# Statistical Methods in Bidding Decision Support for Construction Companies
## 2 Materials and Methods
### Regression Analysis Model  
The main task of the qualitative decision-making model will be to determine the probability of the contractor's success in the tender (winning) and to identify variables that significantly affect the outcome of the tender. A binomial (dichotomous) model is sought in which the explanatory variable \(Y\) is quantified by a zero-one variable. It takes two possible variants described by the codes \"1\"--W (win) and \"0\"--L (loss). If \(p_{i}\) is the probability of the event \(Y_{i}\) = 1, then \(1-p_{i}\) is the probability of the event \(Y_{i}\) = 0. The expected value of the variable \(Y_{i}\) is [58; 59]:  
\[E(Y_{i})=1\cdot p_{i}+0\cdot(1-p_{i})=p_{i} \tag{1}\]  
In binomial models, it is assumed that pi is a function of the vector of values of the explanatory variables \(x_{i}\) for the \(i\)-th object and the parameter vector \(\beta\)[58; 59]:  
\[P_{i}=P(y_{i}=1)=F\left(x_{i}^{T}\beta\right) \tag{2}\]  
Depending on the type of F-function, different types of models are distinguished [60]: a linear probability model, logit model, and probit model. Using the simplest of the binomial models--the linear probability model--has many negative consequences described in the literature [58; 61]. Probit and logit models, on the other hand, as indicated by some authors [60], are similar to each other and in practice one of them is used. Therefore, the search for a binomial model for the phenomenon in question was limited to a logistic regression model. The general form of the logit model is as follows [58; 59]:  
\[Y_{i}^{*}=ln\frac{p_{i}}{1-p_{i}}=\beta_{0}\alpha_{0}+\beta_{1}X_{1i}+\beta_{2 }xX_{2i}+\cdots+\beta_{k}X_{ki}+u_{i} \tag{3}\]  
where:  
\(\beta_{j}\)--structural model parameters,  
\(u_{i}\)--random component,  
\(ln\frac{p_{i}}{1-p_{i}}\)--logit,  
\(Y_{i}^{*}\)--unobservable qualitative variable,  
\(X_{ji}\)--the values of the explanatory variables of the model,  
\(p_{i}\)--the probability of taking the value \"1\" by the dependent variable \(Y_{i}\) calculated from the logistic distribution density function.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{
\begin{tabular}{c} **Record** \\ **(Evaluated** \\ **Tender)** \\ \end{tabular} } & \multicolumn{10}{c}{**Factors**} & \multicolumn{10}{c}{**Result**} \\ \hline i & x\({}_{1}\) & x\({}_{2}\) & x\({}_{3}\) & x\({}_{4}\) & x\({}_{5}\) & x\({}_{6}\) & x\({}_{7}\) & x\({}_{8}\) & x\({}_{9}\) & x\({}_{10}\) & x\({}_{11}\) & x\({}_{12}\) & x\({}_{13}\) & x\({}_{14}\) & x\({}  
\[p_{i}=\frac{e^{x_{i}^{i}\beta}}{1+e^{X_{i}^{i}\beta}}=\frac{1}{1+e^{-X_{i}^{i} \beta}}=\frac{1}{1+e^{-(\beta_{0}+\beta_{1}X_{1i}+\beta_{2}X_{2i}+\dots+\beta_{ k}X_{ki})}} \tag{4}\]  
Unobservable variable \(Y_{i}^{*}\) is defined as a latent variable, as one can observe only the binary variable \(Y_{i}\) in the form:  
\[Y_{i}=\left\{\begin{array}{ll}1;\ Y_{i}^{*}>0\\ 0;\ Y_{i}^{*}\leq 0\end{array}\right. \tag{5}\]  
Logit according to [53], denotes the odds ratio of accepting to not accepting the value \"1\" for the variable \(Y_{i}\). It takes the value zero if \(p_{i}=0.5\). In the case when \(p_{i}<0.5\), the odds ratio takes a negative value, and when \(p_{i}>0.5\), a positive one.