# A Two-Step Algorithm for Handling Block-Wise Missing Data in Multi-Omics
## 4 Results
### Breast Cancer Data  
The breast cancer patient data were sourced from The Cancer Genome Atlas (TCGA) [1]. We utilized multi-omics data (mRNA, miRNA, and DNAAm) as described in [15], accessible via [[https://github.com/hyr0771/DiffRS-net](https://github.com/hyr0771/DiffRS-net)]([https://github.com/hyr0771/DiffRS-net](https://github.com/hyr0771/DiffRS-net)) (accessed date 16 February 2025). This dataset includes four breast cancer subtypes, Basal-like (Basal), Her2-enriched (Her2), Luminal A (LumA), and Luminal B (LumB), which are recognized as the most consistently identified subtypes of human breast cancer [16]. The main characteristics of these subtypes are determined by the expression levels of the estrogen receptor (ER), progesterone receptor (PR), human epidermal growth factor receptor 2 (HER2), and the proliferation marker Ki67 [16; 17]. The dataset comprises 528 samples with 1000 genomic features per omic. Relevant features were selected in [15] to reduce omics dimensionality, placing us in a typical omics scenario where \(n\) is less than \(p\). Sample distribution by cancer subtype is as follows: Basal 87, Her2 31, LumA 284, and LumB 126. We employed cross-entropy loss to model the cancer subtype, incorporating the regularization hyperparameter \(\lambda\) and the norm function \(\Omega_{1}\) to prevent overfitting, particularly in classes with smaller sample sizes like Her2. Regularization helps by penalizing overly complex models, thereby ensuring that the model generalizes well to unseen data and maintains robust performance across different scenarios. To evaluate the effect of missing data blocks, we intentionally introduced random instances where some lacked expression or methylation data, ensuring the missing block structure affected all omics. We systematically assessed the impact by examining scenarios with different percentages of samples having missing blocks, specifically at rates of 5%, 10%, 20%, and 30% (see Figure 1). In each scenario, the dataset was divided into training and testing subsets, with proportions of 2/3 and 1/3, respectively. The model was then trained on the training set and its performance evaluated on the test set. Figure 1: Block-wise missing scenarios in Breast cancer data. In Table 4, we show the performance in the different block-wise missing data scenarios. The accuracy remains high around 80%; only in the 30% scenario does it fall below 75%. Both class specificity and Negative Predictive Value remain high in all classes and in all scenarios. The rest of the metrics (Sensitivity, Positive Predictive Value, and F1) show different behavior depending on the class. In the Basal and LumA classes, the metrics reach high values, worsening as the percentage of missing data increases. The metrics in the Her2 and LumB classes show worse behavior. The Her2 class is the least abundant. This is more evident in the test set, where there are only eight samples. Although in the 5% scenario the F1 metric is acceptable, it does not remain that way in the other scenarios. A different situation appears to be that of the LumB class, which is quite abundant. Classification errors are observed between the LumA and LumB classes, erroneously assigning samples of LumB to LumA. As is well known in biomedicine, these cancer subtypes have weak separation from molecular biomarkers, as highlighted in [16; 17; 18]. In Figure 2, we compare the accuracies between profile-free approaches and our profile-based approach. Profile-free approaches integrate omics applying the model 1 where training data are the subset of available complete observations or all observations where missing data were imputed by the k-nn rule. We highlight that our method outperforms both alternative methods in all scenarios.