  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.
