KCI {CondIndTests}

Kernel conditional independence test.

Description

Tests the null hypothesis that Y and E are independent given X. The distribution of the test statistic under the null hypothesis equals an infinite weighted sum of chi squared variables. This distribution can either be approximated by a gamma distribution or by a Monte Carlo approach. This version includes an implementation of choosing the hyperparameters by Gaussian Process regression.

Usage

KCI(Y, E, X, width = 0, alpha = 0.05, unbiased = FALSE,
  gammaApprox = TRUE, GP = TRUE, nRepBs = 5000, lambda = 0.001,
  thresh = 1e-05, numEig = NROW(Y), verbose = FALSE)

Arguments

Y

A vector of length n or a matrix or dataframe with n rows and p columns.

E

A vector of length n or a matrix or dataframe with n rows and p columns.

X

A matrix or dataframe with n rows and p columns.

width

Kernel width; if it is set to zero, the width is chosen automatically (default: 0).

alpha

Significance level (default: 0.05).

unbiased

A boolean variable that indicates whether a bias correction should be applied (default: FALSE).

gammaApprox

A boolean variable that indicates whether the null distribution is approximated by a Gamma distribution. If it is FALSE, a Monte Carlo approach is used (default: TRUE).

GP

Flag whether to use Gaussian Process regression to choose the hyperparameters

nRepBs

Number of draws for the Monte Carlo approach (default: 500).

lambda

Regularization parameter (default: 1e-03).

thresh

Threshold for eigenvalues. Whenever eigenvalues are computed, they are set to zero if they are smaller than thresh times the maximum eigenvalue (default: 1e-05).

numEig

Number of eigenvalues computed (only relevant for computing the distribution under the hypothesis of conditional independence) (default: length(Y)).

verbose

If TRUE, intermediate output is provided. (default: FALSE).

Value

A list with the following entries:

• testStatistic the statistic Tr(K_(ddot(Y)|X) * K_(E|X))

• criticalValue the critical point at the p-value equal to alpha; obtained by a Monte Carlo approach if gammaApprox = FALSE, otherwise obtained by Gamma approximation.

• pvalue The p-value for the null hypothesis that Y and E are independent given X. It is obtained by a Monte Carlo approach if gammaApprox = FALSE, otherwise obtained by Gamma approximation.

Examples

# Example 1
n <- 100
E <- rnorm(n)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * (X)^2 + rnorm(n)
KCI(Y, E, X)
KCI(Y, X, E)