# Brown-Forsythe test for equality of means in R

## Introduction

ANOVA is a widely used method for comparing the means of *k* independent groups. ANOVA test is valid and
more powerful when it meets three basic assumptions including independence of observations within and between groups,
homogeneity of variances, and normal distribution of observations.

However, most of the time, assumptions of homogeneity of variances and normality are not satisfied, and the inferences derived from ANOVA under these circumstances may not be valid. ANOVA is less powerful (poor control over type I and II error rates) if the assumption of homogeneity of variances is not satisfied.

Brown-Forsythe test for equality of means is an alternative to ANOVA when the assumption of homogeneity of variance is not satisfied for the ANOVA test.

## Brown-Forsythe test hypothesis

*Null hypothesis*: All group means are equal (no variation in means of groups)

H_{0}: μ_{1}=μ_{2}=…=μ_{k} (where *k* is number of groups)

*Alternative hypothesis*: At least, one group mean is different from other groups

H_{a}: All μ are not
equal

Learn more about hypothesis testing and interpretation

## Calculate Brown-Forsythe test in R

We will use the `onewaytests`

R packages for this tutorial.

### Get dataset

We will use the iris dataset to check if the sepal lengths of plant species are significantly different or not.

```
# R version 4.1.2 (2021-11-01)
# load iris dataset
data("iris")
head(iris, 2)
# output
Sepal.Length Sepal.Width Petal.Length Petal.Width Species
1 5.1 3.5 1.4 0.2 setosa
2 4.9 3.0 1.4 0.2 setosa
```

### Check the assumption of homogeneity of variances

We can use ANOVA to compare the means if assumption of homogeneity of variances is satisfied.

We will use the Bartlett’s homogeneity test to check the assumption of equal variance.

```
bartlett.test(Sepal.Length ~ Species, data = iris)
# output
Bartlett test of homogeneity of variances
data: Sepal.Length by Species
Bartletts K-squared = 16.006, df = 2, p-value = 0.0003345
```

Bartlett’s homogeneity test indicates that variances are not equal among the plant species as *p* value
is significant [χ2 = 16.006, *p* = 0.00033].

As the assumption of homogeneity of variances is not satisfied, we will use the Brown-Forsythe test for comparing sepal length means among the iris plant species.

### Perform Brown-Forsythe test for equal means

We will use the `bf.test`

function from `onewaytests`

package to perform the Brown-Forsythe test

Pass the following parameters to `bf.test`

function,

: Specify the formula of the form a ~ b, where a (dependent variable), b (treatment groups)**formula**: data frame for formula**data**

```
library(onewaytests)
bf.test(formula = Sepal.Length ~ Species, data = iris)
# output
Brown-Forsythe Test (alpha = 0.05)
-------------------------------------------------------------
data : Sepal.Length and Species
statistic : 119.2645
num df : 2
denom df : 123.9255
p.value : 1.317059e-29
Result : Difference is statistically significant.
-------------------------------------------------------------
```

The Brown-Forsythe test results indicate that sepal lengths are significantly different [*F _{BF}* = 119.26,

*p*< 0.001] among different iris plant species. Hence, we

__reject the null hypothesis__(as

*p*< 0.05) that group (iris plant species) means are equal.

Note: Brown-Forsythe test for equality of means is different than the Brown-Forsythe test for equality of variances. The later test can be performed using the`brown.forsythe.test`

function from the`vGWAS`

package.

## Enhance your skills with statistical courses using R

## References

- Brown MB, Forsythe AB. Robust tests for the equality of variances. Journal of the American Statistical Association. 1974 Jun 1;69(346):364-7.
- Dag O, Dolgun A, Konar NM. Onewaytests: An R Package for One-Way Tests in Independent Groups Designs. R Journal. 2018 Jul 1;10(1).

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