard error of slope $u\left({\widehat{\beta }}_1\right)$ and the standard error of intercept $u\left({\widehat{\beta }}_0\right).$ That quantities were calculated from the following equations: and The estimator ${{\widehat{\sigma }}_y}^2$ for ${\sigma }^2$ in the variable $y$ is equal to the variance of the regression [46], so it is valid:The fitted regression line for data in Table 1 has the form:

The standard deviation of slope and intercept are respectively $u\left({\widehat{\beta }}_1\right)=0.000254$ µm and $u\left({\widehat{\beta }}_0\right)=0.004752$ µm. The result for residual standard deviation in variable $y$ is equal to ${\widehat{\sigma }}_y=0.029674$ µm and the coefficient of determination is $R^2=1.$ Based on the Shapiro–Wilks test, at the significance level $\alpha =0.05,$ it was concluded that the residuals follow a normal distribution (Table S1). Residual diagnostics were also performed based on the residuals graph, Q-Q normal plot, and residual density graph (Figure S1).

The standard uncertainty $u_0$ associated with the best estimate of a measurand $y_0$ 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: where $u_c\left(L_P\right)$ represents combined standard uncertainty of the probe calibration. Value denoted as ${u(L}_{ULM})$ is standard uncertainty due to the influence of the accuracy of the ULM device. The marks $u\left(L_{LEI}\right)$ and $u\left(L_{LEII}\right)$ stands 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 $5°$ and that the error of the probe position can amount to a maximum of 0.3 mm. The standard uncertainty due to the influence of the resolution of the measuring device is indicated with $u\left(L_N\right).$ 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 $u_0$, calculated in the risk assessment procedure, based on the expression for the combined measurement uncertainty [49] (p. 21) amounts to $u_0^{GUM}=0.1247$ µm. The mark $u_0^{GUM}$ stands for $u_c\left(L_P\right)$ in further text. The measurement uncertainty, which is calculated in this way is the same for all points on the moderate scale $\mathit{x}$. All components that enter the calculation of the combined measurement uncertainty of the probe have the sensitivity coefficient $c=1.$

Another approach to evaluating measurement uncertainty $u_0$ 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 is 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:The uncertainty to be evaluated in the regression model is the uncertainty of the random variable $Y_f$ evaluated for the given value $x_j,j=1,2,\dots ,n_2$ of the explanatory variable $X.$ According to [49,53], this measurement uncertainties can be calculated from the equation: where the standard uncertainty associated with the input quantity $X$ is equal to The numerical calculation yields a value of ${\sigma }_x=0.029626$ for the data from Table 1. According to [52], the standard uncertainty arising from the dependence of the parameters ${\widehat{\beta }}_0$ and ${\widehat{\beta }}_1$ equals:It is important to notice that the expression in equation (13) due to (15) only depends on the squares of the reference values of the moderate scale ${x_j}^2,j=1,2,\dots ,n_2.$ Therefore, the graph of measurement uncertainties, evaluated at the points of the moderate scale $\mathit{x},$ is a parabola with an opening upwards, axisymmetric concerning the direction $x=0$ (Figure 3).

The measurement uncertainty calculated using the expression for the combined measurement uncertainty is the same for all points of the moderate scale $\mathit{x}$. In contrast, the measurement uncertainties calculated by the LRA method are different for different points of the scale. The vector of measurement uncertainties, calculated by the LRA is denoted by ${\mathit{u}}_0^{LRA}=\left(u_0^1,u_0^2,\dots ,u_0^{n_2}\right).$ The point in the middle of the moderate scale has the lowest measurement uncertainty, and the points on the edges have the highest measurement uncertainty. The values of measurement uncertainties for each point of the moderate scale $\mathit{x}$ calculated by LRA are given in Table S2. In comparison with the value $u_0^{GUM},$ the values ${\mathit{u}}_0^{LRA}$ are underestimated.

