Test independence of a data-dependent group of variables with the rest

Given a covariance matrix S of p Gaussian variables and a grouping obtained via thresholding absolute correlations at 1-c using block_diag(), this function tests the null hypothesis of independence between two groups of Gaussian variables.

Usage

selective_p_val(
  S,
  CP,
  k,
  n,
  c,
  d0 = 5,
  tol = 1e-05,
  maxeval = 1e+05,
  mc_iter = 1000
)

Arguments

S

a \(p \times p\) covariance matrix

CP

a vector of length \(p\) with \(i^{th}\) element denoting the group \(i^{th}\) variable belongs to

k

the group to be tested for independence with the remaining variables, i.e. \(P = [i : CP[i]==k]\)

n

sample size

c

a threshold

d0

a natural number; if the number of canonical correlations is greater than d0, Monte Carlo simulation will be used to approximate the p-value for computational convenience; default value is 5

tol

the relative tolerance used to approximate the p-value using selective_p_val_integrate(); default value is 1e-05

maxeval

the maximum number of function evaluations used to approximate the p-value using selective_p_val_integrate(); we recommend using a high value of this to obtain an approximation with high accuracy; default value is 10,000

mc_iter

the number of Monte Carlo iterations used to approximate the p-value; we recommend using a high value of this to obtain an approximation with high accuracy; default value is 1,000

Value

The selective p-value for the test of independence.

Examples

# Simulates a 10 x 5 X from N(0, I)
set.seed(1)
X <- matrix(rnorm(50), 10, 5)

# Compute the correlation matrix of X.
corX <- cor(X)
# Use 'block_diag' to obtain any block diagonal structure
block_diag_structure <- block_diag(corX, c= 0.5)
# test for independence of the variables in group 1 with the remaining variables
selective_p_val(S=cov(X), n=10, CP=block_diag_structure, c=0.5, 
k=1, d0=5, tol = 1e-05, maxeval = 10000, mc_iter=100)
#> [1] 0.2721862