## 4 Results and Discussion
### Linear Regression Model for Reliability Prediction
After three iterations, a model with only four significant variables, with no collinearity effect and with values of VIF sufficiently close to 1 is achieved, presented in Table 4.  
Table 5 summarizes the analysis, presenting the coefficients of determination, standard deviation and Mallow's Cp. The main evaluation criterion is the variation in R\({}^{2}\), with R\({}^{2}\)(pred) being the most important of the values, as it reflects the prediction capacity of the model. The Mallow's CP value is also important in the comparison between models because it evaluates the biases of certain subsets of predictors in the model, having as a goal a value close to the degrees of freedom of the model. The model obtained with four variables presents R\({}^{2}\) (pred) values that are significantly better than those of the other models and a value of 71.4%, which is considered good for the prediction application of a regression model. The equation obtained for the regression model for the reliability of ceramic plates is as follows:  
\[R(t)=-558+8.01\;PT101(t)-1.610\;KPV325(t)-82.0\;PT329(t)+1073\;PT129(t)\]  
The adjusted model had an R\({}^{2}\) of 80.7%, R\({}^{2}\)(aj) of 78.3% and R\({}^{2}\)(pred) of 71.4%. A qualitative and quantitative evaluation of the model led to the identification of the following factors that affect the reliability of ceramic plates:  
* PT101(t)--The loss of vacuum pressure in a vacuum pump. Lower negative vacuum pressures are related to the lower reliability of ceramic plates. The average vacuum pressure loss of 0.1 bar is related to 0.8% lower reliability or, alternatively, a 0.8% increase in the probability of failure. * KPV325(t)--The higher average second-cake-formation valve opening readings are related to lower reliability values of the ceramic plates. An increase in the average percentage of valve openings by 10% is related to a reliability reduction of approximately 16% or, alternatively, an increase in the probability of failure by 16%. * PT329(t)--Regarding the variables related to backwash pressure, the higher the average value measured before the bag filter, the lower the reliability of the ceramic plates. A 0.1-bar increase in backwash pressure before the use of the bag filter leads to a reliability reduction of 8.2% or, similarly, an increase in the probability of failure of 8.2%. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Term** & **Coef** & **Coef SE** & **T Value** & _p Value_ & **VIF** \\ \hline Constant & \(-558\) & \(533\) & \(-1.05\) & \(0.303\) & \\ PT101 & \(8.01\) & \(2.21\) & \(3.62\) & \(0.001\) & \(1.42\) \\ KPV325 & \(-1.61\) & \(0.187\) & \(-8.60\) & \(0.000\) & \(1.22\) \\ PT329 & \(-82.0\) & \(13.8\) & \(-5.96\) & \(0.000\) & \(1.22\) \\ PT129 & \(1073\) & \(409\) & \(2.63\) & \(0.013\) & \(1.53\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Predictor selection for regression model after 3 iterations. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Vars** & **R2** & **R2 (aj)** & **R2 (pred)** & **Mallow’s Cp** & **S** & **U129\_IIT** & **U009\_IIT** & **PT225** & **FT101** & **KPV325** & **FT329** & **PT129** & **FT129** \\ \hline
4 & 80.7 & 78.3 & 71.4 & 3.9 & 5.1743 & & & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \\
6 & 82 & 78.6 & 57.3 & 5.7 & 5.1454 & & & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) \\
8 & 82.4 & 77.6 & 31.4 & 9 & 5.2588 & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Analysis of multiple combinations of predictors for R(t) response. * PT129(t)--For the backwash pressure after bag filtration, higher values are associated with higher reliability. The 0.01-bar reduction in the average value of this pressure is related to 10.7% lower reliability or, alternatively, a 10.7% increase in the probability of failure.