## Survival Model Development
### Survival Function  
Results of survival analyses can be used to determine the probability of survival (reliability) and instantaneous failure rates (hazard). Three distinct functions commonly used in survival analysis are: survival function \(S(t)\) (Equation (1)), probability density function \(f(t)\) (Equation (2)), and hazard or conditional failure rate \(h(t)\) (defined as failure rate at any given time assuming survival up to that time) (Equation (3)) [24]. \[S(t)=P(T>t)=1-F(t)\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **ADT** & **MSL** & **Age** & **Deck Area** \\ \hline ADT & 1.000 & & & \\ MSL & 0.240 & 1.000 & & \\ Age & \(-0.229\) & \(-0.207\) & 1.000 & \\ Deck area & 0.385 & 0.665 & \(-0.217\) & 1.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Correlation between covariates. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Distribution Model** & \(-2\)**Log-Likelihood** & **AIC** \\ \hline Hypertabastic & 637.496 & 647.496 \\ Log-logistic & 649.628 & 659.628 \\ Weibull & 655.688 & 665.688 \\ Lognormal & 712.192 & 722.192 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of Akaike Information Criterion for various models. \[f(t)=\lim_{\Delta t\to 0}P(t<\mathrm{T}<t+\Delta t)/\Delta t \tag{2}\]  
\[h(t)=\lim_{\Delta t\to 0}p(t<T<t+\Delta t|T>t)/\Delta t \tag{3}\]  
where \(T\) indicates the survival time as a random variable, \(t\) is the time, and \(F(t)\) denotes the cumulative probability of failure at various times. \(S(t)=1\) at \(t=0\) and \(S(t)\to 0\) as \(t\rightarrow\infty\). The baseline Hypertabastic probability density function is defined as [18]:  
\[f(t)=\left\{ \begin{array}{l}\mathit{[PERSON]}[W(t)]\left(\alpha t^{2\beta-1} \mathit{[PERSON]}\left(t^{\beta}\right)^{2}-\alpha t^{\beta-1}\mathit{[PERSON]}\left(t ^{\beta}\right)\right)Tanh[W(t)]\text{ for }t>0\\ 0\text{ for }t<0\end{array} \right. \tag{4}\]  
where parameters \(\alpha\) and \(\beta\) are both positive, \(\mathit{[PERSON]}[\bullet]\), \(\mathit{[PERSON]}[\bullet]\) and \(\mathit{[PERSON]}[\bullet]\) are hyperbolic secant, hyperbolic cosecant, and hyperbolic cotangent, respectively; and \(W(t)=\alpha(1-t^{\beta}\mathit{[PERSON]}\left(t^{\beta}\right))/\beta\). The baseline hypertabastic survival function is defined as  
\[S(t)=\mathit{[PERSON]}\left[\alpha(1-t^{\beta}\mathit{[PERSON]}\left(t^{\beta}\right) )/\beta\right] \tag{5}\]  
The baseline hypertabastic failure rate function \(h(t)\) is defined as  
\[h(t)=\alpha\left(t^{2\beta-1}\mathit{[PERSON]}\left(t^{\beta}\right)^{2}-t^{\beta- 1}\mathit{[PERSON]}\left(t^{\beta}\right)\right)Tanh[W(t)] \tag{6}\]  
Using the proposed Hypertabastic accelerated failure time model and after removal of small terms, survival and failure rates can be determined using Equations (7) through (9) [18]:  
\[S\left(t_{g}\right)=\mathit{[PERSON]}\left[\alpha\left(1-t_{g}^{\beta}\mathit{[PERSON] }\left(t_{g}^{\beta}\right)\right)/\beta\right] \tag{7}\]  
\[h\left(t_{g}\right)=\alpha\left(-t_{g}^{\beta-1}\mathit{[PERSON]}\left(t_{g}^{ \beta}\right)\right)Tanh[W\left(t_{g}\right)]e^{\left[c\left(\mathrm{MSL} \right)+d\left(\mathrm{ADT}\right)+h\left(\mathrm{Type}\right)\right]} \tag{8}\]  
\[t_{g}=(\mathrm{AGE})e^{\left[c\left(\mathrm{MSL}\right)+d\left(\mathrm{ADT} \right)+h\left(\mathrm{Type}\right)\right]} \tag{9}\]  
Parameters \"AGE\", \"MSL\", and \"ADT\" indicate the age of the bridge superstructure, maximum span length, and average daily traffic, respectively. \"Type\" is a discrete variable representing the type of bridge superstructure. This variable is equal to 1 for steel, 0 for concrete, and 2 when both types of superstructures are combined (i.e., when superstructure types are not distinguished). The parameters \(\alpha\), \(\beta\), \(c\), \(d\), and \(h\) are determined using the method of maximum likelihood. The log-likelihood function was maximized within the Mathematica software. Detailed equations and procedures used in finding these parameters are described by [PERSON] et al. [18].