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# Levene-Brown-Forsythe test

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# Levene-Brown-Forsythe test

This test is a variant on the standard one-way analysis of variance (ANOVA) test, but is applied when the variances of the groups are not homogeneous. It simply involves a centering transformation of the data {yij} in each sample group by subtracting each value from the group mean, median or trimmed mean. Hence the new variable z, is obtained as:

where the ~ symbol indicates one of the set {mean, median, trimmed mean} and the dot subscript indicates summation over that subscript. The test statistic, W, is then defined as follows:

where the number of groups is k and the number of samples in the ith group is Ni.

The test statistic is then compared to the critical values of the F-distribution with (k-1) and (N-k) degrees of freedom (df).

Example: Comparison of a batch of gears

In the NIST test data example [NIST] a total of 100 measurements were made on 10 batches of gears in a manufacturing process. The value for W based on this data using the median as the transform, was 1.7059, and df=9 and 90. Looking up the F-distribution for this value (e.g. using the Excel expression =(1-FDIST(W,df1,df2))*100 we find a probability level of 90.09%. Assuming we are looking at a 95% level to determine the significance of the ratio, we conclude that the 10 batches do not have significant differences in their pattern of variation.

References

[BRO1] Brown M B, Forsythe A B (1974) Robust Tests for Equality of Variances. J of the American Statistical Association, 69, 364–367

[NIST] NIST/Sematech: Levene test: https://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm