**The Brown Forsythe test is the best test to compare the variance of 2 or more samples.**

## Introduction

Published in 1974 following the work of Morton B. Brown and Alan B. Forsythe, the Brown-Forsythe test is a parametric approach for testing the equal variances of 2 or more samples. It is the best test to compare variances, much more robust than the Fisher^{1}test. It has the interest of being little sensitive to the non-normality of the data.

## The principle

This test is a generalization of the Levene test. It’s based on the variance ratio. But to increase the robustness of the test, it transforms them through the median. The idea is thus to be able to better smooth the values by avoiding being too dependent on the extreme values that ” *blur* ” the results.

## Step 1: Assumptions

The Brown Forsythe test compares the Variance of 2 or more under population. It confronts the following two hypotheses:

**H0 ****Σ _{1}^{2} = σ_{2}^{2} =… = Σ_{k}^{2}**

**H1: Σ _{1}^{2} ≠ σ_{2}^{2 }≠… ≠ σ_{k}^{2}**

## Step 2: Transform variables

The first step in this test is to transform the variables to make them more robust. They are transformed with the following formula:

**X ‘ _{ik} = i x_{ik} – x_{k} I**

- X ‘
_{ik}: Each of the transformed values of all our samples - x
_{ik}: Each of the ”*original*” values of all our samples - X
_{k}: The median of the K-sample

## Step 3: Practical value

Practical value is a bit like the **ANOVA**. Then we calculate the ratio between the mean squares explained and the residual means squares on the transformed value. The practical value is calculated according to the formula:

**n**The individual number of the K sample_{k}:**N:**total number of individuals**K:**the number of samples**μ**average of the processed values of the K sample_{k}:**μ:**The average of all the processed values of the samples**x**Each of the transformed values of the samples K_{k}:

## Step 4: Critical value

The critical value follows Fisher Snedecor’s law. It should be noted that, under the hypothesis, a bilateral test is always carried out as we compare more than 2 samples.

**DDL**K – 1, where k is the number of samples._{1}:**DDL**n_{2}:_{ }-K, where n is the total number of individuals.

It is calculated either with the Fisher table or with the EXCEL spreadsheet via the formula:

**Critical value = F.INV (1-α; dof _{1} ); dof _{2})**

## Step 5: P-Value

The **P-Value** Also follows Fisher’s law. It is calculated in Excel using the formula:

**P-Value = law. F. (Practical value; dof _{1} ); dof _{2})**

## Step 6: Interpretation

Result | Statistical conclusion | Practical conclusion |
---|---|---|

Practical value < Critical value | We retain H0 | There is no significant difference between the different samples at risk α to be wrong |

Practical value> Critical value | We reject H0 | There is a significant difference between the different samples at risk α to be wrong |

Result | Statistical conclusion | Practical conclusion |
---|---|---|

p-value > α | We retain H0 | There is no difference between the samples with a risk of being wrong with p-value% |

p-Value ≤ α | We reject H0 | There is a difference between the samples with a risk of being wrong of p-value% |