## 3 Results and Discussion
#### 3.1.3 Model Validation  
Based on post hoc analysis using [PERSON]'s test, HM\({}_{1-\delta}\), was chosen for validation using independent data sets of different isolates of _S. aureus_ and _E. coli_ obtained at different time periods and grown similarly in milk at a temperature range of 15-35 \({}^{\circ}\)C (\(n_{\rm Sa}=164\)) and 10-37 \({}^{\circ}\)C (\(n_{Ec}=134\)). The calculated values of \(B_{f}\) and \(A_{f}\) through the above-mentioned data sets are summarized in Table 6. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **HM\({}_{\bf 2-5}\)\({}^{\bf Sa}\)** & **HM\({}_{\bf 2-5}\)\({}^{\bf Ec}\)** & **HM\({}_{\bf 2-5}\)\({}^{\bf LAB}\)** \\ \hline \(SSE\) [log CFU mL\({}^{-1}\)]\({}^{2}\) & 14.587 & 9.528 & 6.786 \\ \(RMSE\) [log CFU mL\({}^{-1}\)] & 0.279 & 0.230 & 0.201 \\ \(R_{\it def}\)\({}^{2}\) & 0.979 & 0.986 & 0.988 \\ \hline \end{tabular}
\end{table} 
Table 4: Goodness-of-fit indices for the growth prediction of single cultures of _S. aureus_ (Sa), _E. coli_, (Ec) and Fresco (LAB) in milk by tertiary one-step HM\({}_{1-\delta}\). \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **HM\({}_{\bf 2-5}\)\({}^{\bf Sa}\)** & **HM\({}_{\bf 2-5}\)\({}^{\bf Ec}\)** & **HM\({}_{\bf 2-5}\)\({}^{\bf LAB}\)** \\ \hline \(x_{0,10}\)\({}^{1}\) [log CFU mL\({}^{-1}\)] & 3.74 \(\pm\) 0.05 & 3.05 \(\pm\) 0.03 & 4.33 \(\pm\) 0.02 \\ \(x_{0,12}\) [log CFU mL\({}^{-1}\)] & 2.14 \(\pm\) 0.03 & 3.27 \(\pm\) 0.03 & 4.29 \(\pm\) 0.04 \\ \(x_{0,15}\) [log CFU mL\({}^{-1}\)] & 3.90 \(\pm\) 0.05 & 2.77 \(\pm\) 0.08 & 4.35 \(\pm\) 0.03 \\ \(x_{0,18}\) [log CFU mL\({}^{-1}\)] & 3.26 \(\pm\) 0.02 & 3.21 \(\pm\) 0.03 & 4.30 \(\pm\) 0.03 \\ \(x_{0,21}\) [log CFU mL\({}^{-1}\)] & 3.51 \(\pm\) 0.03 & 3.25 \(\pm\) 0.03 & 4.58 \(\pm\) 0.04 \\ \(x_{0,25}\) [log CFU mL\({}^{-1}\)] & 3.15 \(\pm\) 0.03 & 3.13 \(\pm\) 0.03 & 5.62 \(\pm\) 0.04 \\ \(x_{0,20}\) [log CFU mL\({}^{-1}\)] & 2.72 \(\pm\) 0.05 & 3.20 \(\pm\) 0.02 & 3.91 \(\pm\) 0.04 \\ \(x_{0,35}\) [log CFU mL\({}^{-1}\)] & 3.71 \(\pm\) 0.04 & 3.35 \(\pm\) 0.04 & 4.26 \(\pm\) 0.03 \\ \(x_{0,37}\) [log CFU mL\({}^{-1}\)] & 3.01 \(\pm\) 0.08 & 3.04 \(\pm\) 0.04 & 4.32 \(\pm\) 0.07 \\ \(x_{\rm max}\)\({}^{2}\) [log CFU mL\({}^{-1}\)] & 8.08 \(\pm\) 0.02 & 8.39 \(\pm\) 0.02 & 9.11 \(\pm\) 0.01 \\ \hline \(b_{\rm T}\) [\({}^{\circ}\)C\({}^{-1}\) h\({}^{-0.5}\)] & 0.0409 \(\pm\) 0.0001 & 0.0421 \(\pm\) 0.0002 & 0.0384 \(\pm\) 0.0002 \\ \(T_{\rm min}\) [\({}^{\circ}\)C] & 5.02 \(\pm\) 0.01 & 4.16 \(\pm\) 0.04 & 1.11 \(\pm\) 0.07 \\ \(b_{\rm\lambda}\) [\({}^{\circ}\)C\({}^{-1}\) h\({}^{-0.5}\)] & 0.0302 \(\pm\) 0.0002 & 0.0493 \(\pm\) 0.0001 & 0.0343 \(\pm\) 0.0009 \\ \hline \end{tabular} \({}^{\dagger}\)\(x_{0}\) initial concentration of microorganisms (\(x_{0}=\) log \(N_{0}\)); the second numerical subscript is the temperature. \({}^{2}\)\(b_{\rm T}\) and \(b_{\rm\lambda}\) are regression coefficients, \(T_{\rm min}\) is theoretical minimum temperature for growth.  
\end{table} 
Table 3: Parameters of HM\({}_{1-\delta}\) with a 95% confidence limit determined by the one-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 c} \hline
**Parameter** & **HM\({}_{\bf 2-1}\)\({}^{\bf Sa}\)** & **HM\({}_{\bf 2-1}\)\({}^{\bf Ec}\)** & **HM\({}_{\bf 2-1}\)\({}^{\bf LAB}\)** \\ \hline \(CRC\) & 0.0800 & 0.0458 & 0.0264 \\ \(\Delta SSE_{m}\) & 0.1244 & 0.0631 & 0.0377 \\ \hline \end{tabular} \(\overline{\text{CR}_{T}}\) is the critical range for [PERSON]’s test, \(\Delta\text{SSE}_{m}\) is statistical difference between the modeling of one- and two-step approaches.  
\end{table} 
Table 5: Critical range of the \(SSE\) means and the differences between each pair of \(SSE\) means by comparing the one- and two-step approaches. A validation factor \(B_{f}\) equal to one indicates that observations are equally distributed above and below predictions, producing a perfect agreement. In our validation study, the \(B_{f}\) for \(S\).