When calibrating devices, there is usually no historical data from previous measurements about the slope and intercept of the regression line or their standard errors. A well-performed calibration yields a slope close to one and an intercept close to zero. For the measurer, it would be extremely difficult, to determine, based on prior beliefs, what values can have a slope and an intercept. Especially when it is taken into consideration that these values should be given to at least two decimal places. That kind of guessing would inevitably lead to an incorrect risk assessment. It is an even bigger problem if the data comes from other non-metrology models, where the values for slope and intercept can be any. Therefore, the risk assessment problem for regression is reformulated so that the calibration data shown in Table 1 are used to determine the data $y_0$ and $u_0$ that are included in the expression for the prior distribution $g_0\left(\eta \right).$ The input parameter $u_m$ of the likelihood function $h\left({\eta }_m|\eta \right)$ is assumed to be the standard measurement uncertainty of a future inspection process. The behavior of global consumer’s and producer’s risk was tested for three different cases: for $u_m=u_0/2,$ for $u_m=u_0,$ and $u_m=2u_0.$

In that regard, the contribution of this paper is a proposal to indicate the data for the regression line obtained during the calibration procedure, thereby enabling the measurer to be guided by these data in future measurements. This would allow traceability in the risk assessment.

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 $\pm 30$ µm amounts to $0.6$ µm. Based on this information, it is assumed that in the first observed model M1, the range of tolerance interval $∆T=T_U-T_L$ for each point of the moderate scale $\mathit{x}$ is $∆T=0.6$ µm. Individual values of the upper and lower limits of the tolerance interval were determined for each reference value $\mathit{x}$ of the measuring range. Through these points, a straight line can be drawn. These lines are parallel and symmetrically positioned relative to the line $y=x$, i.e., symmetrically relative to the line obtained for a perfect measurement. Therefore, can be spoken about the upper and lower tolerance lines. 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 $\mathit{x},$ of the moderate area, is equal to $∆T=4u_0^{GUM}.$ Model M2, where the range of the tolerance interval is $∆T=4u_0^{GUM},$ is also linearized. Model M3, where $∆T=6{\mathit{u}}_0^{LRA},$ was analyzed as well. In this model, the tolerance interval is placed symmetrically around the $y=x$ line, but the range of tolerance interval at the edges of the measurement area is wider than the range of tolerance interval in the middle of the scale. This model is non-linearized and can be spoken about as an upper and lower tolerance curve. These tolerance interval ranges $∆T=4u_0^{GUM}$ and $∆T=6{\mathit{u}}_0^{LRA}$ were chosen according to the 4-sigma and 6-sigma rules.

The global consumer’s and producer’s risk also was estimated for model M4. This model favors the minimal value of measurement uncertainty calculated using LRA. For that model $∆T=6\min \left({\mathit{u}}_0^{\mathit{L}\mathit{R}\mathit{A}}\right)$ is valid. Model M4 is artificially linearized based on model M3. An example of a linearized and non-linearized model is 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 $y=x$ line that describes the perfect measurement. Simultaneous tolerance intervals are constructed relative to the fitted regression line [54]. When constructing a two-sided simultaneous tolerance interval, according to [54], for the lower and upper limits of the tolerance interval are valid: and where $k_2$ is a constant which can be calculated using the formula given in [55] or using tabulated values in [56]. It follows from equations (16-17) that is:If the simultaneous tolerance interval were to be constructed around the line $y=x,$ as is the case with the other described tolerance intervals, then, due to (9), it would be valid that $∆T=0$. For this reason, simultaneous tolerance intervals cannot be applied in the risk assessment procedure for regression models defined in this paper.

If the tolerance interval is within the acceptance interval, then the maximum range of the acceptance interval $∆A=A_U-A_L$ is equal to $∆A=1.2∆T$, in all models (Figure 1a). If the acceptance interval is within the tolerance interval, then the maximum range of the acceptance interval is $∆A=0.8∆T$ (Figure 1c). For the shared risk, $∆A=∆T$ applies (Figure 1b). The range of the acceptance interval was chosen so that the global producer’s risk surface intersects with the global consumer’s risk surface. For simplicity and transparency, the basic data on the analyzed models are given in Table 3.

Finally, it should be emphasized that the risk assessment was carried out for the regression line even though the data from Table 1 were collected during the calibration procedure. During the calibration procedure, the measurement uncertainty determined according to the LRA, instead of for $y_{fj},j=1,2,\dots ,n_2$ from equation (13), would be determined for the values of the moderate scale $\mathit{x}.$ The equation for $\mathit{x}$ is obtained from the expression (12) for the regression line [53] (p. 872). Furthermore, to calculate the risk for the calibration procedure, it is necessary to invert the tolerance and acceptance intervals constructed for the regression line to obtain the tolerance and acceptance intervals for the explanatory variable [57] (p.29).

The behavior of the global producer’s risk $R_P$ and global consumer’s risk $R_C$ along the moderate scale depends on the values of the slope ${\widehat{\beta }}_1$ and intercept ${\widehat{\beta }}_0$ of the regression line. The risk curves for $R_P$ and $R_C$ along the moderate scale are parabolas with an upward opening. The minimum of these parabolas is located at the intersection of the regression line $y_f={\widehat{\beta }}_0+{\widehat{\beta }}_{1}x$ and the line $y=x$. The position of the minimum at the moderate scale is indicated by $x_{min}$ and is represented by the following expression: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 $x_{min}=16.12$ µm (Figure 5).

Parabolas are translated to the right when $x_{min}>0,$ i.e., if ${\widehat{\beta }}_0>0$ and ${\widehat{\beta }}_1<1$, or if, as is the case here, ${\widehat{\beta }}_0<0$ and ${\widehat{\beta }}_1>1$. 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 regression line’s length located on the right side of the intersection of the regression line and the $y=x$ line, in contrast to the proportion located on the left side of the intersection of the regression line and the $y=x$ line. Therefore, the calculated global 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 ${\widehat{\beta }}_0>0$ and ${\widehat{\beta }}_1>1$, or ${\widehat{\beta }}_0<0$ and ${\widehat{\beta }}_1<1,$ the parabolas would be translated to the left. The minimal values for the parabolas of the global risk of producers and consumers then would be located at the point of the moderate scale $x_{min}<0,$ and would be calculated from equation (34).

The line marked with $y_s$ and given by the equation: is axisymmetric to the regression line. These lines are axisymmetric concerning $y=x.$ For the ${\widehat{\beta }}_1>1$ worth $2-{\widehat{\beta }}_1<1,$ and conversely, for the ${\widehat{\beta }}_1<1$ worth $2-{\widehat{\beta }}_1>1.$ The global risks of producers and consumers calculated for the line $y_s$ are equal to the risks calculated for the regression line from equation (10) up to the order of magnitude ranging from ${10}^{-4}$ do ${10}^{-15},$ depending on the observed model and assuming that the line $y_s$ is obtained under identical conditions as the regression line, with identical values for $u_0$ and $u_m$ and identical values for tolerance intervals and acceptance intervals. The same applies to conformance probability.

Due to the narrower range of the tolerance interval ∆T, the M2 model exhibits higher values of the global risk of producers and consumers across the entire moderate scale than the M1 model. 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, in the point $x_7=0$. At this point on the moderate scale, the width of the tolerance interval for both models is equal and amounts to $6\min \left({\mathit{u}}_0^{LRA}\right)$ (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, $x_{min}.$

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 a prominent parabola shape compared to models M3 and M4.

The maximal value for conformance probability is reached precisely at the point $x_{min}$ where the global risk of the producer $R_P$ and the global risk of the consumer $R_C$ have a minimal value (Figure 6).

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, $x_{min}.$ 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 $p_C$ drop to 72% for the M3 model, or even 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 $x_1=-30$ µm, 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 lines, the acceptance and tolerance line (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 $∆T=3{{\mathit{u}}_0}^{LRA}$ is set. In that scenario, the regression line intersects the lower acceptance curve $A_L$ and the lower tolerance curve $T_L$. 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 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, a tolerance interval width was set to $∆T=6{{\mathit{u}}_0}^{LRA}$ in the M3 model. The anomaly is not completely resolved by this. The regression line for $r\approx 0.9770$ intersects the lower acceptance curve $A_L$. The iso-risk curves for the global producer’s risk $R_P$ diverge from the parabola graph on the left edge of the moderate scale (Figure 5b). For this reason, the range of tolerance intervals should be further expanded. The deviation from the parabola graph is even more noticeable with the M4 model. With this model, the regression line intersects the lower line of the acceptance interval $A_L$ already by $r\approx 0.8497.$ In both models, M3 and M4, anomalies occur for the values of the multiplicative factor $r$ which is related to the model of minimization of global consumer’s risk (Figure 1c). The specified values of the multiplicative factor $r$ were calculated numerically. Also, although the regression line in models M3 and M4 intersects the lower acceptance curve (line) $A_L$, it fails to intersect the lower tolerance curve (line) $T_L$. For the M2 model, it is sufficient to set $∆T=4{u_0}^{GUM}$ to prevent the regression line from intersecting either the tolerance interval or acceptance interval.

