## 6 Comparison of Regression Models and Recommendations  
In the following chapter, the data of the systematically carried out experimental investigation serves for the analysis of different regression models with the objective of lifetime predictions. Several approaches as introduced earlier are compared and used to investigate the aforementioned goals. Finally, a new sigmoid function is found to be a more appropriate choice to relate stress to failure times due to the use of presence of horizontal asymptotes for low and high relative load levels. The fits of the power function on the experimental data are shown in Figure 9a for Product 1, and in Figure 10a for Product 2. Figures 9b and 10b show the corresponding fits based on the rate-theory function for Products 1 and 2, respectively. Both regression models can fit well the experimental data. The fits agree well with the predicted stress versus time-to-failure curve as obtained by the creep-rate based approach proposed in [24]. The creep-rate based approach reveals a no-failure limit at around 63% in the case of P1, while, in the case of P2, it occurs to be at around 70%. Figure 8: Schematic view of the intersection point used to determine time-to-failure. Figure 9: Product 1: Fit of a power function (**a**), and fit of rate-theory based function (**b**). Both models as well as the logarithmic regression model are unable to reproduce (at least approximately) the physical behavior at high relative load levels. The Powell-Eyring function on the other hand has the advantage of capturing the two asymptotes that are most likely present in this type of tests. This model has only three parameters, two representing the asymptotic values and one describing the speed of the transition. The fit of Equation (4) on the experimental data are shown in Figure 11a for Product 1, and in Figure 12a for Product 2. For all figures, the dashed lines represent the confidence interval of the prediction at level of 95%. If the three parameters are considered independent, a value for \(\kappa_{0}\) which is lower than the 100% is obtained. This is due to the lack of data at high relative load levels, especially for P2. Fixing \(\kappa_{0}=100\%\), which is justified by the fact that 100% load level will lead to theoretically instantaneous failure, further reduces the number of independent variables to two and enforces the \"correct\" asymptotic behavior at high loads. Finally, the load levels below which (on average) failure will not be observed even at infinite times are determined for the Powell-Eyring model to be 66% for P1 and 75% for P2. Figure 11: Product 1: Fit of Powell–Eyring equation on stress versus TTF data (**a**) and comparison of sigmoid function vs. linear regression model for the TTF predictions (**b**). Figure 10: Product 2: Fit of a power function (**a**), and fit of rate-theory based function (**b**). As an alternative, the proposed sigmoid function as introduced in Equation (5) is evaluated. This sigmoid function for the stress versus time-to-failure plot is a nonlinear function in logarithmic time. It is well known that the relationship between stress and TTF is characterized by more than one domain [21], which can not be captured by a single linear function. As discussed for the Powell-Eyring model, the proposed sigmoid function approaches the \"true\" asymptotic behavior for high (horizontal asymptote at 100%) and low stress levels (horizontal asymptote at the threshold to damage development). In the intermediate (second) domain, this function is characterized by an almost linear (in logarithmic time). This is the domain in which typically most of the experimental TTF data points come to rest. As a consequence, the fit of the regression model is stabilized by adding tests at high load levels. These are relatively quick tests albeit complicated by the inherently high scatter that may lead to failure already during loading.