# A Guideline for Implementing a Robust Optimization of a Complex Multi-Stage Manufacturing Process
## 4 Statistical Modeling and Non-Destructive Testing Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Symbol** & **Estimate** & **St. Error** & \(p\)**-Value (\(t\) Test)** \\ \hline Intercept & \(\beta_{0}\) & 373.19 & 40.33 & \textless{}0.0001 \\ Worker-St1-\#1 & \(\beta_{11}\) & 1.84 & 17.88 & 0.9194 \\ Worker-St1-\#2 & \(\beta_{12}\) & 6.30 & 13.08 & 0.6364 \\ Worker-St1-\#3 & \(\beta_{13}\) & 0.0 & & \\ Worker-St9-\#1 & \(\beta_{21}\) & \(-19.57\) & 20.51 & 0.3535 \\ Worker-St9-\#2 & \(\beta_{22}\) & \(-29.87\) & 39.17 & 0.4561 \\ Worker-St9-\#3 & \(\beta_{23}\) & \(-54.51\) & 26.77 & 0.0576 \\ Worker-St9-\#4 & \(\beta_{24}\) & 0.0 & & \\ Worker-St11-\#1 & \(\beta_{31}\) & \(-41.22\) & 29.42 & 0.1791 \\ Worker-St11-\#2 & \(\beta_{32}\) & \(-18.45\) & 27.47 & 0.5109 \\ Worker-St11-\#3 & \(\beta_{33}\) & 3.49 & 24.15 & 0.8867 \\ Worker-St11-\#4 & \(\beta_{34}\) & 0.0 & & \\ Dicing-Saw-1 & \(\beta_{41}\) & \(-11.53\) & 6.88 & 0.1117 \\ Dicing-Saw-2 & \(\beta_{42}\) & 0.0 & & \\ \#Blade-usage-1 & \(\beta_{51}\) & 5.38 & 10.15 & 0.6031 \\ \#Blade-usage-2 & \(\beta_{52}\) & \(-7.11\) & 10.56 & 0.5098 \\ \#Blade-usage-3 & \(\beta_{53}\) & \(-9.42\) & 9.59 & 0.3397 \\ \#Blade-usage-4 & \(\beta_{54}\) & 5.89 & 9.66 & 0.5502 \\ \#Blade-usage-5 & \(\beta_{55}\) & 0.0 & & \\ \#days-St2-3 & \(\beta_{6}\) & 0.97 & 4.94 & 0.8466 \\ \#days-St5-6 & \(\beta_{7}\) & 0.13 & 7.15 & 0.9858 \\ \#days-St8-9 & \(\beta_{8}\) & \(-0.59\) & 3.14 & 0.8513 \\ \#days-St10-11 & \(\beta_{9}\) & 11.67 & 5.94 & 0.0660 \\ Tm-Flateness- & \(\beta_{10}\) & 4.30 & 3.28 & 0.2072 \\ St5 & \(\beta_{11,1}\) & \(-28.32\) & 33.45 & 0.4090 \\ Kapton\({}^{\circledR}\)-St4 & \(\beta_{11,2}\) & 0.0 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model results. Response Y = Sensitivity; Formula (5). Table 1, standard errors are quite large, and this implies that some estimated coefficients, even though related to main process variables, were not significant. In addition, the statistical model (Formula (5)) includes both quantitative and categorical variables, therefore the model estimates (Table 1) are reported by considering the number of levels for each categorical variable, (e.g., the number of coefficient estimates are equal to the number of degrees of freedom of each variable). Thus, the estimation is carried out through the cornered point _criterium_, by which a categorical level is assumed as the reference level, and it is posed to \"0.0\" (Table 1). The second statistical model, applied in this study, is the logit model (Formula (4)). In this specific situation, the response binary variable Y is defined considering the first response variable within the process, namely the measurement process implemented during the manufacturing process, and located at stage #15, that consists in the capacitance test. A lower or a greater mean capacitance out of the specification (350-400) pF implies the transducer is a scrap (FAIL = 1). In case of compliance, the transducer passes to the manufacturing of the next stage #16 (PASS = 0). Therefore, when considering the binary response variable Capacitance PASS/FAIL (performed at stage #15), the estimated logistic regression model for the compliance of the us probe is as the following:  
\[ \begin{split}\text{Ybinary}=\theta_{0}+\theta_{1t}\text{:}X_{1}+ \theta_{2i}\text{:}X_{2}+\theta_{3i}\text{:}X_{3}+\theta_{4j}\text{:}X_{4}+ \theta_{5}\text{:}X_{5}+\theta_{6}\text{:}X_{6}+\theta_{7}\text{:}X_{7j}\\ \text{j}=1,2,3,1=1,2,3,4,t=1,2,3,4,5;\end{split}  \tag{6}\]  
The likelihood ratio test obtained for the estimated logit model (6) is equal to 21.33 (14 df), \(p\)-value 0.0933. This result allows for rejecting the null hypothesis that can be expressed through: \"all the estimated effects are null or negligible\".