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I've been messing around with the Shannon index for diversity $ H' = -\sum_i^R {p_i \ln{p_i}} $, and it's associated 'evenness' measure $E = H'/H_{\text{max}}$. I'm trying to come up with a way of tracking the composition/distribution of a sample of categorical (financial) data over time. I've seen a way to test differences in Shannon index themselves, but I would like to see if the 'evenness' has changed significantly (as a sort of diagnostic).

Is there any $t$-test or similar that I could use to implement this?

i.e., I want to be able to test: $$H_0:E_A = E_B \\ H_1: E_A \neq E_B$$

Edit: I've been able to work out the variance for the evenness statistic, simply: $$ s^2_E = \frac{s^2_H}{(\ln{S})^2} $$ but now my problem is in finding the degrees of freedom. According to the link above, the degrees of freedom should be (replacing the $s^2_H$'s with $s^2_E$) $$ \nu = \frac{(s^2_{E_A} + s^2_{E_B})^2}{\frac{(s^2_{E_A})^2}{N_A} + \frac{(s^2_{E_B})^2}{N_B}} $$ Is this the correct formula?

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I've been able to use the sample variances of the evenness parameter as described in my question in calculating the degrees of freedom of the test.

To answer my question fully, the $t$-test as described in the link above is suitable when adapted for evenness. I've been using this test along with a $\chi^2$ goodness-of-fit test to test for differences in shape and relative abundance between samples.

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A paired (if the individuals are the same for the 2 dates) t-test should work for the Eveness as well. If the assumptions are not respected, you can go for its non-parametric counterpart, the Wilcoxon signed-ranked test.

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  • $\begingroup$ Thank you, do I have to transform the test statistic (as I am manipulating the actual parameter being tested)? $\endgroup$ – davkav9 Jun 17 at 9:31
  • $\begingroup$ The tests that I mentionned above are widely implemented in softwares such as R and Python, you just need to import your data and find the right function (t.test() in R, with paired = TRUE). It will allow you assess the significance without computing the test. $\endgroup$ – Circus pygargus Jun 17 at 10:22
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    $\begingroup$ I'm doing this from a theoretical perspective at the minute, but it seems correct to me to use the variances of the evenness parameter, but I only have it for the SI iteself. I'm wondering if there is some transformation of this I should do $\endgroup$ – davkav9 Jun 17 at 10:30

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