Discussion of ‘Gene hunting with hidden Markov model knockoffs’

Abstract

1. On the motivation The knockoff generation algorithm of Sesia et al. (2019), compounded with the knockoff+ filter of Barber & Candès (2015), provably controls the false discovery rate of conditionally independent variables, where X = ( X j ) j = 1 p and Y denote the predictor and outcome variables, respectively. The false discovery proportion, also known as the false selection proportion, is defined as F D P = V / R ⁠ , where R is the number of variables selected and V is the number of variables falsely selected. A false selection is defined by Sesia et al. (2019) as a selected X j such that Y ∣ X − j is independent of X j ⁠ . Their knockoff algorithm provably controls the false discovery rate, f d r = E ( f d p ) ⁠ , at some user-selected magnitude q ⁠ . The fundamental idea of the method is to generate another set of variables, the knockoffs, that are exchangeable with the original X j ⁠ , except that conditionally on X they are uncorrelated with Y ⁠ . The method then proceeds to compute a test statistic that captures the difference between the apparent dependence of Y and X j and that of Y and the knockoff X ~ j ⁠ . The following points are crucial for our discussion: (i) The false discovery rate is an expectation with respect to the variability in X and Y ⁠ , i.e., it is a random design guarantee. (ii) The procedure is model-free, i.e., nonparametric, with respect to prediction, in that nothing is assumed about the form of Y ∣ X ⁠ . (iii) The framework assumes full knowledge of F X ⁠ , the joint distribution of predictors, marginalized over Y ⁠ ; in practice this is based on the assumption that F X can be estimated well enough to be treated as known. (iv) The method aims at good variable selection, not prediction, estimation or ranking. We think of the method of Sesia et al. (2019) as an adaptation of that in Candès et al. (2018) to genome-wide association studies. The first difference between the two approaches is that the nonnull variables in Candès et al. (2018) are those that belong to the minimal set that renders all other variables independent, i.e., the nonnull set is the Markov blanket of X on Y ⁠ , whereas in Sesia et al. (2019) the nonnull variables are those with nonnull partial correlations. Under some minimal conditions that exclude perfectly dependent variables, these two sets are the same. Secondly, in Candès et al. (2018), F X is multivariate Gaussian, while in Sesia et al. (2019) it is a hidden Markov model. An important contribution of each paper is an algorithm for sampling knockoffs from the assumed model.