What is F-test?
When to you use F-test?
Comparing two variances is useful in several cases, including:
When you want to perform a two samples t-test to check the equality of the variances of the two samples
When you want to compare the variability of a new measurement method to an old one. Does the new method reduce the variability of the measure?
Research questions and statistical hypotheses
Typical research questions are:
- whether the variance of group A (\(\sigma^2_A\)) is equal to the variance of group B (\(\sigma^2_B\))?
- whether the variance of group A (\(\sigma^2_A\)) is less than the variance of group B (\(\sigma^2_B\))?
- whether the variance of group A (\(\sigma^2_A\)) is greather than the variance of group B (\(\sigma^2_B\))?
In statistics, we can define the corresponding null hypothesis (\(H_0\)) as follow:
- \(H_0: \sigma^2_A = \sigma^2_B\)
- \(H_0: \sigma^2_A \leq \sigma^2_B\)
- \(H_0: \sigma^2_A \geq \sigma^2_B\)
The corresponding alternative hypotheses (\(H_a\)) are as follow:
- \(H_a: \sigma^2_A \ne \sigma^2_B\) (different)
- \(H_a: \sigma^2_A > \sigma^2_B\) (greater)
- \(H_a: \sigma^2_A < \sigma^2_B\) (less)
Note that:
- Hypotheses 1) are called two-tailed tests
- Hypotheses 2) and 3) are called one-tailed tests
Formula of F-test
The test statistic can be obtained by computing the ratio of the two variances \(S_A^2\) and \(S_B^2\).
\[F = \frac{S_A^2}{S_B^2}\]
The degrees of freedom are \(n_A - 1\) (for the numerator) and \(n_B - 1\) (for the denominator).
Note that, the more this ratio deviates from 1, the stronger the evidence for unequal population variances.
Note that, the F-test requires the two samples to be normally distributed.
Compute F-test in R
R function
The R function var.test() can be used to compare two variances as follow:
# Method 1
var.test(values ~ groups, data,
alternative = "two.sided")
# or Method 2
var.test(x, y, alternative = "two.sided")- x,y: numeric vectors
- alternative: the alternative hypothesis. Allowed value is one of “two.sided” (default), “greater” or “less”.
Import and check your data into R
To import your data, use the following R code:
# If .txt tab file, use this
my_data <- read.delim(file.choose())
# Or, if .csv file, use this
my_data <- read.csv(file.choose())Here, we’ll use the built-in R data set named ToothGrowth:
# Store the data in the variable my_data
my_data <- ToothGrowthTo have an idea of what the data look like, we start by displaying a random sample of 10 rows using the function sample_n()[in dplyr package]:
library("dplyr")
sample_n(my_data, 10) len supp dose
50 27.3 OJ 1.0
18 14.5 VC 1.0
41 19.7 OJ 1.0
10 7.0 VC 0.5
42 23.3 OJ 1.0
23 33.9 VC 2.0
28 21.5 VC 2.0
9 5.2 VC 0.5
25 26.4 VC 2.0
59 29.4 OJ 2.0We want to test the equality of variances between the two groups OJ and VC in the column “supp”.
Preleminary test to check F-test assumptions
F-test is very sensitive to departure from the normal assumption. You need to check whether the data is normally distributed before using the F-test.
Shapiro-Wilk test can be used to test whether the normal assumption holds. It’s also possible to use Q-Q plot (quantile-quantile plot) to graphically evaluate the normality of a variable. Q-Q plot draws the correlation between a given sample and the normal distribution.
If there is doubt about normality, the better choice is to use Levene’s test or Fligner-Killeen test, which are less sensitive to departure from normal assumption.
Compute F-test
# F-test
res.ftest <- var.test(len ~ supp, data = my_data)
res.ftest
F test to compare two variances
data: len by supp
F = 0.6386, num df = 29, denom df = 29, p-value = 0.2331
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.3039488 1.3416857
sample estimates:
ratio of variances
0.6385951 Interpretation of the result
Access to the values returned by var.test() function
The function var.test() returns a list containing the following components:
- statistic: the value of the F test statistic.
- parameter: the degrees of the freedom of the F distribution of the test statistic.
- p.value: the p-value of the test.
- conf.int: a confidence interval for the ratio of the population variances.
- estimate: the ratio of the sample variances
The format of the R code to use for getting these values is as follow:
# ratio of variances
res.ftest$estimateratio of variances
0.6385951 # p-value of the test
res.ftest$p.value[1] 0.2331433Infos
This analysis has been performed using R software (ver. 3.2.4).