## 4 Objective Evaluation Methodology for Passenger Acceptance
### Stochastic Estimation and Evaluation of the Acceptance  
A modeling method of the driver acceptance evaluation model based on a stochastic estimation is proposed to estimate and evaluate passenger acceptance with the suggested parameter sets; this modeling method fundamentally follows the well-known theory of binary logistic regression. By comparing the acceptance models developed based on the different objective parameter sets, the interpretation performances of each parameter set can be validated. As stated in reference [26], regression methods have become an integral component of any data analysis concerned with describing the relationship between a response variable and one or more explanatory variables. In Figure 2, an example of stochastic estimations with a single variable is shown to show the suggested schematic methodology of the acceptance estimation for the objective evaluation. For this single-variable estimation model, it can be stated that the model is more deterministic when the absolute values of the constant (\(\beta_{0}\)) and coefficient (\(\beta_{1}\)) are higher. In this study, the subjective assessment results, the outcome variables, were recorded in binary with various continuous variables, which were the measured parameters on the vehicle behaviors and relationships. Additionally, specific parameter sets were derived for driver acceptance. Thus, the acceptance model was presented based on multi-variables. Therefore, acceptance can be objectively evaluated by the developed acceptance evaluation model with the measured values for the given parameter sets. Logistic regression models are the most frequently used models for analyses of this kind of binary data, and the experimentally interpretable model to describe the relationship between an outcome variable and a set of explanatory variables can be derived as follows. In this study, dozens of explanatory parameters were used to describe emotional acceptance. To derive a multi-variable logistic regression model, a collection of \(p\) variables denoted by the vector \(\mathbf{x}^{T}\) was considered and in Equation (4):  
\[\mathbf{x}^{T}=\left\{x_{1},\quad x_{2},\quad\ldots,\quad x_{p}\right\} \tag{4}\]  
where it is assumed that each of the variables in the vector \(\mathbf{X}^{T}\) are interval-scaled. By defining the conditional probability, the outcome denoted by \(\Pr(Y=1\left|\mathbf{x}\right. )=\pi(\mathbf{x})\), the logit of the multiple logistic regression model is given by Equation (5). \[g(\mathbf{x})=\ln\left(\frac{\pi(\mathbf{x})}{1-\pi(\mathbf{x})}\right)=\beta _{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\cdots+\beta_{p}x_{p} \tag{5}\]  
Then, the logistic regression model based on conditional probability \((\pi(\mathbf{x}))\) is formulated by Equation (6). \[ \begin{array}{ll}\pi(\mathbf{x})&=\Pr(Y=1\left|\mathbf{x}\right.)\\ &=\frac{\varepsilon^{\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\cdots+\beta_{p} x_{p}}}{1+\varepsilon^{\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\cdots+\beta_{p} x_{p}}}=\frac{\varepsilon^{\beta(\mathbf{x})}}{1+\varepsilon^{\beta(\mathbf{x})}} \end{array}  \tag{6}\]  
Here, it is assumed that there was a sample of n observations of the pairs \((\mathbf{x}_{i},y_{i})\), where \(i=1,2,\cdots,n\). To fit the model in (6), the unknown parameter set of vector \(\beta^{T}\) and the \(p+1\) coefficients of each defined explanatory variable in Equation (7) were obtained. \[\beta^{T}=\left\{\beta_{0},\quad\beta_{1},\quad\ldots,\quad\beta_{p}\right\} \tag{7}\]  
Then, for the objective evaluation, the acceptance was discerned via outcome value of the model in (6) with respect to a critical value, 0.5, as shown in Equation (8). \[\text{Decision criteria}\left\{ \begin{array}{ll}\text{Accept}:&\pi(\mathbf{x })\geq 0.5\\ \text{Reject}:&\pi(\mathbf{x})<0.5\end{array} \right. \tag{8}\]  
Finally, the values of \(\beta^{T}\) were derived by using the maximum likelihood method [27; 28]. A receiver operating characteristics (ROC) curve was also utilized for visualizing, organizing, and selecting classifiers based on their performances [29].