## 4 Application
### Background  
The COVID-19 pandemic has been the top concern and roadblock since 2019 in more than 150 countries around the world. This disease has had extreme impacts on the health and lives of many on a global scale. In the fight to overcome COVID-19, it is essential to quickly detect the infected patients. Investigation through radiography images is the basic procedure used for diagnosing abnormalities in infected patients. Several deep learning algorithms have been proposed for the detection of COVID-19 on CT scans. [PERSON] et al. provided the model output to radiologists, and demonstrated that AI assistance significantly improves radiologist diagnostic accuracy from 85% to 90% in distinguishing COVID classes from non-COVID classes [30]. [PERSON] et al. [31] demonstrated the possibility of using deep CNNs including ResNet18, ResNet50, and DenseNet-121 to classify COVID-19 disease using X-ray images. They achieved a sensitivity of 98% and a specificity of 90% on 5000 chest X-ray images. In this study, the data were sourced from [31]. We downloaded 576 COVID images directly from their work. We also collected 2000 chest X-ray images of healthy individuals from their database and used these images as non-COVID cases. The database created by [PERSON] et al. [31] also includes other diseases such as pneumonia. The goal in our work was to understand the difference between COVID and healthy chest X-ray images. \begin{table}
\begin{tabular}{l l c c c c c} \hline \hline
**Training Sample Size: 1000** & **Algorithms** &
\begin{tabular}{c} **Test AUC** \\ **Bagging** \\ \end{tabular} & **Logistic** & **RF** & **iRF** & **NN** \\ \hline \multirow{6}{*}{\(12\times 12\)} & All Var. & 0.54 & 0.52 & 0.51 & 0.53 & 0.51 \\  & I-score Modules & & & & & \\  & \(2\times 2\) window, 121 mod. & 0.77 & 0.68 & 0.61 & 0.72 & 0.74 \\  & \(3\times 3\) window, 100 mod. & 0.78 & 0.69 & 0.63 & 0.72 & 0.76 \\  & \(4\times 4\) window, 81 mod. & 0.78 & 0.72 & 0.64 & 0.72 & 0.76 \\ \hline \multirow{6}{*}{\(14\times 14\)} & All Var. & 0.54 & 0.52 & 0.50 & 0.52 & 0.51 \\  & I-score Modules & & & & & \\  & \(2\times 2\) window, 169 mod. & 0.77 & 0.64 & 0.59 & 0.70 & 0.72 \\  & \(3\times 3\) window, 144 mod. & 0.77 & 0.70 & 0.60 & 0.70 & 0.73 \\  & \(4\times 4\) window, 121 mod. & 0.77 & 0.70 & 0.62 & 0.71 & 0.73 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The simulation results for Model (12). The theoretical prediction rate was calculated using Equations (13) as 75%. In other words, we expected the prediction performance on the out-of-sample test set to be approximately 75% on average. For each experiment below, the out-of-sample test set had 1000 sample data points and the performance was calculated using the AUC from the ROC (see Section 4.4 for detailed discussion of AUC values). Continuing from Table 6, we fixed the in-sample training size to 1000 data points and we allowed the of variables in the toy data to be 100 or 200. From 49 variables, this is a more challenging situation because it lowers the chance of finding the correct variable modules. Therefore, we did not consider other diseases. We present the summary of these data in Table 8. We first randomly selected 60 images each from the COVID and non-COVID class and used these images as the out-of-sample test set. We did not use the images in the test set until the end after training and validation were complete. For the remaining images, we used data augmentation technique by adding noise drawn from normal distribution. This produced a total of 5000 COVID cases and 5000 non-COVID cases. These 10,000 images composed the training and validating sets (tr. and val. in Table 8, respectively). These 10,000 images have two classes: COVID and non-COVID. Figure 6 shows that for healthy individuals, the chest areas are clear in the X-ray images. However, the images for COVID cases are not as clear. This is an indication that there are inflammatory cells or other related body fluids in the chest.