## 4 Test Results
### Discriminative Function Analysis  
The last stage of multidimensional statistical inference was the discriminatory analysis. Its main purpose was to identify the mathematical functions that allow for the best discrimination between the clusters as described in the preceding paragraph. Discriminant functions answer the question which variables are the best predictors of group membership. In addition, the discriminatory analysis was to verify the validity of the adopted groups. In the first place, discriminatory analysis required the canonical analysis to separate the present clusters by mathematical functions [40][39], starting from the review of variables in the model. The results of the discriminatory analysis are summarized in Table 5.  
The discriminant analysis was performed on transformed variables obtained during the search for principal components (PCA). The results shown in Table 5 indicate that the parameters expressed by variable Z2 (the lowest value of partial Willks' Lambda), responsible for the structural properties of the recycled mixture, provided the most overall discrimination (\(p<0.05\)). The second most discrimination was provided by variable Z1 (rheological properties), responsible for MCAS viscoelastic properties. As for variable Z3, its effect on the group discrimination was marginal. Variable Z3 bonds moisture resistance parameters and a portion of the ITS variation. Its statistically insignificant contribution to the discrimination was due to the low diversity of moisture resistance results across the dataset. With the initial knowledge of the contribution of the given variables on the discrimination, it was necessary to determine the canonical coefficients, on the basis of which the discriminant functions (classification functions) were computed. Arbitrarily, two significant functions were used. The predictive validity of the functions was assessed in the \(\chi^{2}\) test and summarized in Table 6.  
The results in Table 6 indicate that both discriminant functions were statistically significant (\(p<0.05\)) and thus could be used for further statistical inference. The raw values of the canonical functions required for the final determination of discriminant functions, important for the classification of other cases of recyclable MCAS mixes with waste materials, are shown in Table 7.  
It follows from the results above that the first canonical function (V_1) explained about 74% of the variation whereas the second function (V_2) explained 26% of discrimination. The canonical discriminant function had the form (2):  
\[\left\{ \begin{array}{l}\text{V\_1}=0.032\times\text{Z1}+0.695\times\text{Z2 }-0.019\times\text{Z3}-0.395\times\text{Z4}\\ \text{V\_2}=-0.3\times\text{Z1}+0.07\times\text{Z2}-0.112\times\text{Z3}-0.25 \times\text{Z4}\end{array} \right. \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Removed Roots**} & \multicolumn{6}{c}{**Chi-Squared Tests of Roots**} \\ \cline{2-7}  & **Eigenvalue** & **Canonical R** & **Wilks’ Lambda** & \(\chi^{2}\) & **df** & \(p\) \\ \hline
0 & 3.029339 & 0.867076 & 0.120106 & 179.0874 & 8 & 0.00000 \\
1 & 1.066332 & 0.718367 & 0.483949 & 61.3280 & 3 & 0.00000 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Chi-squared test results for subsequent roots. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(N=89\)} & \multicolumn{6}{c}{**Summary of Discriminant Function Analysis F (8.166) = 39.124 \(p<0.0000\)**} \\ \cline{2-7}  & **Wilks’ Lambda** & **Partial Wilks’ Lambda** & **F to Remove (2.83)** & \(p\) & **Tolerance** & **1-Tolerance (R2)** \\ \hline Z1 & 0.239562 & 0.501359 & 41.2750 & 0.00000 & 0.959353 & 0.040647 \\ Z2 & 0.460546 & 0.260792 & 117.6309 & 0.00000 & 0.893245 & 0.106755 \\ Z3 & 0.121989 & 0.984565 & 0.6506 & 0.524386 & 0.991939 & 0.008061 \\ Z4 & 0.150245 & 0.799401 The discriminant (classification) functions allowed direct classification of each mixture case and its projection to the results space used in the experiment.