I would like to know what kind of distribution have biodiversity measure of Shannon-Wiener Function and which statistical test should I use to compare a set of such indexes. I have the same question concerning the Jaccard coefficient of similarity.
Magurran and colaborators described how to get confidence intervals, based on literature around 70's. One possibility is to carry out a bootstrap using the original values. But the underlying, more conceptual concern, is the meaning of this indices. I think that they are not much information about functionality of the community (or ensemble). They are broadly used with weak links to the species' function.
Maybe you can test if the dataset for each set of values fit a normal distribution, and then use a t-test.
In my opinion, both answers are right. It is just a number, so you can use any test, assuming the assumptions are met. I don't know too much about the Shannon, but Simpson's index is bounded between 0 and 1 so possibly should be arc sine transformed. The key thing is though, you can then only test for differences in the diversity index between sites or times. It can be a bit meaningless ecologically, especially as different indices may give different results. A multivariate analysis of community might be better. Possibly ANOSIM in Primer and R?
The Shannon Wienner function measures homogeneity of quantiotative distribution of species in a community, so it can be used for simplified indication of processes which lead from a "balanced" distribution to a striking dominance of few species or even of one species and vice versa. But at the same time it puts on one level large species and small species of very different ecologic properties and significance. It can, in a very simplified way, indicate course of a trend of changes, but tells nothing about the reasons of these changes. Every trend consists of small steps, which may be not "statistically" significant, but really do exist. Interpretation of the structural changes in a community indicated by Shannon index or simiilar indices must be always based on a good knowledge of species ecology and factors, which caused the observed changes. According to my opinion just this knowledge is the essential problem, not the satisfying the demands of statistical formalism.
Tce Jaccard´s index measures similarity of composition of two entities, in ecology most frequently of the communities. it can be based on presence/absence data or on different quantitative data. Similarly as in the case od Shannon´s index, it gets a real meaning in set of many compared entities. It can describe a trend in transition from one state into other/s. This also can be statisticaly insignificant, but can have a rational interpretation. If such interpretation exists, the statistical "significance" is a purely formal criteriion. Comparisom of results obtained with more similarity indices combined woth different aglomerative techniques clearly shows, how easily it can be manipulated with results.
You can compare Shannon index values in various ways.
In general, you can analyse Shannon values with any statistical test for which the assumptions are met, as already stated by Richard Stafford here. So, for example, if you have SERIES of Shannon values that met the assumptions of parametric tests such as t-tests, ANOVAs, regressions, etc., teher is no reason to do not use them; if assumptions of parametric tests are not met, you should apply the respective non-parametric options (but it has been demonstrated that if the Shannon index is calculated for a number of samples, the values tend to be normally distributed).
However, if you are interested in comparing TWO VALUES of Shannon index (i.e. a single value for community A and a single value for community B), it is obvious that, with only these two values, you cannot apply a “standard” statistical test. In such a situation, you can use an “ad hoc” t-test procedure described by Zar (Biostatistical Analysis) and Magurran (Ecological Diversity and Its Measurement) to asses if the two values are significantly different, as indicated here by Mariano Ordano.
Finally, you can decide to compare values of Shannon index using permutation procedures, as already pointed out by Mariano Ordano. The free software PAST (http://folk.uio.no/ohammer/past/) allow you to calculate Shannon index and to assess if a couple of values are significantly different by using both the procedure reported by Zar and Magurran and permutation procedure.
I found the permutation procedure available in PAST very useful, because it can be applied to any kind of diversity indices (PADT actually allows its use for a huge number of indices),and comparing results provided by several indices with different properties is generally very instructive.
I would use randomisation tests as advised by Mariano... both for Shannon and Jaccard
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