# Integrating multi-source block-wise missing data in model selection

arXiv: https://arxiv.org/abs/1901.03797 · HTML: https://mlsanity.com/paper/1901.03797/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jan 2019.

## Built on by (most cited first, 11 of 11)

- [Multi-source Learning via Completion of Block-wise Overlapping Noisy Matrices](https://mlsanity.com/paper/2105.10360/) (2021)
- [Representation Retrieval Learning for Heterogeneous Data Integration](https://mlsanity.com/paper/2503.09494/) (2025)
- [Covariate-assisted Sparse Tensor Completion](https://mlsanity.com/paper/2103.06428/) (2021)
- [Orthogonalized Kernel Debiased Machine Learning for Multimodal Data Analysis](https://mlsanity.com/paper/2103.07088/) (2021)
- [FragmGAN: Generative Adversarial Nets for Fragmentary Data Imputation and Prediction](https://mlsanity.com/paper/2203.04692/) (2022)
- [Prediction approaches for partly missing multi-omics covariate data: A literature review and an empirical comparison study](https://mlsanity.com/paper/2302.03991/) (2023)
- [Using Interpretable Machine Learning to Massively Increase the Number of Antibody-Virus Interactions Across Studies](https://mlsanity.com/paper/2206.14566/) (2022)
- [Multi-task Learning for Heterogeneous Multi-source Block-Wise Missing Data](https://mlsanity.com/paper/2505.24413/) (2025)
- [Distributionally Robust Transfer Learning with Structurally Missing Covariates, with Application to Cross-National Cardiac Arrest Prediction](https://mlsanity.com/paper/2605.24212/) (2026)
- [Statistical Inference for High-Dimensional Linear Regression with Blockwise Missing Data](https://mlsanity.com/paper/2106.03344/) (2021)
- [Shapley Value Estimation for Multi-Site Data with Blockwise-Missing Features](https://mlsanity.com/paper/2609.14902/) (2026)

## Abstract (arXiv)

For multi-source data, blocks of variable information from certain sources are likely missing. Existing methods for handling missing data do not take structures of block-wise missing data into consideration. In this paper, we propose a Multiple Block-wise Imputation (MBI) approach, which incorporates imputations based on both complete and incomplete observations. Specifically, for a given missing pattern group, the imputations in MBI incorporate more samples from groups with fewer observed variables in addition to the group with complete observations. We propose to construct estimating equations based on all available information, and optimally integrate informative estimating functions to achieve efficient estimators. We show that the proposed method has estimation and model selection consistency under both fixed-dimensional and high-dimensional settings. Moreover, the proposed estimator is asymptotically more efficient than the estimator based on a single imputation from complete observations only. In addition, the proposed method is not restricted to missing completely at random. Numerical studies and ADNI data application confirm that the proposed method outperforms existing variable selection methods under various missing mechanisms.