Talk:Brown–Forsythe test

What, in the following formula, cut and pasted from the article, are 'z_{\cdot j}' and 'z_{\cdot\cdot}'?

$$ F = \frac{(N-p)}{(p-1)} \frac{\sum_{j=1}^{p} n_j (z_{\cdot j}-z_{\cdot\cdot})^2} {\sum_{j=1}^{p}\sum_{i=1}^{n_j} (z_{ij}-z_{\cdot j})^2} $$

Briancady413 (talk) 20:59, 1 April 2009 (UTC)
 * See the definitions at Levene's test; they are the group means and overall mean. Btyner (talk) 00:12, 2 April 2009 (UTC)

Sqrt50850 —Preceding undated comment added 12:06, 8 April 2009 (UTC). To clarify Briancady- the double dot (..) is the mean of the overall sample (inter-group mean). The single dot (.) is the mean of the samples within the group.

Odd number of observations
The page now says, "In order to correct for the artificial zeros that come about with odd numbers of observations in a group, any zij that equals zero is replaced by the next smallest zij in group j. The Brown–Forsythe test statistic is the model F statistic from a one way ANOVA on zij.". I have not seen this adjustment in texts books. Reference? Pros and cons? HarveyMotulsky (talk) 17:52, 3 February 2010 (UTC)

I did not add that sentence to the main article so I am not sure where that idea came from. As far as I know, one of the first articles mentioning the low power of Brown-Forsythe for small odd sample size is Martin (1976, see pp.555). O'Brien (1978) is one of the first articles suggesting removal of a random observation from a sample with a small odd number of observations (see pp. 335). Hines and O'Hara Hines (2000) suggest removal of the smallest zij from sample with an odd number of observations (which is equal to zero). They also propose a transformation for even sample sizes, which is to replace the two smallest zij's (say, zi1 and zi2) into (0, sqrt{2}zi1). If that is what the original author was referring to, then the statement in the main article could be misleading and needs to be corrected.

References:

Martin, C.G. (1976) Comment on Levy's "an empirical comparison of the Z-variance and Box-Scheffé tests for homogeneity of variance". Psychometrika, 41, 551--556.

O'Brien, R.G. (1978) Robust techniques for testing heterogeneity of variance effects in factorial designs. Psychometrika, 43, 327--344.

Hines, W.G.S. and O'Hara Hines, R.J. (2000) Increased power with modified forms of the Levene (med) test for heterogeneity of variance. Biometrics, 56, 451--454.

Statshop (talk) 03:29, 5 February 2010 (UTC)

Comparison to Levene's Test
I am not an expert but, if I understand correctly, this test is very similar to Levene's test, however the nomenclature in the equations is different. Would it be possible for someone to edit the equation to make them consistent. I know it's not a big deal but it would be nice to have them matching, as this would help understanding for someone such as me.

The equation in this article:
 * $$F = \frac{(N-p)}{(p-1)} \frac{\sum_{j=1}^{p} n_j (\tilde{z}_{\cdot j}-\tilde{z}_{\cdot\cdot})^2} {\sum_{j=1}^{p}\sum_{i=1}^{n_j} (z_{ij}-\tilde{z}_{\cdot j})^2}$$

and in Levene's test
 * $$W = \frac{(N-k)}{(k-1)} \cdot \frac{\sum_{i=1}^k N_i (Z_{i\cdot}-Z_{\cdot\cdot})^2} {\sum_{i=1}^k \sum_{j=1}^{N_i} (Z_{ij}-Z_{i\cdot})^2},$$

Here I think p and k are equivalent, z and Z are equivalent, n and N are equivalent, and i and j are swapped.

Peterhull90 (talk) 10:13, 5 June 2019 (UTC)

Brown-Forsythe test for Means
The article seems to be only about the Brown-Forsythe test for equality of variances, but there is also a Brown-Forsythe test for equality of means. The original source for that is:

Brown, M. B., & Forsythe, A. B. (1974). The small sample behavior of some statistics which test the equality of several means. Technometrics, 16(1), 129–132. https://doi.org/10.1080/00401706.1974.10489158 Stikpet (talk) 15:04, 4 December 2022 (UTC)