## 6 Comparison of Regression Models and Recommendations
However, if data points are obtained that lie below the linear regression line, i.e., form a concave curve approaching the asymptotic value 1 (short times), they will improve the long-term extrapolations in the convex part of the sigmoid function (large times) where data points are limited due to the required long test durations. Naturally, the second asymptote for low load levels can not be estimated if experimental data are restricted to the approximately linear range around the inflection point. This property of sigmoid functions is discussed in detail by [PERSON] et al. [29]. Figure 11b shows results for Product 1, where the circles represent all tests that failed, and the diamonds the tests that were running at the time the analysis was performed. During fitting, only data points that actually failed are considered, i.e., results obtained at 95%, 75%, 65%, and 60% load levels. At the time the analysis was performed two tests at 60% were still running (diamond marker). After the analysis was performed, one of the two running tests at 60% failed. The corresponding point is marked in the plot (star), but not used for the analysis. The data obtained from the sustained load tests at 85% (marked with stars) are excluded from the fit, and were used to validate the extrapolation method. As it can be seen, the experimentally obtained failure times are in a very good agreement with the predicted times. Based on the fit where the \(\kappa_{\infty}\) parameter is optimized (parameter \(\kappa_{\infty}\) defines asymptotic value), the proposed sigmoid function captures reasonably well the experimental data and flattens out approaching a horizontal asymptote at \(\kappa_{\infty}=0.599\) (magenta curve). That means that fit defines a mean relative load level of 59.9% as a level below which no failure occurs in a theoretically infinite product lifetime. For a given finite required service lifetime, the safe load level can be directly obtained by evaluating the function at a  
Figure 12: Product 2: 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**). given lifetime. Additionally, fits were obtained with two pre-defined asymptotic load levels, i.e., fixed asymptotes at 40% (blue) and 0% (black), respectively. The asymptotic load level is the most uncertain but also most sensitive model parameter. Thus, providing a (conservative) estimate significantly improves the stability of the model and reduces the requirement for very long tests. The fit with a fixed asymptote at 40% is chosen as an engineering approach following the common assumption of linear creep (no damage) in a load range of 0-40% [22; 32]. The fixed asymptote at 0% is selected to ensure an infinite time-to-failure for unloaded systems. As expected, this approach (black line) is more conservative than the curve with optimized asymptote (magenta line) or fixed asymptote at 40% (blue line), and still shows higher sustained load strength than the previously promoted approach with a linear model which can be qualified as overly conservative due to its non-physical properties. It has to be noted here that the linear trend line (green line) is fitted to all failure times (circles). According to [PERSON] et al. [10], a linear trend line should be fitted for the intermediate range (from high load levels to load levels that did not fail in 2000 h) and results in the largest reduction of load levels at 50 years. Finally, the proposed new approach clearly shows in Figure 11b all limitations of the simple power-law approach (green line), even though it captures most of the available data (65-80%). The proposed function defines an optimized asymptotic relative load level at about 60% for the investigated product P1. The linear fit including all observed failure times defines a level below which no failure occurs in 50 years at about 47%.