Yes, you can just make comments based on the raw values or you can compare them using ANOVA and post-hoc tests, which compare values using their standard errors and take into account the fact that multiple comparisons may increase Type 1 errors :)

Just a few notes:

Try SHE Analysis as a first stop (Magurran).

Each index will report on a particular aspect of diversity, e.g., some will be biased by rare species others will suppress the effects of rare species. Just remember to compare like with like. Do not mix your indices in the comparison.

Some of the metrics will be ordinal or continuous within a defined range (from 0 to 1). You will need to determine if you apply parametric or non-parametric methods to your indices.

I would probably try a PERMANOVA, or using a non-parametric analysis with ranks.

Depending on your sample size you make the choice of the most appropriate index. For small samplings the use of Simpson's Index is more adequate than Shannon (Magurran, 2004). Often, the use of several indexes hinder to solve a problem. For the comparison between the indexes, I would use an ANOVA, with Tukey or Duncan test at 5%.

For stastastical analysis you mast conferm about how many tails or sampling sites or diversity of individual and or species then, use ANOVA, t test, p , f test SD, and significant values also determine.

Hi All,  I agree with Eddy Cannella.  Early research I conducted was looking at the distribution of the shrimp Sicyonia ingentis in relation to two different types of sewage outfall off southern California (one industrial the other domestic).  The statistics were tough to do at the time.  I didn't have all the cool statistical programs in 1986.  Good luck.

Hi all.

I think you can also use Hill numbers or the effective number of species with the iNEXT (online or R package), with these analyzes you can have its confidence intervals (95%). If your goal is to measure alpha and beta diversity, it is more important to get the effective number of species, as these values are more comparative. The works of Lou Jost and Anne Chao can serve you a lot.

Success and regards

http://onlinelibrary.wiley.com/doi/10.1111/2041-210X.12349/full

http://onlinelibrary.wiley.com/doi/10.1890/06-1736.1/full

http://onlinelibrary.wiley.com/doi/10.1111/2041-210X.12613/full

http://onlinelibrary.wiley.com/doi/10.1111/j.2006.0030-1299.14714.x/full

http://dx.doi.org/10.1111/j.2006.0030-1299.14714.x/full

You should start identifying the nature of the distribution of values of the index, i.e. Shannon index values are stricktly positive and continuos, so you should use some GLM model which allows for Gamma distribution.

Diversity (Shannon values) = a + Beta*Site + Beta*Season,

Diversity(Shannon ~ Gamma(µ, τ)

 E(Shannon) = µ_i   &   Var(Shannon) = µ_i2 / τ

ANOVA approach can ONLY be ok, if you have enough replication, otherwise It will assume normality and homogeneity, in the distribution of values of the Shannon index, and residuals, which we know is not true for small samples. Let me know if you need assistance.

Are you sure that statistical significance is even relevant to your study?

If you have three sites and three seasons but only nine values for each index, then you have no replication and cannot proceed to statistical hypothesis testing. Are your nine values sample diversities or did you proceed (perhaps through Zahl's jackknife alternative to Pielou's "pooled quadrate method") to get community diversities? If all you have is sample diversities, then you would need a statistical test (and so you would need replication) if you wish to compare between sites or seasons. However, given sufficient replication you could find the community diversities and it is always certain that those are different, so tests for statistical significance are irrelevant. The question would then be whether or not the degree of difference amongst the community diversities was large enough to be ecologically important. However, ecological importance and statistical significance are entirely unrelated.

Also: If you do have the data for replicate measures of sample diversity but not enough replicates to determine community diversity, I would recommend caution in testing for differences among sites or seasons. You would be in a situation analogous to comparing species richnesses without sufficient data to determine asymptotic richnesses. In the latter setting, the standard thing is to resort to rarefaction but it has long been known that rarefying too far can reverse the rank order of communities: an area of smooth sand may show many species in the first few samples but not many more as sampling proceeds, whereas a patchy seabed may yield fewer species to the first sample but many extra ones to additional sampling. In short, rarefaction can completely muddle your results, which is never a good thing! The same process operates with the Shannon index (which should anyway be replaced with Exponential Shannon -- one of the Hill Numbers) and other diversity measures, though the problem has too rarely been discussed for anything but species richness.

Hi. I would suggest you as a first step to plot species accumulation curves with your data, in order to see if 9 samples are enough, or if you need to take more samples. After that, before choosing a statistical test, you should explore your data, in order to see if they adjust to normality, homocedasticity, etc.

However, I agree with the previous post by TC Kenchington, in that if you only have 9 samples for two factors (site and season), you do not have replications, so no statistical comparison is possible.  Therefore, if the goal of your research is to make spatiotemporal comparisons of diversity of shrimps, the first thing you should do is go to the field and take more samples. If your research is mainly descriptive, you could limit yourself to discuss the values of each site. 

Re-reading my earlier answer, I see that I made at least one error:

Community diversities always differ from one community to the next but all we can actually have is an estimate of the diversity for each community and tests of significant differences among estimates are meaningful. That being said, I doubt that you have enough data to prepare replicate estimates of community diversity. It is unusual enough to collect enough to even prepare one such estimate.

Tests for significant differences among estimates of community diversity might be problematic anyway. I doubt that anyone has looked at their error structure. You would probably have to resort to simulating the estimation process and generating some sort of approximation to a probability density. That's way beyond my skill level!

