BootRWCor_SD: Boootstrap multiple testing method of Romano & Wolf (2005) for correlations, with stepdown procedure.

Description

Multiple testing method based on the evaluation of quantile by bootstrap in the initial dataset (Romano & Wolf (2005)), with stepdown procedure.

Usage

BootRWCor_SD(
  data,
  alpha = 0.05,
  stat_test = "empirical",
  Nboot = 1000,
  vect = FALSE,
  logical = TRUE,
  arr.ind = FALSE
)

Arguments

data

matrix of observations

alpha

level of multiple testing

stat_test

4 test statistics are available:

'empirical': \(\sqrt{n}*abs(corr)\)
'fisher': \(\sqrt{n-3}*1/2*\log( (1+corr)/(1-corr) )\)
'student': \(\sqrt{n-2}*abs(corr)/\sqrt(1-corr^2)\)
'2nd.order': \(\sqrt{n}*mean(Y)/sd(Y)\) with \(Y=(X_i-mean(X_i))(X_j-mean(X_j))\)

Nboot

number of iterations for Bootstrap quantile evaluation

vect

if TRUE returns a vector of TRUE/FALSE values, corresponding to vectorize(cor(data)); if FALSE, returns an array containing TRUE/FALSE values for each entry of the correlation matrix

logical

if TRUE, returns either a vector or a matrix where each element is equal to TRUE if the corresponding null hypothesis is rejected, and to FALSE if it is not rejected if FALSE, returns a list of successive p-values : element [[i+1]] of the list giving the p-values evaluated on the non-rejected hypothesis at step [[i]]; p-values are either as a vector or a list depending on vect

arr.ind

if TRUE, returns the indexes of the significant correlations, with respect to level alpha

Value

Returns

an array containing indexes \(\lbrace(i,j),\,i<j\rbrace\) for which correlation between variables \(i\) and \(j\) is significant, if arr.ind=TRUE.

References

Romano, J. P., & Wolf, M. (2005). Exact and approximate stepdown methods for multiple hypothesis testing. Journal of the American Statistical Association, 100(469), 94-108.

Roux, M. (2018). Graph inference by multiple testing with application to Neuroimaging, Ph.D., Universit<U+00E9> Grenoble Alpes, France, https://tel.archives-ouvertes.fr/tel-01971574v1.

Examples

Run this code

# NOT RUN {
 
n <- 100
p <- 10
corr_theo <- diag(1,p)
corr_theo[1,3] <- 0.5
corr_theo[3,1] <- 0.5
data <- MASS::mvrnorm(n,rep(0,p),corr_theo)
alpha <- 0.05
# significant correlations:
BootRWCor_SD(data,alpha,stat_test='empirical', arr.ind=TRUE)
# successive p-values
res <- BootRWCor_SD(data,stat_test='empirical', logical=FALSE)
lapply(res,FUN=function(x){round(x,2)})
# succesive rejections
lapply(res,FUN=function(x){whichCor(x<alpha)})  
# }