There are different types of tests that can be utilized to assess the equality of variances.

  1. F-test:- Used for two groups variance comparison. Data must be normally distributed.

  2. Bartlett’s test:- Used for two or more groups variance comparison. Data must be normally distributed.

  3. Levene’s test:- An alternative to Bartlett’s test for non-normally distributed data.

  4. Fligner-Killeen’s test:- A non-parametric test for non-normal data.

Two sample equality of variances in R

1. F-test in R

ToothGrowth data set we used for F test calculation.

Let’s see the 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 ...

Total 60 observations and 3 variables and the variable supp contains two groups.

Shapiro Test:

shapiro.test(ToothGrowth$len)

    Shapiro-Wilk normality test

data:  ToothGrowth$len
W = 0.96743, p-value = 0.1091

The p value is greater than 0.05, we can assume the normality.

res.ftest <- var.test(len ~ supp, data = ToothGrowth)
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

The p-value of 0.2331 is greater than the significance level of 0.05. We can conclude that there is no significant difference between the two variances.

Compare more than two sample variances in R

2. Bartlett’s test in R

We are using PlantGrowth dataset contains 30 observations and 2 variables.

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 ...

The column group contains 3 factor variables ctrl, trt1, and trt2. Before doing Bartlett’s test let’s check the normality assumption.

Shapiro Test:

shapiro.test(PlantGrowth$weight)

    Shapiro-Wilk normality test

data:  PlantGrowth$weight
W = 0.98268, p-value = 0.8915

We can assume that data is normally distributed.

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

The p-value is 0.2371 is greater than the significance level of 0.05. We can conclude that there is no significant difference between the tested sample variances.

3. Levene’s test in R

Levene test function is from car package, let’s load the library.

Warning: package 'car' was built under R version 4.2.1
Loading required package: carData
Warning: package 'carData' was built under R version 4.2.1
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

The p-value is 0.3412 is greater than the significance level of 0.05. We can conclude that there is no significant difference between the tested sample variances.

Levene’s test with multiple independent variables can check based on ToothGrowth dataset,

ToothGrowth dataset dose column stored as numeric variable let’s convert into factor variable first,

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

4. Fligner-Killeen test in R

Will make use of the same data set,

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

The p-value is 0.3088 is greater than the significance level of 0.05. We can conclude that there is no significant difference was observed between the tested sample variances.