Depending on the multiplicative factor $r_k$ where $k=1,2,\dots ,n_3,$ according to equation (24), the guard band $w_k$ can have a positive or negative value or be equal to zero. If $w_k<0,$ it is about the model of the minimization of global producer’s risk (Figure 1a). If $w_k=0,$ it is a shared risk model (Figure 1b). For $w_k>0,$ it is a model for minimizing the global consumer risk (Figure 1c). It is natural to observe the behavior of the global risk of producers and consumers along the guard band axis. In general, if it is $w_{k_1}<w_{k_2}$ then for $k_1\ne k_2,k_1<k_2$ and $k_1,k_2ϵ\left\{1,\dots ,n_3\right\}$ holds that $R_{C_{k_1}}>R_{C_{k_2}}$ and $R_{P_{k_1}}<R_{P_{k_2}}.$ Simplified, the consumer’s risk decreases along the guard band axis as depicted in Figure 8a, whereas the producer’s risk increases along the guard band axis as depicted in Figure 8b.

Figure 8 shows the curves of the minimum of the global producer’s and consumer’s risk calculated numerically along the guard band axis at points $\left(x_{min},w_k\right),k=1,2,\dots ,n_3$. These are the curves where the risks, for each model are the least. Due to the described construction of acceptance and tolerance intervals, risks are assessed for intervals of varying lengths along the guard band axis. The values of measurement uncertainties calculated using LRA, ${\mathit{u}}_0^{LRA},$ are considerably smaller in comparison with $u_0^{GUM}.$ It is therefore pointless to compare all models for the same tolerance interval $∆T=0.6$ µm given by the manufacturer. For the measurement uncertainty ${\mathit{u}}_0^{LRA},$ the tolerance interval selected in this manner will be too wide, resulting in negligible risk values $R_C$ and $R_P$ for models M3 and M4, while the conformance probability will be equal to one. The M4 model’s curve of minimum for global producer’s risk intersects the curve of minimum of the M1 model (Figure 8b).

The conformance probability for the curve of minimum in each model is determined by the straight line along the guard band axis. These values remain the same for all points $\left(x_{min},w_k\right),k=1,2,\dots ,n_3.$ For models M1, M2, M3 and M4 respectively worth: $p_C\approx 0.9838$, $p_C\approx 0.9545,$ $p_C\approx 0.9973$ and $p_C\approx 0.9971$.

The described behavior of the global producer’s risk $R_P$ and the global consumer’s risk $R_C$ along the moderate scale and the guard band axis can be easily noticed by observing the risk surfaces shown in Figure 9. From the picture, it is evident that there are areas where the global risk of the producer and the global risk of the consumer of models M3 and M4 are lower than the risk of the models M1 and M2.

The conformance probability curves along the moderate scale are as in Figure 6, but the conformance probability surfaces were evaluated for intervals of different ranges along the guard band scale (Figure 10).

The behavior of the global producer’s risk $R_P$ and the global consumer’s risk $R_c$ across all models were tested for three different assumed values of measurement uncertainty $u_m$ of a futured inspection process: for $u_m=u_0/2,$ $u_m=u_0$ and for $u_m=2u_0.$

With the increase in measurement uncertainty $u_m$, the global consumer’s risk $R_C$ also increase (Figure 11).

Additionally, with the increase in measurement uncertainty value $u_m$, the producer’s risk also rises $R_P$ (Figure 12).

It is evident that, in addition to the range of the tolerance interval to the occurrence of anomalies in the behavior of the global producer’s risk $R_P$, measurement uncertainty $u_m$ also has an impact. Deviation from the parabolic curve becomes more pronounced, (Figure 12c,d). Graphs can completely alter their behavior, as is the case in model M2. When it is worth that $u_m=2u_0$, the graph of the global producer’s risk $R_P$ is a parabola with a downward opening (Figure 12b). For $u_m=2u_0,$ the regression line is not outside the tolerance interval or the acceptance interval, in none of the mentioned case. To avoid anomalies in the behavior of the risk curves for $R_P$, it is necessary to expand the ranges of tolerance intervals. For model M2, this range should be set to at least $∆T=4.4{u_0}^{GUM}$, for model M3 to $∆T=7.4{{\mathit{u}}_0}^{LRA}$ and for model M4 to $∆T=7.7{{\mathit{u}}_0}^{LRA}$. This prevents anomalies in the graphs for all values of $r_k,k=1,2,\dots ,n_3$.

According to equation (33), the expression for conformance probability $p_C$ is independent of the measurement uncertainty $u_m$ of a future inspection process. Therefore, the graphs for conformance probability for all tested values of measurement uncertainty $u_m$ are as shown in Figure 6, and for the 3D case, as shown in Figure 10.