# Statistical Methods in Bidding Decision Support for Construction Companies
## 3 Results and Discussion  
### LOG Model--The Model Using Regression Analysis  
Using logit regression, an attempt was made to estimate the qualitative variable \(Y\), also trying to explain which factors, with what strength and in what direction, affect the chance of a tender success (\(\gamma\)). The parameter estimates are summarized in Table 3.  
By analyzing the obtained results with the assumed significance level \(\alpha\) = 0.1, only two variables significantly affect the model: \(\mathrm{x_{3}}\)--contractual conditions and \(\mathrm{x_{6}}\)--need for work. However, the \(p\) value for the variables \(\mathrm{x_{12}}\)--time to prepare the offer and \(\mathrm{x_{15}}\)--the degree of difficulty of the works, are slightly higher than 0.1, so it was decided to include these variables and recalculate the model. The parameter estimates for the logistic regression model (with four explanatory variables) are summarized in Table 4.  
Finally, three variables were left (the non-significant variable \(\mathrm{x_{12}}\)--time to prepare an offer, was discarded) and recalculations were made. The parameter estimates for the logistic regression model (with three explanatory variables) are summarized in Table 5.  
The form of the proposed logit model (LOG model) is as follows:  
\[\hat{Y}_{i}=ln\frac{p_{i}}{1-p_{i}}=-0.9532\cdot x_{3}-2.2877\cdot x_{6}+0.6012 \cdot x_{15}+15.9217 \tag{12}\]  
This means that the probability \(p_{i}\) (that is, situation \(Y_{i}=1\)) is estimated as:  
\[\hat{p}_{i}=\frac{\exp(-0.9532\cdot x_{3}-2.2877\cdot x_{6}+0.6012\cdot x_{15}+ 15.9217)}{1+\exp(-0.9532\cdot x_{3}-2.2877\cdot x_{6}+0.6012\cdot x_{15}+15.9217)} \tag{13}\]  
Statistical verification of the logit model consisting in determining the degree of the model fitting the data and testing the statistical significance of the parameters was successful. The odds quotient is 9.62 and is higher than 1 which means that the classification  
significance level \(\alpha=0.1\), that only four variables significantly discriminate between groups (x\({}_{3}\), x\({}_{5}\), x\({}_{6}\), x\({}_{13}\)). The results for the model and the evaluation of all 15 input variables (step 0) are given in Table 6.  
By analyzing the obtained results with the assumed significance level \(\alpha=0.1\), only four variables (x\({}_{3}\); x\({}_{5}\); x\({}_{6}\); x\({}_{13}\)) discriminated significantly between groups. The model parameters are as follows: Wilks' Lambda = 0.41344, the corresponding \(F\) statistic (15.72) = 6.8100, and \(p<0.0000\). During the first step of the analysis, the variable x\({}_{2}\) was removed--the least significantly discriminating group. Subsequent steps (\(k=2\),..., 15) made it possible to select the most significant variables (Table 7). Finally, six input variables, x\({}_{1}\)--type of works, x\({}_{3}\)--contractual conditions, x\({}_{5}\)--project value, x\({}_{6}\)--need for work, x\({}_{13}\)--possible participation of subcontractors, x\({}_{15}\)--the degree of difficulty of the works, discriminate significantly between groups. The model parameters are as follows: Wilks' Lambda = 0.47047; the corresponding statistic \(F\) (6.81) = 15.195; \(p<0.0000\). It is worth noting that the smaller the value of Wilks' Lambda (from the range \(<\)0, 1>) the better the discriminating power the model has. In the analyzed example (0.47047), it is acceptable. Tolerance coefficient \(T_{k}\) determines the proportion of the variance of the variable \(x_{k}\) that is not explained by the variables in the model. If \(T_{k}\) coefficient takes a value smaller than the default 0.01, the variable is more than 99% redundant with other variables in the model. Entering variables with low tolerance coefficients into the model may cause its large inaccuracy. In the model under consideration, the \(T_{k}\) coefficients for the assumed variables exceed the value of 0.5.