## 4 Results and Discussion
### Regression model  
Shelf lives of experimental harvests of living lettuce were regressed with _t_(1), _t_(2), _t_(3), _t_(4) and _t_(5). The regression models of living lettuce (Table 2) pointed to the critical importance of low temperatures on the 1 st day after harvest. Each 1 degree-day on the 1 st day of storage reduced shelf life by 0.22 days (Pr (\(>\)\(|\)t\(|\)) of 0.0227). The same number of input nodes used to construct regression models of butter lettuce indicated that the degree-day of the 1 st day positively affected, and those of the 2 nd and 5 th days negatively affected the shelf lives (Table 2). It was our intention to use regression to compare and confirm the results of 2-stage NN models. It was unexpected to observe that high temperatures on day 1 benefited the RSL of butter lettuce (Table 2). The reason is unknown and needs to be studied further. Inspection of quantile-quantile plot of S-PLUS indicated the errors (residuals) are normal. Figure 4: A 2-stage experimental model was validated with a commercial harvest of 26 cases of living lettuce. The predicted errors were calculated by specifying the future average temperature (_fa_) as 21°C. \begin{table}
\begin{tabular}{|l|l|l|l|l|} \hline Product type & & Value & Std. Error & Pr (\(>\)|t|) \\ \hline Living\({}^{\rm Z}\) & (Intercept) & 19.3833 & 0.8117 & 0.0000 \\ \hline  & \(t(1)^{\rm Y}\) & -0.2217 & 0.0983 & 0.0277 \\ \hline  & \(t(2)\) & -0.1555 & 0.1188 & 0.1956 \\ \hline  & \(t(3)\) & 0.0120 & 0.1602 & 0.9408 \\ \hline  & \(t(4)\) & -0.3431 & 0.1772 & 0.0575 \\ \hline  & \(t(5)\) & -0.2008 & 0.1435 & 0.1671 \\ \hline  & \({\rm R}^{\rm 2}\ =\ 0.64\) & & & \\  & (n=66) & & & \\ \hline  & & & & \\ \hline Butter\({}^{\rm X}\) & (Intercept) & 12.2640 & 0.5353 & 0.0000 \\ \hline  & \(t(1)\) & 0.6149 & 0.2505 & 0.0193 \\ \hline  & \(t(2)\) & -1.1463 & 0.2655 & 0.0002 \\ \hline  & \(t(3)\) & 0.3851 & 0.5507 & 0.4898 \\ \hline  & \(t(4)\) & 1.6504 & 1.1373 & 0.1571 \\ \hline  & \(t(5)\) & -2.1689 & 0.9530 & 0.0302 \\ \hline  & \({\rm R}^{\rm 2}\ =\ 0.75\) & & & \\  & (n=38) & & & \\ \hline  & & & & \\ \hline \end{tabular} \({}^{\rm Z}\) Living lettuce: Individually wrapped lettuce with roots; trained with experimental data (\({\rm R}^{\rm 2}\) = 0.64, n = 66); validated with commercial data (\({\rm R}^{\rm 2}\) = 0.43, n = 26).  
\({}^{\rm Y}\)\(t(1)\), \(t(2)\), \(t(3)\), \(t(4)\) and \(t(5)\) indicate the daily average temperatures on the first, second, third, fourth and fifth day in storage.  
\({}^{\rm X}\) Butter lettuce: Individually wrapped lettuce without roots; trained with experimental data (\({\rm R}^{\rm 2}\) = 0.75, n = 38); validated with commercial data (\({\rm R}^{\rm 2}\) = 0.79, n = 9).  
\end{table} 
Table 2: Linear regression models for shelf life of living and butter lettuce. This study provided evidence that 2-stage NN models are applicable to commercial situations. The data covered a broad range of pre-harvest conditions and post-harvest treatments. Models established with such a broad data base have flexibility to predict RSL under most foreseeable conditions. It could be questioned whether use of simple NNs is just as good as 2-stage NNs to predict RSL. In our experimental design, when both types of NN models were trained and tested using identical data sets, the results seemed to indicate two advantages in using 2-stage over simple NNs. First, when historical records were used to generate future temperatures and associated shelf lives, our 2-stage NN models improved RSL predictions to achieve R\({}^{2}\) of 0.61 compared with simple NN R\({}^{2}\) of 0.57 (Fig. 2). Secondly, although future temperatures (_fa_) are unknown on the day (_c_) when a prediction is to be made, end users normally have a temperature range in mind. For example, _fa_ may be 1\({}^{\circ}\)C in storage, 4\({}^{\circ}\)C in household refrigerator, and so on. Such temperature ranges can be specified by end-users to make real-life predictions using 2-stage NN models.