I have community abundance data where each species has a mean and a standard deviation (determined by replicate samples). I am comparing the diversity of two communities. The various diversity metrics (shannon, simpson, ...) use the mean abundances to calculate diversity, but that means that the uncertainty information provided by the standard deviations is lost. Is there a way to include the standard deviations in the diversity calculation? In other words, is there a way to propagate uncertainty through the diversity metric calculation?
I know how to propagate error of sums (for $C = A + B$, then $S_C^2 = S_A^2 + S_B^2$)
And how to propagate error of mult/div (for $C = A * B$, then $(S_C/C)^2 = (S_A/A)^2 + (S_B/B)^2$
And I have some vague memories that to propagate uncertainty through more complex equations (like the diversity metrics that have exponents) I'd need to remember some calculus. This seems like a common problem though, so I'm assuming an ecologist or statistician has already figured it out. However, all I can find are ways to estimate uncertainty using bootstrapping of a single sample, not by incorporating known standard deviations. Is there an equation (or even better R package) that can do these diversity uncertainty calculations?
(note: I only have the mean $\pm$ sd, not the original abundances of each replicate. So it is not possible to get a conservative sd by repeating the diversity calculation on each replicate and calculating mean and sd of the replicate diversities.)