## 3 Results and Discussion
### Statistical Modeling  
#### 3.3.1 Fitted Models  
Although the main output target of the statistical analysis was the maximum peak strength of the unconfined compression strength test (\(q_{u}\)) and its dependence on \(t_{FA}\), \(t_{L}\), and \(c_{SH}\), other parameters were analyzed to find any possible relationships. Among the analyzed parameters are the actual concentration of the mixture (\(c_{A}\), given by Equation (1)), the variation mass of the specimens on the molding and testing days (\(\Delta m\)), and the variation in environmental temperature on the molding and testing days (\(\Delta t\)). Based on an initial statistical analysis (hypothesis testing), the following conclusions were drawn:  
* The actual concentration (\(c_{A}\)) of the mixture is highly dependent on the variable \(t_{L}\) (r = 0.841, \(p<0.1\%\)); therefore, its inclusion as an input parameter to the statistical model together with \(t_{L}\) is not possible;
* There is no significant correlation between \(q_{u}\) values and the mass variation (\(\Delta m\)) during curing. The \(p\)-value associated with the hypothesis of no correlation between these parameters is equal to 22.1%; therefore, there is no evidence of correlation between \(q_{u}\) and \(\Delta m\);
* The factor that most contributes to the maximum peak strength response is \(t_{FA}\), followed at some distance by the \(c_{SH}\) and the \(t_{L}\). The \(t_{L}\) is the least important factor of the three in explaining the response, although it is still significant (\(p\)-value = 4.9%).  
A significance level of 5% was considered in all statistical analyses. A subsequent statistical regression analysis led to two statistical models which can be used to predict the strength response. They are denoted here as Model 1 and Model 2. A logarithm transformation to the response data is needed to fit a regression model properly.  
Figure 13: Unconfined compression strength of all mixtures tested after 7 days of curing, along with the strength of the untreated tailings compacted at 90%, 95%, and 100% degree of compaction (Standard Proctor).  
In Model 1, the three input variables intervened (\(t_{FA}\), \(t_{L}\), and \(c_{SH}\)), as expressed in Equation (6). The obtained model is loglinear, as all interactions between variables and non-linearities were not significant (\(p\)-value \(>\) 10% for all interactions analyzed). Figure 14 shows the growth trend of \(q_{u}\) according to Model 1, based on the variables fly ash content (\(t_{FA}\)) and SH concentration (\(c_{SH}\)), which are the most significant variables driving \(q_{u}\). In this case, the variable \(t_{L}\) was fixed at 0.17. Higher strengths are in the upper right corner: there is an increase in \(q_{u}\) with both input variables.  
\[\ln(q_{u})=0.728+1.67\ t_{FA}+4.19\ t_{L}+0.0187\ c_{SH} \tag{6}\]  
Model 2 has the advantage of using only two variables, as expressed in Equation (7). Figure 15 shows the growth trend of \(q_{u}\) based on the variables fly ash content (\(t_{FA}\)) and actual sodium hydroxide concentration (\(c_{A}\)) calculated by Equation (1).  
\[\ln(q_{u})=1.390+1.67\ t_{FA}+0.131\ c_{A} \tag{7}\]  
Figure 14: The maximum peak strength as a function of \(t_{FA}\) and \(c_{SH}\).  
Figure 15: The maximum peak strength as a function of \(t_{FA}\) and \(c_{A}\).