The R Book

Variance Ratio Test

How do we know if two variances are significantly different from one another? We need to carry out Fisher's F test, the ratio of the two variances (see p. 224). Here is a function to print the p value (p. 290) associated with a comparison of the larger and smaller variances:

variance.ratio<-function(x,y) {
  v1<-var(x)
  v2<-var(y)
  if (var(x) > var(y)) {
  vr<-var(x)/var(y)
  df1<-length(x)-1
  df2<-length(y)-1}
else { vr<-var(y)/var(x)
  df1<-length(y)-1
  df2<-length(x)-1}
2*(1-pf(vr,df1,df2)) }

The last line of our function works out the probability of getting an F ratio as big as vr or bigger by chance alone if the two variances were really the same, using the cumulative probability of the F distribution, which is an R function called pf. We need to supply pf with three arguments: the size of the variance ratio (vr), the number of degrees of freedom in the numerator (9) and the number of degrees of freedom in the denominator (also 9).

Here are some data to test our function. They are normally distributed random numbers but the first set has a variance of 4 and the second a variance of 16 (i.e. standard deviations of 2 and 4, respectively):

a<-rnorm(10,15,2)
b<-rnorm(10,15,4)

Here is our function in action:

variance.ratio(a,b)

[1]  0.01593334

We can compare our p with the p-value given by the built-in function called var.test

var.test(a,b) F test to compare two variances data: a and b F = 0.1748, num df = 9, denom df = 9, p-value = 0.01593 alternative hypothesis: ...