MTRItest {HDLSSkST}

k-Sample MTRI Test of Equal Distributions

Description

Performs the distribution free exact k-sample test for equality of multivariate distributions in the HDLSS regime. This test is a multiscale approach based on RI test, where the results for different number of partitions are aggregated judiciously.

Usage

MTRItest(M, labels, sizes, k_max, multTest = "Holm", s_psi = 1, s_h = 1, 
lb = 1, n_sts = 1000, alpha = 0.05)

Arguments

M

n\times d observations matrix of pooled sample, the observations should be grouped by their respective classes

labels

length n vector of membership index of observations

sizes

vector of sample sizes

k_max

maximum value of total number of clusters which is required for the test

multTest

"HOlm"(default) or "BenHoch"; different multiple tests

s_psi

function required for clustering, 1 for t^2, 2 for 1-\exp(-t), 3 for 1-\exp(-t^2), 4 for \log(1+t), 5 for t

s_h

function required for clustering, 1 for \sqrt t, 2 for t

lb

each observation is partitioned into some numbers of smaller vectors of same length lb, default: 1

n_sts

number of simulation of the test statistic, default: 1000

alpha

numeric, confidence level \alpha, default: 0.05

Value

MTRItest returns a list containing the following items:

RIvec

a vector of the Rand indices based on different number of clusters

Pvalues

a vector of RI test p-values based on different number of clusters

decisionMTRI

if returns 1, reject the null hypothesis and if returns 0, fails to reject the null hypothesis

contTabs

a list of the observed contingency table based on different number of clusters

mulTestdec

a vector of 0s and 1s. 0: fails to reject the corresponding hypothesis and 1: reject the corresponding hypothesis

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<[e-mail]>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

Sture Holm (1979). A simple sequentially rejective multiple test procedure, Scandinavian journal of statistics, 65-70, doi:10.2307/4615733.

Yoav Benjamini and Yosef Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing, Journal of the Royal statistical society: series B (Methodological) 57.1: 289-300, doi: 10.2307/2346101.

Examples

  # muiltivariate normal distribution:
  # generate data with dimension d = 500
  set.seed(151)
  n1=n2=n3=n4=10
  d = 500
  I1 <- matrix(rnorm(n1*d,mean=0,sd=1),n1,d)
  I2 <- matrix(rnorm(n2*d,mean=0.5,sd=1),n2,d) 
  I3 <- matrix(rnorm(n3*d,mean=1,sd=1),n3,d) 
  I4 <- matrix(rnorm(n4*d,mean=1.5,sd=1),n4,d)
  levels <- c(rep(0,n1), rep(1,n2), rep(2,n3), rep(3,n4)) 
  X <- as.matrix(rbind(I1,I2,I3,I4)) 
  #MTRI test:
  results <- MTRItest(X, levels, c(n1,n2,n3,n4), 8)
  
  ## outputs:
  results$RIvec
  #[1] 0.25641026 0.14871795 0.00000000 0.03076923 0.05128205 0.08333333 0.10384615

  results$Pvalues
  #[1] 0 0 0 0 0 0 0

  results$decisionMTRI
  #[1] 1

  results$contTabs
  #$contTabs[[1]]
  #     [,1] [,2]
  #[1,]   10    0
  #[2,]   10    0
  #[3,]    0   10
  #[4,]    0   10

  #$contTabs[[2]]
  #     [,1] [,2] [,3]
  #[1,]   10    0    0
  #[2,]    0   10    0
  #[3,]    0   10    0
  #[4,]    0    0   10

  #$contTabs[[3]]
  #     [,1] [,2] [,3] [,4]
  #[1,]   10    0    0    0
  #[2,]    0   10    0    0
  #[3,]    0    0   10    0
  #[4,]    0    0    0   10

  #$contTabs[[4]]
  #     [,1] [,2] [,3] [,4] [,5]
  #[1,]   10    0    0    0    0
  #[2,]    0   10    0    0    0
  #[3,]    0    0    4    6    0
  #[4,]    0    0    0    0   10

  #$contTabs[[5]]
  #     [,1] [,2] [,3] [,4] [,5] [,6]
  #[1,]   10    0    0    0    0    0
  #[2,]    0   10    0    0    0    0
  #[3,]    0    0    4    6    0    0
  #[4,]    0    0    0    0    8    2

  #$contTabs[[6]]
  #     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
  #[1,]   10    0    0    0    0    0    0
  #[2,]    0    5    5    0    0    0    0
  #[3,]    0    0    0    4    6    0    0
  #[4,]    0    0    0    0    0    8    2

  #$contTabs[[7]]
  #     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
  #[1,]    8    2    0    0    0    0    0    0
  #[2,]    0    0    5    5    0    0    0    0
  #[3,]    0    0    0    0    4    6    0    0
  #[4,]    0    0    0    0    0    0    8    2


  results$mulTestdec
  #[1] 1 1 1 1 1 1 1