## 3 Research Method  
### Logistic Regression Analysis  
Regression analysis is the most widely used method to analyze the correlation between one dependent variable and several independent variables [12; 13]. Typical regression analysis assumes a linear relationship between the independent variables and the dependent variable. This is not suitable for dealing with binary events, such as ground subsidence, where the dependent variable is represented by the incidence of subsidence [14]. Meanwhile, logistic regression is a model that estimates the logistic regression coefficient by assuming that the relationship between the dependent and independent variables is nonlinear. This is suitable for analyzing the relationship between the dependent and independent variables that have only two values [15].  
Therefore, logistic regression was chosen as the first method for analyzing ground subsidence in urban areas in Korea. The relationship between the dependent and independent variables was assessed. The dependent variable was whether or not ground subsidence occurred. The independent variables consisted of factors influencing ground subsidence. These factors included pipe diameter, pipe type, pipeline age or year of installation, removal system, buried depth, and surrounding roads. This analysis was conducted using the collected sewer pipe attributes and ground subsidence history data. As for the dichotomous dependent variable, such as ground depression, \(E_{(yx)}\) refers to the probability of ground subsidence occurring given the independent variable \(x\). If the range of \(x\) is from \(-\infty\) to \(+\infty\), the value is a probability, so the \(p\)-value ranges from 0 to 1. \(P^{\prime}\) has a model of a curve close to an S-shape, and this characteristic has the advantage of converting the incidence of ground subsidence into a probability. Therefore, it can be expressed as Equation (3).  
\[E_{(yx)}=\frac{\exp(B_{0}+B_{1}x)}{1+\exp(B_{0}+B_{1}x)} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Buried Depth** & **Number of Sewers** & \begin{tabular}{c} **Ratio** \\ **(\%)** \\ \end{tabular} & \
\begin{tabular}{c} **Number of Sewers** \\ **Closest to Subsidence** \\ \end{tabular} &
\begin{tabular}{c} **Ratio** \\ **(\%)** \\ \end{tabular} \\ \hline
5 m or more & 1554 & 0.4 & 18 & 0.5 \\
3–5 m & 6261 & 1.6 & 40 & 1.2 \\
2–3 m & 16,386 & 4.3 & 108 & 3.3 \\
1–2 m & 105,108 & 27.7 & 1115 & 34.2 \\
1 m or less & 250,743 & 66.0 & 1984 & 60.8 \\ Total & 380,052 & 100 & 3265 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Sewer pipes’ buried depth in the target region.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Removal System** & **Number** & \begin{tabular}{c} **Ratio** \\ **(\%)** \\ \end{tabular} & \
\begin{tabular}{c} **Number of Sewers** \\ **Closest to Subsidence** \\ \end{tabular} &
\begin{tabular}{c} **Ratio** \\ **(\%)** \\ \end{tabular} \\ \hline Combined Pipe & 318,083 & 92.7 & 2992 & 92.8 \\ Storm Pipe & 8274 & 2.4 & 80 & 2.5 \\ Sewer Pipe & 16,748 & 4.9 & 153 & 4.7 \\ Sum & 343,105 & 100 & 3225 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Sewer removal system in the target region.  
The logistic function above is nonlinear for \(B_{0}\) and \(B_{1}x\); however, assuming E(\(yx\)) = \(P\), it is possible to linearize the equation as follows.  
\[P^{\prime}=\ln\left(\frac{p}{1-p}\right)=\ln(\frac{E_{(yx)}}{1-E_{(yx)}}) \tag{4}\]  
By substituting Equation (3) into Equation (4), it is possible to linearize the equation as follows.  
\[P^{\prime}=B_{0}+B_{1x} \tag{5}\]  
Such a transformation is called logistic transformation, and \(P^{\prime}\) is called Logit [16].