Fisher’s F-Test in R Programming

Fisher’s F test calculates the ratio between the larger variance and the smaller variance. We use the F test when we want to check whether the means of three or more groups are different or not. F-test is used to assess whether the variances of two populations (A and B) are equal. The method is simple; it consists of taking the ratio between the larger variance and the smaller variance. var.test() function in R Programming performs an F-test between 2 normal populations with the hypothesis that variances of the 2 populations are equal.

Formula for Fisher’s F-Test

F = Larger Sample Variance / Smaller Sample Variance

Implementation in R

• To test the equality of variances between the two samples use var.test(x, y)
• To compare two variance use var.test(x, y, alternative = “two.sided”)

Syntax: var.test(x, y, alternative = “two.sided”) Parameters: x, y: numeric vectors alternative: a character string specifying the alternative hypothesis.

Example 1:

Let we have two samples x, y. The R function var.test() can be used to compare two variances as follow: 

r

# Taking two samples
x <- rnorm(249, mean = 20)
y <- rnorm(79, mean = 30)
# var test in R
var.test(x, y, alternative = "two.sided")

Output:

       F test to compare two variances
data:  x and y
F = 0.88707, num df = 248, denom df = 78, p-value = 0.4901
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.6071405 1.2521004
sample estimates:
ratio of variances  
        0.8870677 

It returns the following:

1. the value of the F test statistic.
2. the degrees of freedom of the F distribution of the test statistic.
3. the p-value of the test is 0.4901
4. a confidence interval for the ratio of the population variances.
5. the ratio of the sample variances is 0.8870677

The p-value of the F-test is p = 0.4901 which is greater than the alpha level of 0.05. In conclusion, there is no difference between the two samples.

Example 2:

Let us have two random samples from two random populations. Test whether two populations have the same variance. 

r

# Taking two random samples
A = c(16, 17, 25, 26, 32,
        34, 38, 40, 42)
B = c(600, 590, 590, 630, 610, 630)
# var test in R
var.test(A, B, alternative = "two.sided")

Output:

       F test to compare two variances
data:  A and B
F = 0.27252, num df = 8, denom df = 5, p-value = 0.1012
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.04033118 1.31282683
sample estimates:
ratio of variances  
        0.2725248

It returns the following:

1. the value of the F test statistic.
2. the degrees of freedom of the F distribution of the test statistic.
3. the p-value of the test is 0.1012
4. 95% confidence interval for the ratio of the population variances.
5. the ratio of the sample variances is 0.2725248

The p-value of the F-test is p = 0.1012 which is greater than the alpha level of 0.05. In conclusion, there is no difference between the two samples. 

Example 3:

 Let us have two random samples. 

r

# Taking two random samples
x = c(25, 29, 35, 46, 58, 66, 68)
y = c(14, 16, 24, 28, 32, 35,
          37, 42, 43, 45, 47)
# var test in R
var.test(x, y)

Output:

       F test to compare two variances
data:  x and y
F = 2.4081, num df = 6, denom df = 10, p-value = 0.2105
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.5913612 13.1514157
sample estimates:
ratio of variances  
           2.4081

It returns the following:

1. the value of the F test statistic.
2. the degrees of freedom of the F distribution of the test statistic.
3. the p-value of the test is 0.2105
4. 95% confidence interval for the ratio of the population variances.
5. the ratio of the sample variances is 2.4081

The p-value of the F-test is p = 0.2105 which is greater than the alpha level of 0.05. In conclusion, there is no difference between the two samples.