# Testing when the profile of a sample has changed - degrees of freedom?

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?

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.