# Brown-Forsythe F* Test

The Brown-Forsythe test is useful when the variances across the different groups are not equal. This test uses the statistic  and is based on the following property.

Property 1: If F* is defined as follows

then F* ~ F(k – 1, df) where the degrees of freedom (also referred to as df*) are

and

With the same sized samples for each group, F* = F, but the denominator degrees of freedom will be different. When the ANOVA assumptions are satisfied, F* is slightly less powerful than the standard F test, but it is still an unbiased, valid test. When variances are unequal F will be biased, especially when the cell sizes are unequal; in this case F* remains unbiased but valid.

Example 1: Repeat Example 2 of Basic Concepts for ANOVA using the Brown-Forsythe F* test.

Figure 1 – Brown-Forsythe F* test for Example 1

We start by running the Anova: Single Factor data analysis on the data in the range A3:D11 in Figure 3 of Basic Concepts for ANOVA. The result is shown on the left side of Figure 1. We then add the total sample size (cell G11) using the formula =SUM(G7:G10).

We next build the two tables on the right of Figure 1. Cells in the range N7:N10 contain the numerators of the formulas for the mj described above. The sum of these (in cell N11) is the denominator of the quotient that produces F*. Cells in the range O7:O10 contain the mj. Cells in the range P7:P10 contain the values in the denominator of the formula for df. The reciprocal of the sum of these values is df (in cell P11). Figure 2 contains some of the key formulas for the implementation.

Figure 2 – Representative formulas in Figure 1

Note that since df*is not an integer, we elect to use F_DIST_RT instead F.DIST.RT to get a more exact p-value.

Since variances of the data are quite similar and the samples are of equal size, the F and p-values from Brown-Forsythe are not much different from those in the standard ANOVA of Example 2 of Basic Concepts for ANOVA.

Real Statistics Excel Function: The Real Statistics Resource Pack contains the following supplemental functions where R1 is the data without headings, organized by columns:

BFTEST(R1) = p-value of the Brown-Forsythe’s test on the data in R1

FSTAR(R1) = F* for the Brown-Forsythe’s test on the data in R1

DFSTAR(R1) = df* for the Brown-Forsythe’s test on the data in R1

For Example 1, we have BFTEST(A4:D11) = .044935, FSTAR(A4:D11) = 3.0556 and DFSTAR(A4:D11) = 27.5895 (where A4:D11 refers to Figure 3 of Basic Concepts for ANOVA). If the last sample element in Method 1 and the last two sample elements in Method 4 are deleted (i.e. the data in Example 3 of Basic Concepts for ANOVA), then BFTEST(A4:D11) = .074804, and so this time there is no significant difference between the four methods.

Finally, the following array function combines all of the above functions:

FSTAR_TEST(R1, lab): outputs a column range with the values F*, df1, df2 and p-value for Brown-Forsythe’s F* test for the data in ranges R1.

If lab = TRUE a column of labels is added to the output, while if lab = FALSE (default) no labels are added.

Real Statistics Data Analysis Tool: The Real Statistics Resource Pack provides access to Brown-Forsythe’s F-star test via the One Factor Anova data analysis tool, as described in the following example.

Example 2: Repeat Example 1 using the data on the left side of Figure 3.

Figure 3 – Brown-Forsythe data and dialog box

Enter Ctrl-m and double click on Analysis of Variance. Select Anova: one factor on the dialog box that appears. Now fill in the dialog box that appears as shown on the right side of Figure 3.

The output is shown in Figure 4.

Figure 4 – Brown-Forsythe F* data analysis

### 17 Responses to Brown-Forsythe F* Test

1. Sarah Laurello says:

I just installed the Real Stats Add In on Excel. Following the above example, which value do I enter to use the BFTEST? Above it says (A4:D11); however there is not A4-D11 for this example. I have the same question about the FSTAR function and the DFSTAR function. Thank you!

