## 4 Case Study and Results Discussion
### Analysis 2: Covariate Selection Sensitivity
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**LL**} & \multirow{2}{*}{**Beta**} & \multirow{2}{*}{**Eta**} & \multirow{2}{*}{**Data**} & \multirow{2}{*}{**Tint**} & \multirow{2}{*}{**Text**} & \multirow{2}{*}{**Diff**} & \multirow{2}{*}{**IFS**} & \multirow{2}{*}{**C2H4**} & \multirow{2}{*}{**C2H6**} & \multirow{2}{*}{**CH4**} & \multirow{2}{*}{**GTF**} & \multirow{2}{*}{**CO2**} & \multirow{2}{*}{**Co**} & \multirow{2}{*}{**TGC**} & \multirow{2}{*}{**TGC**} & \multirow{2}{*}{**TGC**} & \multirow{2}{*}{**CO**} \\ \hline
752.7 & 2.5 & 170,543 & 0.00 & & & & & & & & & & \\
752.7 & 2.5 & 170,542 & & 0.13 & & & & & & & & & \\
752.7 & 2.5 & 170,542 & & 0.00 & & & & & & & & \\
752.7 & 2.5 & 170,542 & & & 0.16 & & & & & & & \\
752.7 & 2.5 & 170,543 & & & 0.00 & & & & & & & \\
748.5 & **2.8** & **150,534** & & & & & **0.79** & & & & & \\
752.7 & 2.5 & 170,542 & & & & & 0.00 & & & & & \\
752.3 & 2.6 & 156,825 & & & & & & 0.99 & & & & \\
752.2 & 2.6 & 158,074 & & & & & & 0.76 & & & & \\
752.7 & 2.5 & 170,543 & & & & & & & 0.13 & & \\
752.7 & 2.5 & 170,542 & & & & & & & & 0.00 & & \\
752.2 & 2.6 & 157,132 & & & & & & & & & 0.88 & \\
752.7 & 2.5 & 170,543 & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Log-likelihood scores and covariate weight values when using only one covariate. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & **Tint** & **Diff** & **C2H4** & **C2H6** & **CH4** & **GTF** & **TGC** & **TGC** & **TGC** & **R.Die** & **\%** \\ \hline Mean & 0.0551 & 0.0467 & 0.0629 & 0.0009 & 0.0011 & 0.0010 & 0.0056 & 0.0011 & 7.9905 & 0.3413 \\ STDV & 0.123 & 0.061 & 0.033 & 0.001 & 0.003 & 0.002 & 0.008 & 0.003 & 5.579 & 0.259 \\ CV & 223 & 130 & **51** & 141 & 223 & 223 & 148 & 223 & **69** & **76** \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The mean, standard deviation, and coefficient of variation for the covariate weights when utilizing 5 folds. To check if these results were significant, three tests were taken into account to assess the significance of the covariates on the reduction in the log-likelihood score: the chi-squared test for the difference in the LL scores, the LRT, and Akaike values, as shown in Table 14.  
The principal findings of this test indicated that adding any covariates to the model led to a significant improvement in the reduction in the LL score. Furthermore, the chi-squared test revealed that the covariate producing significant changes in the model was C\({}_{2}\)H\({}_{4}\) (\(p\leq 0.05\)), a finding supported by the lowest Akaike value when this covariate was introduced in the model. Additionally, the LRT test indicated that % Hum could also introduce significant changes (\(p\leq 0.05\)). Consequently, models considering only C\({}_{2}\)H\({}_{4}\) and both C\({}_{2}\)H\({}_{4}\) and % Hum were used for further testing. Continuing with the covariate selection process, a correlation analysis was applied to identify and exclude correlated features that contributed minimally to the model. Figure 7 presents the \(p\)-values from the Pearson correlation test, where a significance level below 5% suggests correlations between features (\(p\)-value \(\leq 0.05\)). The results revealed that multiple covariates exhibited a high degree of correlation, which could be addressed by removing certain features. The key question was which features should be removed. This decision was crucial since various combinations of feature removal were possible. For instance, consider the temperature covariates: Tint and Diff were correlated because Diff represented the difference between Tint and Text. One possible solution could be to remove these two covariates and use only Text. However, based on the previous analysis presented in Table 10, both Tint and Diff had higher covariate weights than Text, making this selection less advisable. Alternatively, should Tint or Diff be removed from the model? To address this and similar cases objectively, features with high CV were removed until no correlation was detected.