Controlled variable Selection with Model-X Knockoffs

This vignette illustrates the basic usage of the knockoff package with Model-X knockoffs. In this scenario we assume that the distribution of the predictors is known (or that it can be well approximated), but we make no assumptions on the conditional distribution of the response. For simplicity, we will use synthetic data constructed from a linear model such that the response only depends on a small fraction of the variables.

set.seed(1234)

# Problem parameters
n = 200           # number of observations
p = 200           # number of variables
k = 60            # number of variables with nonzero coefficients
amplitude = 4.5   # signal amplitude (for noise level = 1)

# Generate the variables from a multivariate normal distribution
mu = rep(0,p)
rho = 0.25
Sigma = toeplitz(rho^(0:(p-1)))
X = matrix(rnorm(n*p),n) %*% chol(Sigma)

# Generate the response from a linear model
nonzero = sample(p, k)
beta = amplitude * (1:p %in% nonzero) / sqrt(n)
y.sample = function(X) X %*% beta + rnorm(n)
y = y.sample(X)

First examples

To begin, we call knockoff.filter with all the default settings.

library(knockoff)
result = knockoff.filter(X, y)

We can display the results with

print(result)

## Call:
## knockoff.filter(X = X, y = y)
## 
## Selected variables:
##  [1]   8  11  15  27  45  50  51  60  66  68  71  81  87  88  99 101 111 112 114
## [20] 134 135 146 150 152 153 158 160 161 162 164 166 172 177 179 181

The default value for the target false discovery rate is 0.1. In this experiment the false discovery proportion is

fdp = function(selected) sum(beta[selected] == 0) / max(1, length(selected))
fdp(result$selected)

## [1] 0.02857143

By default, the knockoff filter creates model-X second-order Gaussian knockoffs. This construction estimates from the data the mean μ and the covariance Σ of the rows of X, instead of using the true parameters (μ, Σ) from which the variables were sampled.

The knockoff package also includes other knockoff construction methods, all of which have names prefixed withknockoff.create. In the next snippet, we generate knockoffs using the true model parameters.

gaussian_knockoffs = function(X) create.gaussian(X, mu, Sigma)
result = knockoff.filter(X, y, knockoffs=gaussian_knockoffs)
print(result)

## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs)
## 
## Selected variables:
##  [1]  11  15  27  50  60  66  82  83  94  99 114 134 135 141 146 150 153 160 161
## [20] 162 165 166 172 179 181

Now the false discovery proportion is

fdp(result$selected)

## [1] 0

By default, the knockoff filter uses a test statistic based on the lasso. Specifically, it uses the statistic stat.glmnet_coefdiff, which computes W_j = |Z_j| − |Z̃_j| where Z_j and Z̃_j are the lasso coefficient estimates for the jth variable and its knockoff, respectively. The value of the regularization parameter λ is selected by cross-validation and computed with glmnet.

Several other built-in statistics are available, all of which have names prefixed with stat. For example, we can use statistics based on random forests. In addition to choosing different statistics, we can also vary the target FDR level (e.g. we now increase it to 0.2).

result = knockoff.filter(X, y, knockoffs = gaussian_knockoffs, statistic = stat.random_forest, fdr=0.2)
print(result)

## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs, 
##     statistic = stat.random_forest, fdr = 0.2)
## 
## Selected variables:
## [1]  68  87 114 152 158 161

fdp(result$selected)

## [1] 0

User-defined test statistics

In addition to using the predefined test statistics, it is also possible to use your own custom test statistics. To illustrate this functionality, we implement one of the simplest test statistics from the original knockoff filter paper, namely W_j = |X_j^⊤ ⋅ y| − |X̃_j^⊤ ⋅ y|.

my_knockoff_stat = function(X, X_k, y) {
  abs(t(X) %*% y) - abs(t(X_k) %*% y)
}
result = knockoff.filter(X, y, knockoffs = gaussian_knockoffs, statistic = my_knockoff_stat)
print(result)

## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs, 
##     statistic = my_knockoff_stat)
## 
## Selected variables:
##  [1]  11  12  50  54  60  66  68  83  85  87  88  94  99 114 134 146 158 160 161
## [20] 166 177 179 181

fdp(result$selected)

## [1] 0.08695652

As another example, we show how to customize the grid of λ’s used to compute the lasso path in the default test statistic.

my_lasso_stat = function(...) stat.glmnet_coefdiff(..., nlambda=100)
result = knockoff.filter(X, y, knockoffs = gaussian_knockoffs, statistic = my_lasso_stat)
print(result)

## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs, 
##     statistic = my_lasso_stat)
## 
## Selected variables:
##  [1]  15  27  60  66  67  71  83  99 111 114 134 135 141 153 158 160 161 165 172
## [20] 177 179 186 193

fdp(result$selected)

## [1] 0

The nlambda parameter is passed by stat.glmnet_coefdiff to the glmnet, which is used to compute the lasso path. For more information about this and other parameters, see the documentation for stat.glmnet_coefdiff or glmnet.glmnet.

User-defined knockoff generation functions

In addition to using the predefined procedures for construction knockoff variables, it is also possible to create your own knockoffs. To illustrate this functionality, we implement a simple wrapper for the construction of second-order Model-X knockoffs.

create_knockoffs = function(X) {
  create.second_order(X, shrink=T)
}
result = knockoff.filter(X, y, knockoffs=create_knockoffs)
print(result)

## Call:
## knockoff.filter(X = X, y = y, knockoffs = create_knockoffs)
## 
## Selected variables:
##  [1]  11  12  27  41  50  51  56  60  66  68  71  80  83  87  88  94  99 101 111
## [20] 112 114 132 134 135 140 141 142 146 150 152 153 156 158 160 161 164 165 166
## [39] 167 168 172 177 179 181 182 187 193

fdp(result$selected)

## [1] 0.06382979

Approximate vs Full SDP knockoffs

The knockoff package supports two main styles of knockoff variables, semidefinite programming (SDP) knockoffs (the default) and equi-correlated knockoffs. Though more computationally expensive, the SDP knockoffs are statistically superior by having higher power. To create SDP knockoffs, this package relies on the R library [Rdsdp][Rdsdp] to efficiently solve the semidefinite program. In high-dimensional settings, this program becomes computationally intractable. A solution is then offered by approximate SDP (ASDP) knockoffs, which address this issue by solving a simpler relaxed problem based on a block-diagonal approximation of the covariance matrix. By default, the knockoff filter uses SDP knockoffs if p < 500 and ASDP knockoffs otherwise.

In this example we generate second-order Gaussian knockoffs using the estimated model parameters and the full SDP construction. Then, we run the knockoff filter as usual.

gaussian_knockoffs = function(X) create.second_order(X, method='sdp', shrink=T)
result = knockoff.filter(X, y, knockoffs = gaussian_knockoffs)
print(result)

## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs)
## 
## Selected variables:
##  [1]   8  11  12  15  27  48  50  51  60  66  68  80  83  87  88  94  99 101 111
## [20] 112 114 132 134 135 140 141 146 150 152 153 158 160 161 164 165 166 167 168
## [39] 172 177 179 181 193

fdp(result$selected)

## [1] 0.04651163

Equi-correlated knockoffs

Equicorrelated knockoffs offer a computationally cheaper alternative to SDP knockoffs, at the cost of lower statistical power. In this example we generate second-order Gaussian knockoffs using the estimated model parameters and the equicorrelated construction. Then we run the knockoff filter.

gaussian_knockoffs = function(X) create.second_order(X, method='equi', shrink=T)
result = knockoff.filter(X, y, knockoffs = gaussian_knockoffs)
print(result)

## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs)
## 
## Selected variables:
##  [1]   8  11  12  27  50  51  60  66  67  68  80  82  83  87  88  94  99 101 111
## [20] 112 114 134 135 140 141 146 150 152 153 160 161 162 164 165 166 168 172 177
## [39] 179 181 182 193

fdp(result$selected)

## [1] 0