## 2 Regression Models  
For the analysis of time-to-failure test data, [PERSON] and [PERSON] introduced a linear relationship between stress (relative load level) and logarithmic time-to-failure [10; 11; 16; 20]. This corresponds to a logarithmic time-to-failure function as formulated in Equation (1):  
\[y=a\cdot\ln(t_{f})+b \tag{1}\]  
with \(y=N/N_{100\%}\) the load level, \(N\) the sustained load, \(N_{100\%}\) the pull-out capacity obtained in short-term reference tests, and \(t_{f}\) the failure time. Equation (1) uses the natural logarithm of time which corresponds to an exponential decay of the load levels with time, parameter \(a=\) the half life of decade, and parameter \(b\) the scale factor which also represents the theoretical load level at instantaneous time, \(b=\lim\limits_{t_{f}\to 0}y\). This conservative method works relatively well for relatively high load levels, but can not accurately predict failure times for lower load levels. This logarithmic regression model violates the physical asymptotic properties for low and high loads and is solely able to represent data in the transitional domain. For very short load durations approaching \(t=0\), the model predicts relative sustained load levels larger than 1.0, i.e., sustained load strengths that exceed the short-term strength [21]. On the other hand, the logarithmic model will predict finite failure times also for very low load levels, and even for unloaded specimens since the regression line will intersect the abscissa. This is not consistent with the established assumptions in concrete design that assumes linear creep, i.e., the absence of creep damage, up to approximately 40% of the short-term strength/capacity [21; 22; 23]. In the literature, a number of alternative regression models can be found that are either derived from theory or represent empirical models that are more or less infused by physical concepts. Very recently, the authors demonstrated that the logarithm of the creep rate of bonded anchor systems under sustained load is proportional to the logarithm of the failure time [24]. In combination with the power-law relationship between creep rate and relative applied stress level, a power-law function for the relation of failure time on the load level is obtained:  
\[\ln(y)=\ln(b)+n\cdot\ln(t_{f}) \tag{2}\]  
where \(y=N/N_{100\%}\), \(b=y(\lim_{t_{f}\to 0})\), and \(n\) is the slope in the log-log domain which translates to an exponent in linear plot. Coincidentally, this formulation (Equation (2)) agrees with the power-law regression model proposed by [PERSON] et al. [25] which does not suffer from the more severe restriction of the logarithmic model, i.e., the regression model will always approach infinite failure time for an unloaded specimen. However, this linear regression model in logarithmic load versus logarithmic time domain still violates the upper asymptote. Given the limitation to test data at relatively high load levels, this shortcoming negatively influences the stability of the regression as it imposes an non-physical trend (linear in a log-log plot) onto the data that tend to approach asymptotically one for short failure times, i.e., exhibits initially a concave shape. Additionally, [PERSON] et al. [24] introduced a function based on the rate-theory. This can be formulated in the form of \(N/N_{100}=f(t_{f})\) as  
\[y=\kappa_{\infty}+\sinh^{-1}\left[\frac{(b\cdot t_{f})^{n}}{c}\right] \tag{3}\]where \(y=N/N_{100\%}\), \(\kappa_{\infty}\), \(b\), \(c\), and \(n\) parameters to be fitted, and representing the asymptotic load level (\(\kappa_{\infty}\)), the onset of the asymptotic value (\(b\) and \(c\)), and the slope (\(n\)) of the decreasing load levels versus time. This model takes into account a terminal asymptote, which is related to the creep limit, but nevertheless it still violates the asymptotic physical behavior for high load levels, i.e., \(\lim\limits_{t_{f}\to 0}y=\infty\).