To all of you people: Thank you so much for your recommendations, they will be very helpful. I have just a couple of questions about them. To those who recommended hill's numbers or effective number of species: would you consider this analysis as a substitute for the conventional statistic test in my particular sampling and data scenario? how suitable did this approach have been in your study field?. 

To Eliecer Rodrigo Diaz: I'm interested in a wider explanation regarding the procedure of GLM model and the equations below that you kindly provided. 

Again, thanks a lot. Greetings and best regards. 

The Hill Numbers are alternatives to indices like Shannon or Simpson. Indeed,species richness is the zeroth order Hill Number, while the most generally useful of the series is Exponential Shannon, which is simply e raised to the power of the Shannon index value for the sample or community in question. That is the 1st order Hill Number. The 2nd order Hill Number is the inverse of the Simpson Index (D) -- not to be confused with the Gini-Simpson index, which is 1-D.

When Jost, in 2006, returned attention to the work that Renyi had published in 1961 and Hill had followed up in 1973, he stressed the "effective number of species" perspective on these numbers. The "effective number of species" is numerically equal to a Hill Number and to whichever Hill Number you choose as being the most relevant. (You can pick any order from zero to infinity and you are not limited to integers!)

I recommend using both species richness and exponential Shannon, unless you have a strong argument for preferring another of the many alternatives. But each of those only gives you a measurement scale (equivalent to metres, when measuring length). You still need care in determining what you are measuring (which is a whole lot more complex than it looks at first sight) and, if you need hypothesis testing at all, you still need a way to test your null hypothesis.

It is important that when performing statistical analyzes to compare the diversity of species using any diversity index (Shannon, Simpson, Pielou, etc.), it is important to take into account how the respective sampling were made. I am in agreement with the previous comments of Gabriela Echevarria in which she suggests that you as a first step to plot species accumulation curves with your data, in order to see if 9 samples are enough and explore your data, in order to see if they adjust to normality, homocedasticity, etc. 

However, all the samples that you realize have to be standardized, that is to say, to have the same unit of effort to be able to make the statistical comparisons, both spatial and temporal. If the sampling does not meet these requirements, any statistical analysis you perform will present bias or errors. Now, if you are looking for descriptive analysis, for example: presence-absence of temporal and / or spatial species, there is no problem.

Depending on what you are looking for using biotic indices, ANOVA, T-test and Duncan multiple test (Post-hoc) will be OK...........

SHE analysis would be a good way to understand the nature of covariation between the indices and get a sensible hypothesis that can be addressed:

http://www.pisces-conservation.com/sdrhelp/index.html?she.htm

You would test for normality within your comparison groups; if normal, run your stats; if not normal, then transform the data, check for normality, run the stats on the transformed data.

But with small numbers of reps it's really a roll of the dice. Some would argue against using statistics for such small sample size. Yet many published studies use standard stats even with small N.....and for practical, ethical, or other reasons, small sample size is what sometimes we have. If you can detect differences with small sample size....then the difference is likely important.

Trevor made an important point above....if you have just 9 samples then the advice to ask a different question is valid....and species accumulation curves may be a place to start. You could try rank abundance plots (Fig 8 in this paper: Article Sampling Biological Communities

I would also consider Trevor John Kenchington first response as to the robustness of the data. You do not need to or desire statistical significance with informatics such as diversity measures. They are descriptors of community structure first and foremost and should be used to characterise populations so that you interpretations of ecological processes are relevant to those populations.

All these other answers are much more technically advanced than what I am about to offer, but there is an easy to use t-test for comparing two diversity scores (from Zar 1996), which automatically calculate with this excel sheet. I found the guts of it online (no reference recorded) and then modified it to suit my needs. Perhaps it could be of value to someone with simple needs.

Without replicates from each site, you did not sample the sites. A really exhaustive census of each would allow you to calculate and compare diversity index scores from each site: if these are different, then the diversity at each site IS different...but I don't think that's what you did! Note too that an index isn't the actual diversity, it's an indication only. For site A H'=6 and site B H'=3 does NOT mean that site A has twice the diversity of site B. Statistics require an estimate of the sampling error. I only glanced at the excel spreadsheet based (?) on Zar's approach, but it looks as though it too fails to require replicates--I'd avoid this! Check out Anne Magurran's book on measuring diversity and download EstimateS. And don't feel badly: diversity indices get misused all of the time!

In your case, as you do not have any replicates when testing for the interaction (c.f. Trevor and Peter's answers), there is no statistical method to test for its effect. If you only restrict your analyses to simple effects, you will still be quite limited: only 3 replicates per effect. Then, as mentionned by Peter, indices include a level of uncertainty that can vary from one observation to the next depending on the number of individuals and species/taxa found in each. That makes simple statistical tests on them unreliable and statistically incorrect: you need to had a correction for this uncertainty, as is routinely done in meta-analyses when analysing summary statistics from previously published papers.

In your case, estimating the bootstrap confidence intervals of your indices for each observation to refine your interpretation of the effects is probably be the best option. Many softwares (e.g. PAST, EstimateS) and R packages (BiodiversityR, vegan) for biodiversity analysis include this possibility.

The sbdiv function from the simboot R package will also allow you to test for differences in the Shannon and Simpson indices, as well as Hill number, between according to a factor levels (but not two at the same time).

Most diversity indexes behave non-linearly, while most statistical tests assume linearity. So no, you cannot just use ANOVA and t-tests to compare them.

This question is too vague. What features of the index do you want to compare?

Taking selected parameters variation can be determined by ANOVA