# Measurements of Human Perception of Train Vibration
## 5 Regression Analysis  
The primary objective of multiple regression, a term initially coined by [PERSON] in 1908 [19], is to quantify the relationship between multiple independent (explanatory) variables and a dependent (criterion, response) variable. While both Pearson correlation and basic linear regression assess the linear relationship between two statistical variables, significant distinctions exist between these two methods. Pearson correlation gauges the strength and direction of the linear association between two numeric variables without assuming causality.  
The focus of the analysis in this article was primarily to ascertain whether a cause-and-effect relationship exists between train speed and the WODL index. To enable such predictions, it is prudent to verify whether the WODL indicator exhibits a normal distribution. In Figure 14, a check for the normal distribution of the WODLz indicator (in the vertical direction) is presented.  
As can be seen from Figure 14, the normality of the distribution is quite good. Then, the linear regression function was checked to determine if a linear relationship is possible. The results are summarized in Table 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**N = 87** & **B** & **Standard Error from b** & \(p\) \\ \hline Intercept & 130.71 & 7.39 & 0.00000 \\ Train speed & 0.21 & 0.103 & 0.04328 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The possibility of using a linear equation in regression.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Train Type** & **Train Speed [km/h]** & **Value of WODL [-]** \\ \hline Old passenger train & 64 & 9.28 \\ Old passenger train & 82 & 16.65 \\ Technical train & 44 & 4.52 \\ Non-integrated high-speed train & 80 & 4.92 \\ Cargo train & 31 & 4.94 \\ New integrated high-speed train & 101 & 13.66 \\ Old integrated high-speed train & 92 & 6.21 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Exceedances of comfort level according to different types of train.  
The \(p\)-value is below 0.05, which means that the hypothesis of a linear relation between train speed and the value of the WODLz index can be recognized as true, and this relation is as follows:  
\[\text{WODLz}=0.21\text{S}+130.71, \tag{6}\]  
where S is the train speed.  
Although the \(p\)-value is quite good, the correlation is weak (see Figure 15).  
However, when treating train types separately and endeavoring to achieve an improved linear fit during the linearity checking stage, \(p\)-values above 0.05 were obtained, signalling a rejection of the linearity hypothesis. The most favourable \(p\)-value, nearing 0.05 and equating to 0.07, was observed for high-speed trains.  
The subsequent analysis aimed to determine whether if speed influences on the WODL index or this influence is more closely associated with the type of train. For this analysis, multiple regression, specifically utilizing the stepwise regression method, was employed. Initially, the investigation sought to establish the presence of a linear relationship between WODLz, train speed, and the type of train. The outcomes of this analysis are outlined in Table 5.  
Figure 14: Checking the normality of the distribution.  
Figure 15: Linear regression between train speed and WODLz for all types of trains.  
As can be seen from this multiple regression analysis, there is a relation between train type, train speed, and the value of the WODLz factor, and it is much stronger than for train speed and WODLz alone.