# Nonparametric test for equality of variances

I have two small data sets (sizes $n_1 = 8$ and $n_2 = 21$) which look like they have significantly different variances. I know very little about the underlying distributions, but it's definitely not safe to assume they're normal or anything nice like that, which rules out the F-test. I'm aware I could use one of the other named tests (Bartlett, Brown–Forsythe, ...) although I'm not currently quite sure what they assume about the population distribution, if anything.

Instead, I've tried my hand at using a permutation test: the null hypothesis is that the two datasets have equal variance, so relabel the data points at random and measure the difference in the two variances of the relabelled sets. Out of 1,000,000 attempts, fewer than 40,000 had a larger difference in variance (<4%).

Is it correct to say that, therefore, the difference in variance of the two data sets is significant at the $p < 0.05$ level? If so, is there a well-recognised name for this kind of test?

• What do you mean by relabel the data points? I don't get that part May 8, 2017 at 9:50
• It's just a two-sample permutation test with the difference in variance as the test statistic; I don't know of any special name for that (a ratio rather than a difference would perhaps be more typical for variances). Note that there are fewer than 4.3 million combinations; you could almost as easily compute the exact p-values. May 8, 2017 at 9:50
• @machaz relabel which sample the points are coming from. That's how permutation tests work (at least in the simple cases) May 8, 2017 at 9:50
• @Glen_b thanks.Now its clear.I hadn't rally used permutation tests before. May 8, 2017 at 11:17

Your permutation test setup makes perfect sense - congratulations!

No, as far as I know, there is no established name for this kind of procedure. I'd recommend you describe it as you did here, and potentially refer to any introductory textbook for permutation tests. (You might want to add whether you ran a one-sided or a two-sided test, which is not entirely clear to me from your description. Either one, of course, can be implemented in your permutation test framework.)

• There is an established name for such a procedure: it's called the Levene test for homoegeneity of variances.
– whuber
Aug 25, 2020 at 14:16
• @whuber: thanks, that's good to know. I even used Levene's test a couple of years ago, but apparently only the idea stuck... Aug 25, 2020 at 14:31
• @whuber I never heard Levene's test mentioned in the framework of permutation tests. Can you elaborate in what way Levene's test is equivallent to the permutation test described? Aug 25, 2020 at 16:01
• @COOL I wasn't trying to claim Levene's test is a permutation test. The point is that this question ultimately asks for nonparametric tests of variances and that's what Levene's is. Actually, your inquiry does reflect a deeper insight: many (maybe most?) nonparametric tests based on ranks indeed are equivalent to permutation tests. That is because the null hypotheses usually amount to supposing all possible permutations are equally likely.
– whuber
Aug 25, 2020 at 16:14
• @whuber Ah got it, thanks for the explanations. Aug 25, 2020 at 16:16