Non-parametric Two-sample test

Non-parametric two-sample test

The two-sample test can be additionally performed by providing the two samples to be compared as x and y. We generate the sample \(y = (y_1, \ldots,y_n)\) from a skew-normal distribution \(SN_d(0,I_d, \lambda)\), where \(d=4\), \(n=200\) and \(\lambda= (0.5,\ldots,0.5)\).

library(sn)
library(mvtnorm)
library(QuadratiK)
n <- 200
d <- 4
skewness_y <- 0.5
set.seed(2468)
x_2 <- rmvnorm(n, mean = rep(0,d))
y_2 <- rmsn(n=n, xi=0, Omega = diag(d), alpha=rep(skewness_y,d))

h <- 2
set.seed(2468)
two_test <- kb.test(x=x_2, y=y_2, h=h)
two_test

## 
##  Kernel-based quadratic distance two-sample test 
## U-statistics  Dn          Trace 
## ------------------------------------------------
## Test Statistic:   4.276823    9.843008 
## Critical Value:   0.7745576   1.783123 
## H0 is rejected:   TRUE        TRUE 
## CV method:  subsampling 
## Selected tuning parameter h:  2

For the two-sample case, the summary function provides the results from the test and the descriptive statistics per variable and per group, as similarly described for the \(k\)-sample test. Additionally, it generates the qq-plots comparing the quantiles of the two groups for each variable.

summary_two <- summary(two_test)

## 
##  Kernel-based quadratic distance two-sample test 
##   Statistic Test_Statistic Critical_Value Reject_H0
## 1        Dn       4.276823      0.7745576      TRUE
## 2     Trace       9.843008      1.7831227      TRUE

summary_two$summary_tables

## [[1]]
##             Group 1    Group 2    Overall
## mean    0.021762263  0.3799990  0.2008806
## sd      1.014655344  0.9498167  0.9977884
## median -0.005110155  0.3833061  0.2125618
## IQR     1.471877262  1.1310211  1.3666010
## min    -2.675477796 -2.2219439 -2.6754778
## max     2.300153117  3.1690406  3.1690406
## 
## [[2]]
##            Group 1    Group 2     Overall
## mean   -0.03347117  0.2216529  0.09409085
## sd      1.06408749  1.0304067  1.05383755
## median  0.02476594  0.1717768  0.09272994
## IQR     1.52458343  1.3739349  1.45668193
## min    -3.22222061 -2.6162342 -3.22222061
## max     2.96751758  2.3300745  2.96751758
## 
## [[3]]
##            Group 1    Group 2    Overall
## mean   -0.06473408  0.3312699  0.1332679
## sd      0.93818786  0.9868499  0.9818422
## median -0.07044427  0.4006745  0.1382735
## IQR     1.37135831  1.2185714  1.3854150
## min    -2.86000669 -3.0246026 -3.0246026
## max     2.56476485  2.7590501  2.7590501
## 
## [[4]]
##           Group 1    Group 2     Overall
## mean   -0.1658894  0.2065222  0.02031639
## sd      1.0175325  0.9718613  1.01104987
## median -0.2371959  0.1427746  0.04889195
## IQR     1.3802070  1.2320445  1.32957715
## min    -2.5899601 -2.0159679 -2.58996007
## max     2.7066430  2.6637589  2.70664302

The figure automatically generated by the summary function on the result of the two-sample test displays the qq-plots between the two samples with a table of the standard descriptive statistics for each variable, computed per group and overall.

# two_test_h <- kb.test(x=x_2, y=y_2)

set.seed(2468)
h_test2 <- select_h(x=x_2, y=y_2, alternative="skewness")

h_test2$h_sel