# On the Construction of Knockoffs in Case-Control Studies

## TL;DR

This paper shows that in case-control studies, knockoff variables can be constructed using unlabeled data with different case-control ratios, enabling error control even with limited labeled data.

## Contribution

It introduces a method to construct knockoffs using unlabeled or differently sampled data, broadening the applicability of model-X knockoffs in case-control settings.

## Key findings

- Knockoffs can be built from control-only data.
- Knockoffs can be built from case-only data.
- Type-I error guarantees are maintained with unlabeled data.

## Abstract

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: (1) a population of controls only; (2) a population of cases only; (3) 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 unlabeled data, which is often available more easily than labeled data, while maintaining Type-I error guarantees.