MMD Two-sample Testing in the Presence of Arbitrarily Missing Data

TMLR Paper5147 Authors

Abstract: In many real-world applications, it is common that a proportion of the data may be missing or only partially observed. We develop a novel two-sample testing method based on the Maximum Mean Discrepancy (MMD) which accounts for missing data in both samples, without making assumptions about the missingness mechanism. Our approach is based on deriving the mathematically precise bounds of the MMD test statistic after accounting for all possible missing values. To the best of our knowledge, it is the only two-sample testing method that is guaranteed to control the Type I error for both univariate and multivariate data where data may be arbitrarily missing. Simulation results show that the method has good statistical power, typically for cases where 5% to 10% of the data are missing. We highlight the value of this approach when the data are missing not at random, a context in which either ignoring the missing values or using common imputation methods may not control the Type I error.

Submission Length: Long submission (more than 12 pages of main content)

Changes Since Last Submission:

We have made several changes to the main text in response to the reviewers’ comments. These specific changes are highlighted in the individual official comments to the reviewers.

Furthermore, we have added additional benchmark methods, WMWM (Zeng et al., 2024) and MissForest (Stekhoven & Bühlmann, 2012) as requested by one of the reviewers. Figures 1, 2 and 3 have been updated to include these changes.

We thank the reviewers for their constructive feedback and feel the implemented changes based on their comments and suggestions have strengthened the manuscript.

References:

- Zeng, Yijin and Adams, Niall M and Bodenham, Dean A (2024) On two-sample testing for data with arbitrarily missing values, arXiv preprint arXiv:2403.15327

- Stekhoven, D.J. and Bühlmann, P., 2012. MissForest—non-parametric missing value imputation for mixed-type data. Bioinformatics, 28(1), pp.112-118.

Assigned Action Editor: ~Benjamin_Guedj1

Submission Number: 5147