The function `knockoff.filter`

is a wrapper around several simpler functions that

- Construct knockoff variables (various functions with prefix
`create`

) - Compute the test statistic \(W\) (various functions with prefix
`stat`

) - Compute the threshold for variable selection (
`knockoff.threshold`

)

These functions may be called directly if desired. The purpose of this vignette is to illustrate the flexibility of this package with some examples.

```
set.seed(1234)
library(knockoff)
```

Let us begin by creating some synthetic data. For simplicity, we will use synthetic data constructed from a generalized linear model such that the response only depends on a small fraction of the variables.

```
# Problem parameters
n = 1000 # number of observations
p = 1000 # number of variables
k = 60 # number of variables with nonzero coefficients
amplitude = 7.5 # signal amplitude (for noise level = 1)
# Generate the variables from a multivariate normal distribution
mu = rep(0,p)
rho = 0.10
Sigma = toeplitz(rho^(0:(p-1)))
X = matrix(rnorm(n*p),n) %*% chol(Sigma)
# Generate the response from a logistic model and encode it as a factor.
nonzero = sample(p, k)
beta = amplitude * (1:p %in% nonzero) / sqrt(n)
invlogit = function(x) exp(x) / (1+exp(x))
y.sample = function(x) rbinom(n, prob=invlogit(x %*% beta), size=1)
y = factor(y.sample(X), levels=c(0,1), labels=c("A","B"))
```

Instead of using `knockoff.filter`

directly, we can run the filter manually by calling its main components one by one.

The first step is to generate the knockoff variables for the true Gaussian distribution of the variables.

`X_k = create.gaussian(X, mu, Sigma)`

Then, we compute the knockoff statistics using 10-fold cross-validated lasso

`W = stat.glmnet_coefdiff(X, X_k, y, nfolds=10, family="binomial")`

Now we can compute the rejection threshold

`thres = knockoff.threshold(W, fdr=0.2, offset=1)`

The final step is to select the variables

```
selected = which(W >= thres)
print(selected)
```

```
## [1] 3 12 76 108 172 173 196 210 248 273 281 298 304 334 360 378 426
## [18] 428 451 494 510 543 563 595 596 708 787 837 843 844 931 941 953 959
```

The false discovery proportion is

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

`## [1] 0.3235294`

We show how to manually run the knockoff filter multiple times and compute average quantities. This is particularly useful to estimate the FDR (or the power) for a particular configuration of the knockoff filter on artificial problems.

```
# Optimize the parameters needed for generating Gaussian knockoffs,
# by solving an SDP to minimize correlations with the original variables.
# This calculation requires only the model parameters mu and Sigma,
# not the observed variables X. Therefore, there is no reason to perform it
# more than once for our simulation.
diag_s = create.solve_asdp(Sigma)
# Compute the fdp over 20 iterations
nIterations = 20
fdp_list = sapply(1:nIterations, function(it) {
# Run the knockoff filter manually, using the pre-computed value of diag_s
X_k = create.gaussian(X, mu, Sigma, diag_s=diag_s)
W = stat.glmnet_lambdasmax(X, X_k, y, family="binomial")
t = knockoff.threshold(W, fdr=0.2, offset=1)
selected = which(W >= t)
# Compute and store the fdp
fdp(selected)
})
# Estimate the FDR
mean(fdp_list)
```

`## [1] 0.1204519`

If you want to see some basic usage of the knockoff filter, see the introductory vignette. If you want to see how to use knockoffs for Fixed-X variables, see the Fixed-X vignette.