## 3 Results and Discussion
#### 3.1.2 One-Step Data Analysis and Evaluation of Model Performance
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **HM\({}_{2+\delta}\)\({}^{\mathbf{Sa}}\)** & **HM\({}_{2+\delta}\)\({}^{\mathbf{Ec}}\)** & **HM\({}_{2+\delta}\)\({}^{\mathbf{LAB}}\)** \\ \hline \(x_{0,10}\)\({}^{1}\) [\(\log\) CFU mL\({}^{-1}\)] & 3.46 \(\pm\) 0.09 & 2.99 \(\pm\) 0.01 & 4.34 \(\pm\) 0.10 \\ \(x_{0,12}\) [\(\log\) CFU mL\({}^{-1}\)] & 2.44 \(\pm\) 0.18 & 3.08 \(\pm\) 0.30 & 4.21 \(\pm\) 0.20 \\ \(x_{0,15}\) [\(\log\) CFU mL\({}^{-1}\)] & 4.15 \(\pm\) 0.26 & 2.14 \(\pm\) 0.39 & 4.38 \(\pm\) 0.14 \\ \(x_{0,18}\) [\(\log\) CFU mL\({}^{-1}\)] & 3.27 \(\pm\) 0.12 & 3.45 \(\pm\) 0.15 & 4.38 \(\pm\) 0.10 \\ \(x_{0,21}\) [\(\log\) CFU mL\({}^{-1}\)] & 3.45 \(\pm\) 0.13 & 3.09 \(\pm\) 0.19 & 4.19 \(\pm\) 0.22 \\ \(x_{0,25}\) [\(\log\) CFU mL\({}^{-1}\)] & 3.22 \(\pm\) 0.12 & 3.16 \(\pm\) 0.21 & 5.42 \(\pm\) 0.17 \\ \(x_{0,20}\) [\(\log\) CFU mL\({}^{-1}\)] & 3.33 \(\pm\) 0.12 & 3.30 \(\pm\) 0.14 & 4.24 \(\pm\) 0.09 \\ \(x_{0,38}\) [\(\log\) CFU mL\({}^{-1}\)] & 3.38 \(\pm\) 0.19 & 3.35 \(\pm\) 0.12 & 4.23 \(\pm\) 0.20 \\ \(x_{0,37}\) [\(\log\) CFU mL\({}^{-1}\)] & 2.75 \(\pm\) 0.29 & 3.24 \(\pm\) 0.30 & 4.40 \(\pm\) 0.13 \\ \(x_{\max}\)\({}^{2}\) [\(\log\) CFU mL\({}^{-1}\)] & 8.12 & 8.36 & 9.19 \\ \hline \(b_{T}\) [\({}^{\circ}\)C\({}^{-1}\) h\({}^{-0.5}\)] & 0.0441 \(\pm\) 0.0032 & 0.0400 \(\pm\) 0.0033 & 0.0362 \(\pm\) 0.0031 \\ \(T_{\min}\) [\({}^{\circ}\)C] & 5.67 \(\pm\) 1.23 & 3.09 \(\pm\) 1.75 & \(-\)0.11 \(\pm\) 2.27 \\ \(b_{\lambda}\) [\({}^{\circ}\)C\({}^{-1}\) h\({}^{-0.5}\)] & 0.0297 \(\pm\) 0.0057 & 0.0507 \(\pm\) 0.0134 & 0.0369 \(\pm\) 0.0081 \\ \hline \hline \end{tabular} \({}^{\dagger}\)\(x_{0}\) initial concentration of microorganisms (\(x_{0}\) = log \(N_{0}\)); the second numerical subscript is the temperature. \({}^{2}\)\(x_{\max}\) is the average value of the maximum bacterial counts. The corresponding standard deviations for \(S\). _aureus_, \(E\). _coli_, and LAB Fresco were 0.21, 0.21, and 0.23 log CFU mL\({}^{-1}\), respectively. \(b_{T}\) and \(b_{\lambda}\) are regression coefficient, \(T_{\min}\) is theoretical minimum temperature for growth.  
\end{table} 
Table 1: Parameters of HM\({}_{2+\delta}\) with the 95% confidence limit determined by the two-step analysis method used for the growth prediction of single cultures of \(S\). _aureus_ (Sa), _E. coli_ (Ec), and Fresco (LAB) in milk. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **HM\({}_{2+\delta}\)\({}^{\mathbf{Sa}}\)** & **HM\({}_{2+\delta}\)\({}^{\mathbf{Ec}}\)** & **HM\({}_{2+\delta}\)\({}^{\mathbf{LAB}}\)** \\ \hline \(SSE\) [\(\log\) CFU mL\({}^{-1}\)]\({}^{2}\) & 37.982 & 20.887 & 13.127 \\ \(RMSE\) [\(\log\) CFU mL\({}^{-1}\)] & 0.449 & 0.341 & 0.280 \\ \(R_{\mathit{adj}}\)\({}^{2}\) & 0.944 & 0.969 & 0.977 \\ \hline \hline \end{tabular} \({}^{\dagger}\)Number of data points: \(n_{\text{Sa}}\) = 188; \(n_{\text{Ec}}\) = 180; \(n_{\text{LAB}}\) = 168. \(SSE\) is global sum of square, \(RMSE\) is root mean square error, and \(R_{\mathit{adj}}\)\({}^{2}\) is adjusted determination coefficient.  
\end{table} 
Table 2: Goodness-of-fit indices of the tertiary HM\({}_{2-\delta}\) constructed through two-step data analysis for growth prediction of single cultures of \(S\). _aureus_ (Sa), _E. coli_ (Ec), and Fresco (LAB) in milk. The statistical evaluation _RMSE_, as shown in Table 4, are one-third lower on average for the one-step approach for single bacterial populations compared to the two-step approach. Other indices also support the finding that the one-step approach based on primary HM is superior in the prediction of the growth of all three bacterial populations in this study compared to the two-step modeling approach. For example, statistical differences between the modeling of one- and two-step approaches (\(\Delta SSE_{m}\)) for the tertiary HMs were evaluated by post hoc analysis using [PERSON]'s test, with the results presented in Table 5. The means \(SSE\) (\(SSE_{m}=SSE/n\)) were regarded as statistically significant at the 0.01 level of significance (\(p\)-value < 0.01).