## 1 Introduction
### Survival Analysis  
Survival analysis is a particular form of reliability analysis in which \"time-to-event\" information (such as the age of a bridge at the end of service life or survival time of a patient) is analysed along with their associated contributing parameters (covariates). This is a data-driven approach that can evaluate the effect of covariates on the outcome (time to reach the specified event).  
Survival analysis techniques have long been used in biomedical research. The applications of this approach to bridge engineering, although limited, have expanded in recent years. Recent research efforts have been focused on developing survival models for bridges decks [18; 20; 19]  
In survival analyses, the key variable is the \"survival time\" or \"failure time\". In general, survival time is a non-negative random variable indicating the elapsed time from a reference time to the occurrence of a given event [24]. Examples of survival time include time to recurrence of cancer after administration of a drug, lifetime of an electronic component, etc. Bridge survival analysis could take into account the time that takes it to reach an event such as initiation of corrosion, cracking, or the end of service life. The term \"survival\" is, by definition, equivalent to the term \"reliability\" commonly used in engineering. Similarly, the term \"hazard\" used in survival analyses is analogous to \"failure rate\" in engineering applications.  
Non-parametric and semi-parametric survival models, such as Kaplan-Meier and Cox-regression models, have been widely used to estimate survival time in different fields. As the simplest form of survival analysis, the non-parametric Kaplan-Meier technique does not consider the simultaneous effect of multiple covariates on the outcome. Only the effect of a single covariate (time to reach an event) can be considered. Data on survival times (for all observations) are simply sorted and the survival and hazard rate are calculated based on the number of subjects that have survived at any given time. The semi-parametric Cox model considers the relative influence of covariates on the outcome and assumes proportionality of hazards. The parametric method is the most elaborate form of survival analysis, in which the survival function considers quantified effects of individual covariates. The parametric method can be a good substitute for the Cox model when the assumption of proportionality of hazards is not valid, and when a distribution function is available for the survival time and the baseline hazard function [24]. In such cases, parametric calculations are more descriptive and concise in comparison to non-parametric and semi-parametric methods.  
In this study, parametric survival analyses were conducted using the Hypertabastic distribution function first introduced by [PERSON] et al. [25]. This function has since been used in a variety of biomedical and engineering applications [25; 26]. In a PhD dissertation, [PERSON] [27] and [PERSON] et al. [28] studied the validity and effectiveness of both the accelerated failure time and proportional hazard forms of the Hypertabastic distribution and compared the Hypertabastic with classical models such as lognormal and loglogistic. They concluded that the Hypertabastic model was effective and accurate when compared with classical models.