Two-sample Test for Multivariate Means by Johansen (1980)

Given two multivariate data \(X\) and \(Y\) of same dimension, it tests $$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$$ using the procedure by Johansen (1980) by adapting Welch-James approximation of the degree of freedom for Hotelling's \(T^2\) test.

mean2.1980Johansen(X, Y)

Arguments

X

an \((n_x \times p)\) data matrix of 1st sample.

Y

an \((n_y \times p)\) data matrix of 2nd sample.

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

\(p\)-value under \(H_0\).

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Johansen S (1980). “The Welch-James Approximation to the Distribution of the Residual Sum of Squares in a Weighted Linear Regression.” Biometrika, 67(1), 85. ISSN 00063444.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.1980Johansen(smallX, smallY) # run the test
#> 
#> 	Two-sample Test for Multivariate Means by Johansen (1980)
#> 
#> data:  smallX and smallY
#> T2 = 2.546, p-value = 0.5452
#> alternative hypothesis: true means are different.
#> 

if (FALSE) {
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(mean2.1980Johansen(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.1980Johansen'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))
}