## 4 Case Study and Results Discussion
### Analysis 2: Covariate Selection Sensitivity
In other words, covariates with a smaller CV were prioritized to remain in the model. Using this criteria, Diff, C\({}_{2}\)H\({}_{4}\), R.Die, and % Hum were the combination of features that did not have any correlation, as shown in Figure 8, and were considered for the final model. Considering all the analyses presented thus far, Table 15 displays all the models for comparison, sorted by the LL score. Case 0 represents the model with no covariates. Case A includes only C\({}_{2}\)H\({}_{4}\) as a covariate due to its significance in reducing the LL score, as indicated by the chi-squared test and Akaike Value. Case B is based on covariate selection using expert criteria. Case C comprises covariates showing high significance according to the log-likelihood ratio \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline  & **LL** & **-2 LL** & **No. Var.** & **G** & **df** & \(p\)**-Value** & **LRT** & **df** & \(p\)**-Value** & **Akaike** \\ \hline - & 763.9 & 1527.9 & 0 & & & & & & 1527.86 \\ CO\({}_{2}\) & 752.7 & 1505.3 & 1 & 22.53 & 1 & **0.00** & 31.45 & 15 & **0.01** & 1503.34 \\ Tint & 752.7 & 1505.3 & 2 & 0.00 & 1 & 0.99 & 8.92 & 14 & 0.84 & 1501.33 \\ Dift & 752.7 & 1505.3 & 3 & 0.00 & 1 & 0.99 & 8.92 & 13 & 0.78 & 1499.33 \\ R.Die & 752.7 & 1505.3 & 4 & 0.00 & 1 & 0.98 & 8.92 & 12 & 0.71 & 1497.33 \\ Text & 752.7 & 1505.3 & 5 & 0.00 & 1 & 1.00 & 8.92 & 11 & 0.63 & 1495.33 \\ C\({}_{2}\)H\({}_{6}\) & 752.7 & 1505.3 & 6 & 0.00 & 1 & 1.00 & 8.92 & 10 & 0.54 & 1493.33 \\ CH\({}_{4}\) & 752.3 & 1504.5 & 7 & 0.81 & 1 & 0.37 & 8.92 & 9 & 0.44 & 1490.52 \\ IFS & 752.3 & 1504.5 & 8 & 0.00 & 1 & 1.00 & 8.11 & 8 & 0.42 & 1488.52 \\ Dda & 752.3 & 1504.5 & 9 & 0.00 & 1 & 1.00 & 8.11 & 7 & 0.32 & 1486.52 \\ TGC & 752.1 & 1504.1 & 10 & 0.40 & 1 & 0.53 & 8.11 & 6 & 0.23 & 1484.12 \\ Co & 752.1 & 1504.1 & 11 & 0.00 & 1 & 1.00 & 7.71 & 5 & 0.17 & 1482.12 \\ GTF & 752.1 & 1504.1 & 12 & 0.02 & 1 & 0.88 & 7.71 & 4 & 0.10 & 1480.10 \\ TGC-CO & 752.0 & 1504.0 & 13 & 0.06 & 1 & 0.81 & 7.69 & 3 & 0.05 & 1478.04 \\ **\% Hum** & 752.0 & 1504.0 & 14 & 0.01 & 1 & 0.92 & 7.63 & 2 & **0.02** & 1476.03 \\ **C\({}_{2}\)H\({}_{4}\)** & 748.2 & 1496.4 & 15 & 7.62 & 1 & **0.01** & 7.62 & 1 & **0.01** & **1466.41** \\ \hline \end{tabular}
\end{table} 
Table 14: LL scores, chi-squared test, log-likelihood ratio test, and Akaike values when adding a covariate to the model to check significance. test. Case D includes covariates with substantial weights from Table 10. Case E represents covariates identified through the correlation and CV analysis. Lastly, case F is the model including all covariates (Dda, Text, IFS, CH\({}_{4}\), GTF, CO\({}_{2}\), Co, and TGC-CO are considered but not displayed for table readability). Figure 8: Pearson correlation \(p\)-values of Diff, C\({}_{2}\)H\({}_{4}\), R.die, and % Hum covariates. Figure 7: Pearson correlation \(p\)-values of all covariates. The table shows that all the models we compared (A-F) had much lower LL scores than case 0. This means they fit the data better and could potentially make more accurate predictions. However, there was a trade-off. While adding extra features to the models improved the LL score, it also tended to slightly increase the \(\beta\) values and decrease the \(\eta\) values. Models A, C, and E were particularly interesting. They had the lowest AIC and BIC scores. These scores could be explained by AIC and BIC penalizing models for being too complex. It is important to note that even though model A had the absolute best AIC and BIC scores, its LL score was slightly higher than models C and E. This highlights the challenge of finding an accurate model that is not overly complex.