# A Two-Step Algorithm for Handling Block-Wise Missing Data in Multi-Omics
## 4 Results
### Exposome Data  
The exposure data belong to the ISGlobal Exposome 2021 Data Challenge project [19], which is an initiative aimed at exploring the relationship between environmental exposures and human health. The dataset is publicly available on the same website as the Data Challenge ([[https://github.com/isglobal-exposomeHub/ExposomeDataChallenge2021/](https://github.com/isglobal-exposomeHub/ExposomeDataChallenge2021/)]([https://github.com/isglobal-exposomeHub/ExposomeDataChallenge2021/](https://github.com/isglobal-exposomeHub/ExposomeDataChallenge2021/)), accessed date 16 February 2025). The dataset used in this study is the exposureNA dataset, which contains exposure data with missing values. In this study, we focused on numerical variables, keeping all 1301 samples and a total of 176 numerical variables, which were categorized into five blocks based on their characteristics: Covariates (7), Chemical exposures (94), Indoor air (5), Lifestyles (6), and Outdoor exposures (64). The target variable for our regression model is the z-score body mass index for children aged 6-11 years, standardized by sex and age, and optimized using the mean squared error loss. Overall, our model comprises 176 variables organized into five blocks.  
Similarly to the breast cancer dataset, missing data in blocks were randomly introduced to evaluate their impact on the performance of the regression model. Missing blocks were generated at rates of 5%, 10%, 20%, 30%, 40%, and 50% (see Figure 3). For each different scenario, the dataset was divided into training (2/3) and test (1/3) subsets. The model was trained using the training data and evaluated in the test set to assess its performance.  
To evaluate performance, we used the root mean square error (RMSE), the mean absolute error (MAE), and the Pearson correlation coefficient between true and predicted responses as evaluation metrics (see Table 5 and Figure 4). Performance remained competitive across all metrics and scenarios, with only a slight decrease observed for high percentages (30% or more) of missing data across blocks. Notably, the Pearson correlation was statistically significant (\(p\)-value < 2.2 \(\times\) 10\({}^{-16}\)) in all scenarios.  
Figure 2: Breast cancer data. Comparison of accuracy between the three approaches in the different missing block scenarios: complete (only available complete observations were included), imputed (missing data were imputed and all observations were included), profiles (our method). The 95% confidence intervals are shown.  
Figure 4: Metrics across block-wise missing scenarios in exposure data.  
Figure 3: Block-wise missing scenarios in exposure data.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Scenarios** & **RMSE** & **MAE** & **Correlation** \\ \hline
5\% & 0.714 & 0.526 & 0.751 \\
10\% & 0.754 & 0.546 & 0.754 \\
20\% & 0.765 & 0.544 & 0.752 \\
30\% & 0.803 & 0.583 & 0.722 \\
40\% & 0.806 & 0.579 & 0.720 \\
50\% & 0.808 & 0.583 & 0.723 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Performance metrics in exposure data. RMSE, MAE, correlation.  
To illustrate the distribution of the true values and their predictions across the six scenarios, Figure 5 presents their scatter plots. As expected, based on the metric values, particularly the correlation, there is no significant difference in the distribution of the point clouds among the scenarios. The red line represents the bisector.