# A Prototype Knockoff Filter for Group Selection with FDR Control

## TL;DR

This paper introduces a prototype knockoff filter for group variable selection in linear regression, enhancing computational efficiency and statistical power while controlling FDR, especially when features are spanned by hidden factors.

## Contribution

It extends the Reid-Tibshirani prototype method to a group setting, improving performance and outperforming existing group knockoff filters under certain conditions.

## Key findings

- The prototype knockoff filter outperforms the Reid-Tibshirani and Dai-Barber group knockoff filters in simulations.
- Some knockoff path statistics, like the Lasso path statistic, may reduce power under specific conditions.
- The proposed method maintains FDR control while increasing power in group variable selection.

## Abstract

In many applications, we need to study a linear regression model that consists of a response variable and a large number of potential explanatory variables and determine which variables are truly associated with the response. In 2015, Barber and Candes introduced a new variable selection procedure called the knockoff filter to control the false discovery rate (FDR) and proved that this method achieves exact FDR control. In this paper, we propose a prototype knockoff filter for group selection by extending the Reid-Tibshirani prototype method. Our prototype knockoff filter improves the computational efficiency and statistical power of the Reid-Tibshirani prototype method when it is applied for group selection. In some cases when the group features are spanned by one or a few hidden factors, we demonstrate that the PCA prototype knockoff filter outperforms the Dai-Barber group knockoff filter. We present several numerical experiments to compare our prototype knockoff filter with the Reid-Tibshirani prototype method and the group knockoff filter. We have also conducted some analysis of the knockoff filter. Our analysis reveals that some knockoff path method statistics, including the Lasso path statistic, may lead to loss of power for certain design matrices and a specially designed response even if their signal strengths are still relatively strong.