1 How this chapter is organized?

2 F-Test: Compare two variances in R

• What is F-test?
• When to you use the F-test?
• Research questions and statistical hypotheses
• Formula of F-test
• Compute F-test in R

Read more: —> F-Test: Compare Two Variances in R.

3 Compare multiple sample variances in R

This article describes statistical tests for comparing the variances of two or more samples.

• Compute Bartlett’s test in R
• Compute Levene’s test in R
• Compute Fligner-Killeen test in R

Read more: —> Compare Multiple Sample Variances in R.

4 See also

• R Basics
• Import and Export Data using R
• Preparing and Reshaping Data in R for Easier Analyses
• Data Manipulation in R
• Data visualization
• Descriptive Statistics and Graphics
• Correlation Analyses in R
• Comparing Means in R

5 Infos

This analysis has been performed using R statistical software (ver. 3.2.4).

When to you use the 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:

In statistics, we can define the corresponding null hypothesis (\(H_0\)) as follow:

1. \(H_0: \sigma^2_A = \sigma^2_B\)
2. \(H_0: \sigma^2_A \leq \sigma^2_B\)
3. \(H_0: \sigma^2_A \geq \sigma^2_B\)

The corresponding alternative hypotheses (\(H_a\)) are as follow:

1. \(H_a: \sigma^2_A \ne \sigma^2_B\) (different)
2. \(H_a: \sigma^2_A > \sigma^2_B\) (greater)
3. \(H_a: \sigma^2_A < \sigma^2_B\) (less)

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

# Method 1
var.test(values ~ groups, data, 
         alternative = "two.sided")
# or Method 2
var.test(x, y, alternative = "two.sided")

# If .txt tab file, use this
my_data <- read.delim(file.choose())
# Or, if .csv file, use this
my_data <- read.csv(file.choose())

# Store the data in the variable my_data
my_data <- ToothGrowth

library("dplyr")
sample_n(my_data, 10)

    len supp dose
43 23.6   OJ  1.0
28 21.5   VC  2.0
25 26.4   VC  2.0
56 30.9   OJ  2.0
46 25.2   OJ  1.0
7  11.2   VC  0.5
16 17.3   VC  1.0
4   5.8   VC  0.5
48 21.2   OJ  1.0
37  8.2   OJ  0.5

# 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 

# ratio of variances
res.ftest$estimate

ratio of variances 
         0.6385951 

# p-value of the test
res.ftest$p.value

[1] 0.2331433

Infos

This analysis has been performed using R software (ver. 3.3.2).

Statistical tests for comparing variances

There are many solutions to test for the equality (homogeneity) of variance across groups, including:

• F-test: Compare the variances of two samples. The data must be normally distributed.

• Bartlett’s test: Compare the variances of k samples, where k can be more than two samples. The data must be normally distributed. The Levene test is an alternative to the Bartlett test that is less sensitive to departures from normality.

• Levene’s test: Compare the variances of k samples, where k can be more than two samples. It’s an alternative to the Bartlett’s test that is less sensitive to departures from normality.

• Fligner-Killeen test: a non-parametric test which is very robust against departures from normality.

Statistical hypotheses

For all these tests (Bartlett’s test, Levene’s test or Fligner-Killeen’s test),

• the null hypothesis is that all populations variances are equal;
• the alternative hypothesis is that at least two of them differ.

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 ToothGrowth and PlantGrowth data sets:

# Load the data
data(ToothGrowth)

data(PlantGrowth)

To 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]. First, install dplyr package if you don’t have it: install.packages(“dplyr”).

Show 10 random rows:

set.seed(123)
# Show PlantGrowth
dplyr::sample_n(PlantGrowth, 10)

   weight group
24   5.50  trt2
12   4.17  trt1
25   5.37  trt2
26   5.29  trt2
2    5.58  ctrl
14   3.59  trt1
22   5.12  trt2
13   4.41  trt1
11   4.81  trt1
21   6.31  trt2

# PlantGrowth data structure
str(PlantGrowth)

'data.frame':   30 obs. of  2 variables:
 $ weight: num  4.17 5.58 5.18 6.11 4.5 4.61 5.17 4.53 5.33 5.14 ...
 $ group : Factor w/ 3 levels "ctrl","trt1",..: 1 1 1 1 1 1 1 1 1 1 ...

# Show ToothGrowth
dplyr::sample_n(ToothGrowth, 10)

    len supp dose
28 21.5   VC  2.0
40  9.7   OJ  0.5
34  9.7   OJ  0.5
6  10.0   VC  0.5
51 25.5   OJ  2.0
14 17.3   VC  1.0
3   7.3   VC  0.5
18 14.5   VC  1.0
50 27.3   OJ  1.0
46 25.2   OJ  1.0

# ToothGrowth data structure
str(ToothGrowth)

'data.frame':   60 obs. of  3 variables:
 $ len : num  4.2 11.5 7.3 5.8 6.4 10 11.2 11.2 5.2 7 ...
 $ supp: Factor w/ 2 levels "OJ","VC": 2 2 2 2 2 2 2 2 2 2 ...
 $ dose: num  0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 ...

Note that, R considers the column “dose” [in ToothGrowth data set] as a numeric vector. We want to convert it as a grouping variable (factor).

ToothGrowth$dose <- as.factor(ToothGrowth$dose)

We want to test the equality of variances between groups.

Compute Bartlett’s test in R

The R function bartlett.test() can be used to compute Barlett’s test. The simplified format is as follow:

bartlett.test(formula, data)

• formula: a formula of the form values ~ groups
• data: a matrix or data frame

The function returns a list containing the following component:

To perform the test, we’ll use the PlantGrowth data set, which contains the weight of plants obtained under 3 treatment groups.

• Bartlett’s test with one independent variable:

res <- bartlett.test(weight ~ group, data = PlantGrowth)
res

    Bartlett test of homogeneity of variances

data:  weight by group
Bartlett's K-squared = 2.8786, df = 2, p-value = 0.2371

From the output, it can be seen that the p-value of 0.2370968 is not less than the significance level of 0.05. This means that there is no evidence to suggest that the variance in plant growth is statistically significantly different for the three treatment groups.

• Bartlett’s test with multiple independent variables: the interaction() function must be used to collapse multiple factors into a single variable containing all combinations of the factors.

bartlett.test(len ~ interaction(supp,dose), data=ToothGrowth)

    Bartlett test of homogeneity of variances

data:  len by interaction(supp, dose)
Bartlett's K-squared = 6.9273, df = 5, p-value = 0.2261

Compute Levene’s test in R

As mentioned above, Levene’s test is an alternative to Bartlett’s test when the data is not normally distributed.

The function leveneTest() [in car package] can be used.

library(car)
# Levene's test with one independent variable
leveneTest(weight ~ group, data = PlantGrowth)

Levene's Test for Homogeneity of Variance (center = median)
      Df F value Pr(>F)
group  2  1.1192 0.3412
      27               

# Levene's test with multiple independent variables
leveneTest(len ~ supp*dose, data = ToothGrowth)

Levene's Test for Homogeneity of Variance (center = median)
      Df F value Pr(>F)
group  5  1.7086 0.1484
      54               

Compute Fligner-Killeen test in R

The Fligner-Killeen test is one of the many tests for homogeneity of variances which is most robust against departures from normality.

The R function fligner.test() can be used to compute the test:

fligner.test(weight ~ group, data = PlantGrowth)

    Fligner-Killeen test of homogeneity of variances

data:  weight by group
Fligner-Killeen:med chi-squared = 2.3499, df = 2, p-value = 0.3088

Infos

This analysis has been performed using R software (ver. 3.2.4).