Brown-Forsythe Test for Homogeneity of Variance

Given univariate samples \(X_1~,\ldots,~X_k\), it tests $$H_0 : \sigma_1^2 = \cdots \sigma_k^2\quad vs\quad H_1 : \textrm{at least one equality does not hold}$$ using the procedure by Brown and Forsythe (1974).

vark.1974BF(dlist)

Arguments

dlist

a list of length \(k\) where each element is a sample vector.

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

Brown MB, Forsythe AB (1974). “Robust Tests for the Equality of Variances.” Journal of the American Statistical Association, 69(346), 364--367. ISSN 0162-1459, 1537-274X.

Examples

## CRAN-purpose small example
small1d = list()
for (i in 1:5){ # k=5 sample
  small1d[[i]] = rnorm(20)
}
vark.1974BF(small1d) # run the test
#> 
#> 	Brown-Forsythe Test for Homogeneity of Variance
#> 
#> data:  small1d
#> statistic = 0.98285, p-value = 0.4208
#> alternative hypothesis: at least one of equalities does not hold.
#> 

# \donttest{
## test when k=5 (samples)
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  mylist = list()
  for (j in 1:5){
     mylist[[j]] = rnorm(50)   
  }
  
  counter[i] = ifelse(vark.1974BF(mylist)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'vark.1974BF'\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=""))
#> 
#> * Example for 'vark.1974BF'
#> *
#> * number of rejections   : 47
#> * total number of trials : 1000
#> * empirical Type 1 error : 0.047
# }