Hi Sarah,
You are correct. The range A4:D11 refers to the data in Figure 3 of the Basic Concepts for ANOVA page (where the original example is found). I have now updated the Brown-Forsythe page to make this clearer.
Charles

2. mohamed hzaman says:

hallo,
I have downloaded the real statistics on my excel, it is working very well,, but I do not know how to analyze my data, I have 6 means measurements and need to know through a simple way how to do Tukey HSD table that tell me which one is significant or not to which one,,,
I am working in biology field and my background not so sharp in statistics
Yours
Mohamed

3. Kevin Bluxome says:

Hi Mr. Zaiontz,

Hope you’re doing well. Glad I could be of assistance with the Schierer Ray Hare test for the next release. Regarding the BF test, I have a question. Looking at the way the BF F statistic is calculated, it appears that smaller sample sizes are given larger “weights,” for lack of a better term…the ratio of (1-(group size/total N)) will be larger for smaller groups. If these smaller groups have smaller variances as a whole than the larger groups, it seems that to some extent, the larger weight given to these smaller variances would more or less even out. Conversely, if the smaller groups have the larger variances, wouldn’t that overly inflate the denominator of the F statistic and reduce its value? To me, that would suggest that the BF statistic was being overly conservative in the latter situation (and fairly well on target in the former), but isn’t the Welch F supposed to be generally more accurate (and more conservative) than the BF F statistic with regard to type I error? Am I missing something in my interpretation of the formula? Thanks again for the great work!

• Charles says:

Kevin,

I am very pleased that you like the Real Statistics website and software. Thanks again for proposing the Scheirer Ray Hare test. It is now part of the current release.

The situation that is worse for the usual ANOVA is when the smaller groups have the bigger variances. If BF compensates for this then it is probably doing the right thing.

Yes, generally Welch’s test is preferred over the BF test, esp. when the homogeneity of variance assumption is violated.

Charles

4. hojat alaii says:

thank you very much
I couldn’t find figure 3 in my excel can you help me?

• Charles says:

I’m not sure I understand the problem that you are having. Are you saying that you can’t find the Brown-Forsythe data analysis tool on your version of the Excel Real Statistics Resource Pack? What version of the Real Statistics Resource Pack are you using? To find out enter the formula =VER() in any cell.
Charles

5. Ram Ritter says:

Hello
what’s the right post hoc test that follows a Brown-Forsythe test when the variances across the different groups are not equal? may i use tukey HSD or should it be Games-Howell (as it is after Welch’s test)?

• Charles says:

Ram,
Probably Games-Howell. Why did you decide to use Brown-Forsythe instead of Welch’s test?
Charles

• Ram Ritter says:

I have a problem with the Games-Howell not showing a significant difference between the groups and hoped i would be able to use tukey- HSD as a post HOC after Brown-Forsythe.

• Charles says:

Ram,
You shouldn’t choose the test based on the result that you would like to see.
Charles

6. Gagandeep says:

Hi,

If I have 10 Methods data (i.e. 10 columns of data) and when I run Brown-Forsthe test, I get p < 0.05.

This implies the variances varry.

Now, how do I get to know which columns have same variances and which do not?

• Charles says:

The Brown-Forsythe F* test (at least the version of the test that I describe on the referenced webpage) is not used to test whether the variances are unequal, but instead is used instead of ANOVA when you already know that the variances are unequal.
In my experience, in this case, you are usually better off using Welch’s ANOVA rather than Brown-Forsythe.
If you get a significant result (p < .05) from Brown-Forsythe (or Welch's ANOVA) the commonly used post-hoc test to identify which means are unequal is the Games-Howell test.
Charles

7. John says:

Hello all,

How to report the Welch’s F test. For example this are the results:

statistic df1 df2 sig
Welch 13,458 1 70,168 ,001

• John says:

Statistic: 13,458
df1: 1
df2: 70, 168
sig: ,001

• Charles says:

John,
Report it the same way you would for one-way Anova.
Charles