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
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
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) = .04534, 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.
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