# Can a Levene's test be conducted with only summary statistics?

My research question specifically deals with testing the variances of 3+ groups for significant differences. I'm not looking to test an assumption of homoscedasticity for an ANOVA or any other test, but rather my end goal is the testing of variances. Unfortunately, I only have my own complete dataset but the other groups come from previously published works that only reported summary statistics (N, means, standard deviations)

Unfortunately I cannot ask the researchers for their original data since some of these studies are quite old. I've managed to simplify most of the Levene's test formula to only using what I have but I'm stuck at the principal transformation for the test and can't simplify it further.

Does anyone have any suggestions on how I may either conduct a Levene's with just this information or other tests of variance that might work for this dilemma. I have wondered about the use of the Hartley's test, but so many people warn against it that I'm cautious.

• You must tell us which summary statistics you do have! Nov 19, 2015 at 9:11
• Unfortunately, as I mentioned, I only have Sample Size (N), Mean, and Standard Deviation Nov 19, 2015 at 23:20
• I forgot to mention that I am fairly certain I can assume Normality for some larger samples (~100 individuals, so still not very large) Nov 19, 2015 at 23:23

Does anyone have any suggestions on how I may either conduct a Levene's with just this information

You can't. You need to be able to recover the summary statistics (mean and variance) for the $|Y_{ij} - \bar{Y}_{i\cdot}|$ values, and in general there's no way to do that for the mean and variance of the $Y$s.

or other tests of variance that might work for this dilemma.

Anything based on direct subgroup variances can at least be performed, but the standard tests will have the usual lack of robustness issues.

I don't think that's really possible. Levene's test is a $t$-test (ANOVA for >2 groups) of the absolute values of the deviations of the data from the mean (median for the Brown-Forsythe variant). Thus, you need the data for that.

If you are willing to assume strict normality (it's a big assumption here), you could use Hartley's $F_\max$ test with only variances and $N$'s:
$$F_\max = \frac{s^2_2}{s^2_1}$$

• Thanks! I had already looked at Hatley's and was very cautious. Hatley's is also limited in testing only the two extremes. I did find Bartlett's test which can be run with summary statistics as an omnibus in the same way that Levene's can. Still have trouble with the sensitivity to non-normality, but I can test all samples at the same time. Dec 9, 2015 at 4:38