test for equality of scale

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A non-parametric test of the null hypothesis that samples have been drawn from populations with a common distribution, with the alternative being that the distributions have the same mean (or median) but different scales (and thus different variances).

The test statistic of the 1953 Rosenbaum test is the number of observations in one sample that exceed (or are less than) all the observations from the second sample.

The Mood dispersion test introduced by Mood in 1954 is suitable for symmetric populations. The test is based on replacing the original values with their ranks in the combined sample. Suppose that the samples have sizes m and n, with m+n=N. The test statistic is M, given by where rj is the overall rank in the combined sample of the jth observation in the sample of size m. If m and n are reasonably large, then the transformed statistic z is an observation from an approximate standard normal distribution, where If there are tied ranks (see ranks) then the variance needs a correction.

The Barton–David test assumes a symmetric distribution, as does the Levene test, suggested by Levene in 1960 for comparing several populations. Let the kth observation in the jth of m samples be denoted by yjk and let the mean of the nj observations in this sample be denoted by y¯j. Define zjk by


with the mean of the z-values in the jth sample being denoted by z¯j and the overall mean of the n(=∑jnj) z-values being z¯. The Levene test statistic is W given by If the null hypothesis of equality of variance is correct, then the distribution of W is approximately an F-distribution with (m − 1) and (nm) degrees of freedom.

A drawback of the Levene test is that the sample means may be affected by outliers. The Brown–Forsythe test, suggested in 1974, avoids this problem by working with zjk instead of zjk, where


and Mj is the median of the jth sample.

A related test is the Fligner–Killeen test, suggested in 1976. Let Rjk be the rank of zjk amongst the n ordered zjk-values and define a(j) by: The test statistic, X2, is given by: where Under the null hypothesis of equality of variances, X2 is an observation from a chi-squared distribution with (m − 1) degrees of freedom.

The Siegel–Tukey test introduced by Siegel and Tukey in 1960 uses alternatives to ranks that reflect the spread of the data. Denoting the N ordered values in the combined sample by x(1), x(2),…, x(N−1), x(N), these replacement values are determined as follows:

x(1) → 1, x(N) → 2, x(N−1) → 3, x(2) → 4, x(3) → 5, x(N−2) → 6,….

The test statistic is the sum of these values corresponding to the observations in the smaller sample.

Similarly motivated, the Ansari–Bradley test introduced by Ansari and Bradley in 1962 uses replacement values given by

x(1) → 1, x(N) → 1, x(N−1) → 2, x(2) → 2, x(3) → 3, x(N−2) → 3,…,


Subjects: Probability and Statistics.

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