## 2 Methodology
### Methodology for Identifying a Dominant Risk Factor Using the Cox Regression Model  
The probability density function \(f(t)\) represents the exact time when the event occurs, as follows:  
\[f(t)=\lim_{\Delta t\to 0}\frac{P(t\leq T<t+\Delta t)}{\Delta t} \tag{2}\]  
The cumulative distribution function is defined as \(F(t)=P(T<t)\). It is known that:  
\[f(t)=\frac{d}{dt}F(t) \tag{3}\]  
In survival analysis, a survival function is more frequently used. It is defined as \(S(t)=P(T>t)\), which means the event is later than time \(T\). In addition, \(S(t)=1-F(t)\).  
The hazard function \(h(t)\) is defined as the probability of a subject experiencing an event at time \(T+\Delta t\), under the condition of no event occurring from 0 to \(t\) (which means that the event occurs within a very short period). The following equation shows the mathematical expression for the hazard function:  
\[h(t)=\lim_{\Delta t\to 0}\frac{P(t\leq T<t+\Delta t|T\geq t)}{\Delta t}=\lim_{ \Delta t\to 0}\frac{F(t+\Delta t)-F(t)}{\Delta t}=\frac{f(t)}{S(t)} \tag{4}\]  
From this equation, it is known that \(h(t)\), \(f(t)\), and \(S(t)\) are related. The Cox model with multiple variables is:  
\[h(t)=h_{0}(t)e^{\sum_{i=1}^{n}x_{i}\beta_{i}} \tag{5}\]Equation (5) is a regression model that introduces logarithmic linearity for the hazard function, where \(x\) is a covariate, while \(\beta\) is a regression coefficient. \(h_{0}(t)\) is called the baseline hazard function. As long as \(h_{0}(t)>0\), it can take any shape as a function of \(t\). The hazard ratio (HR) is calculated as the failure ratio of two samples, whose covariates are \(x_{i}^{\prime}\) and \(x_{i}^{\prime\prime}\):  
\[\text{HR}(\text{t},\ x_{1},\ x_{0}\ )=\frac{h_{0}(t)e^{x_{1}^{\prime}\beta_{1}+x_{2} \beta_{2}+\cdots+x_{n}\beta_{n}}}{h_{0}(t)e^{x_{1}^{1}\beta_{1}+x_{2}\beta_{2}+ \cdots+x_{n}\beta_{n}}}=\exp\left[\beta_{i}\big{(}x_{i}^{\prime}-x_{i}^{\prime \prime}\big{)}\right] \tag{6}\]  
The hazard ratio can be interpreted as the ratio of the possibility of degradation for any time duration. If the hazard ratio is higher than 1, then the risk will increase as the variate increases, and vice versa. In this study, for each numeric covariate, \(N\big{(}\mu,\ \sigma^{2}\big{)}\) is calculated, where \(\mu\) is the mean value and \(\sigma^{2}\) is the variance. The covariates are then standardized to \(N(0,\ 1)\) using the following equation:  
\[z_{i}=\frac{x_{i}-\mu}{\sigma} \tag{7}\]  
where \(x_{i}\) is the original value of the covariate and \(z_{i}\) is the standardized value. Thus, for numeric variables, the hazard ratios represent the ratio of risk variety per standardized unit. More specifically, the hazard ratio minus 1 and multiplied by 100 expresses the hazard percent change for \(1\sigma\). For instance, if the hazard ratio is 1.3 and \(\sigma=200\), the risk will be 30% higher when the variate increases by 200.  
In Cox regression analysis, the value of the regression coefficient \(\beta\) is unknown. \(\beta\) can be estimated using the partial likelihood function \(l_{p}(\beta)\). However, if the data were continuously and perfectly measured, tied data (two or more data with the same survival time) would not exist. However, in real cases, because time is measured discretely, tied data are very common. According to Efron (1977) and [PERSON] (2014), if the number of tied data is large, the Efron method should be used (Efron, 1977; Efron, 1977). Therefore, in this study, the Efron method is used to estimate \(\beta\). The partial likelihood function is calculated as follows:  
\[l(\beta)=\prod_{i=1}^{m}\frac{e^{x_{i}+\beta}}{\prod_{k=1}^{d_{i}}(\sum_{j \in R(t_{i})}e^{x_{j}\beta}-\frac{k-1}{d_{i}}\sum_{j\in R(t_{i})}e^{x_{j}\beta })} \tag{8}\]