## 3 Results  
### Model Results for Classification of lrrPCA on PX2, Local, and PXL Data  
Initial randomization testing of all nine models confirmed that model AUC and accuracy results were not the product of chance (\(p\) < 0.01). Using T2 image data alone, modelPX2T2 had an AUC of 0.93 when tested on ProstateX-2 image data; its AUC dropped significantly to 0.49 when tested on the local institutional image data (\(p\) < 0.01). Similarly, modelLocalT2 had an AUC of 0.96 when trained and tested on local T2 image data; its AUC dropped to 0.50 when tested on ProstateX-2 T2 image data (\(p\) < 0.01). These results and corresponding model accuracies are summarized in Table 3. Standard deviations for each cross-validated model run are available in Supplementary Table S4. Using ADC map image data alone, modelPX2 ADC and modelLocalADC had AUCs of 0.91 and 0.82, respectively, when tested on image data from their respective institutions. Their AUCs decreased (0.23-0.49) significantly when tested on the other institution's image data (\(p\) < 0.01). Using both T2 and ADC map image data, modelPX2T2+ADC and modelLocalT2+ADC had AUCs of 0.95 and 0.98, respectively, when tested on image data from their respective institutions. Their AUCs also decreased (0.41-0.46) significantly when tested on the other institution's image data (\(p\) < 0.01). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**ResNet Model**} & **Training Image** & **Testing** & & & & & \\  & **Image** & **Image** & & & & & \\  & **Source** & **Source** & & & & & \\ \hline \hline \multirow{4}{*}{ModelPX2} & \multirow{4}{*}{PX2 (89)} & \multicolumn{2}{c}{**AUC**} & **Accuracy** & **AUC** & **Accuracy** & **AUC** & **Accuracy** \\  & & & PX2 * & 0.93 & 0.91 & 0.91 & 0.88 & 0.95 & 0.90 \\  & & Local \(\dagger\) & 0.49 & 0.53 & 0.23 & 0.48 & 0.46 & 0.55 \\  & & PXL \(\dagger\) & 0.87 & 0.79 & 0.80 & 0.78 & 0.78 & 0.80 \\ \hline \multirow{4}{*}{ModelLoc} & \multirow{4}{*}{Local} & \multirow{4}{*}{0.96} & 0.89 & 0.82 & 0.82 & 0.98 & 0.92 \\  & & PX2 \(\dagger\) & 0.50 & 0.54 & 0.49 & 0.49 & 0.41 & 0.51 \\  & & PXL \(\dagger\) & 0.77 & 0.71 & 0.84 & 0.84 & 0.94 & 0.87 \\ \hline \multirow{4}{*}{ModelPXL} & \multirow{4}{*}{PXL} & \multirow{4}{*}{0.83} & 0.89 & 0.98 & 0.92 & 0.96 & 0.93 \\  & & PX2 \(\dagger\) & 0.92 & 0.92 & 0.85 & 0.91 & 0.85 & 0.93 \\ \cline{1-1}  & & Local \(\dagger\) & 0.96 & 0.86 & 0.88 & 0.92 & 0.99 & 0.92 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Accuracy and area under the receiver operating characteristic curve for models trained on and tested with the entire ProstateX-2, local institutional, and combined ProstateX-2 and local institutional image data using 5-fold cross-validation. \(t\)-tests compared the AUC and accuracies of models trained and tested using the same image data source (labeled \(\dagger\)) to those using different training and testing sources (labeled \(\dagger\)). Statistically significant differences (\(p\) < 0.01) shown by 2-tailed \(t\)-test are underlined. PX2: ProstateX-2 data, Local: local institutional data, PXL: combined prostateX-2 and local institutional data. Model\({}^{\text{PXLT2}}\), representing the multi-institutally trained model, had an AUC of 0.83 when tested on T2 image data from both institutions, and higher AUCs (0.92-0.96) when tested on single-institution image data (_p_ < 0.01). Model\({}^{\text{PXLT2}}\) and model\({}^{\text{PXLT2}}\) + ADC had even higher AUCs (0.96-0.98) when tested on multi-institutional image data using ADC and combined T2 and ADC sequences; their performance slightly decreased (0.85-0.99) when tested on single-institution image data. The single-institutally trained and tested models (model\({}^{\text{PX2}}\) and model\({}^{\text{Local}}\)) tended to have higher AUCs when using T2 images compared with ADC map images; the use of combined T2 and ADC map images did not improve the performance of these models.