Model-X knockoffs is a flexible wrapper method for high-dimensional regression algorithms, which provides guaranteed control of the false discovery rate (FDR). Due to the randomness inherent to the method, different runs of model-X knockoffs on the same dataset often result in different sets of selected variables, which is undesirable in practice. In this article, we introduce a methodology for derandomising model-X knockoffs with provable FDR control. The key insight of our proposed method lies in the discovery that the knockoffs procedure is in essence an e-BH procedure. We make use of this connection and derandomise model-X knockoffs by aggregating the e-values resulting from multiple knockoff realisations. We prove that the derandomised procedure controls the FDR at the desired level, without any additional conditions (in contrast, previously proposed methods for derandomisation are not able to guarantee FDR control). The proposed method is evaluated with numerical experiments, where we find that the derandomised procedure achieves comparable power and dramatically decreased selection variability when compared with model-X knockoffs.
Many contemporary large-scale applications involve building interpretable models linking a large set of potential covariates to a response in a non-linear fashion, such as when the response is binary. Although this modelling problem has been extensively studied, it remains unclear how to control the fraction of false discoveries effectively even in high dimensional logistic regression, not to mention general high dimensional non-linear models. To address such a practical problem, we propose a new framework of ‘model-X’ knockoffs, which reads from a different perspective the knockoff procedure that was originally designed for controlling the false discovery rate in linear models. Whereas the knockoffs procedure is constrained to homoscedastic linear models with n⩾p, the key innovation here is that model-X knockoffs provide valid inference from finite samples in settings in which the conditional distribution of the response is arbitrary and completely unknown. Furthermore, this holds no matter the number of covariates. Correct inference in such a broad setting is achieved by constructing knockoff variables probabilistically instead of geometrically. To do this, our approach requires that the covariates are random (independent and identically distributed rows) with a distribution that is known, although we provide preliminary experimental evidence that our procedure is robust to unknown or estimated distributions. To our knowledge, no other procedure solves the controlled variable selection problem in such generality but, in the restricted settings where competitors exist, we demonstrate the superior power of knockoffs through simulations. Finally, we apply our procedure to data from a case–control study of Crohn's disease in the UK, making twice as many discoveries as the original analysis of the same data.
more » « less- PAR ID:
- 10397977
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- Volume:
- 80
- Issue:
- 3
- ISSN:
- 1369-7412
- Format(s):
- Medium: X Size: p. 551-577
- Size(s):
- p. 551-577
- Sponsoring Org:
- National Science Foundation
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Consider a case–control study in which we have a random sample, constructed in such a way that the proportion of cases in our sample is different from that in the general population—for instance, the sample is constructed to achieve a fixed ratio of cases to controls. Imagine that we wish to determine which of the potentially many covariates under study truly influence the response by applying the new model‐X knockoffs approach. This paper demonstrates that it suffices to design knockoff variables using data that may have a different ratio of cases to controls. For example, the knockoff variables can be constructed using the distribution of the original variables under any of the following scenarios: (a) a population of controls only; (b) a population of cases only; and (c) a population of cases and controls mixed in an arbitrary proportion (irrespective of the fraction of cases in the sample at hand). The consequence is that knockoff variables may be constructed using unlabelled data, which are often available more easily than labelled data, while maintaining Type‐I error guarantees.
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