A PERMANOVA is better since there are multiple groups :)

Andrew Paul McKenzie Pegman Thank you for your answer, a PERMANOVA for sure could be useful but I'm not sure it'll solve my problems. It helps me inquire about variance between groups and variance within groups but I also have variance within each point (the confidence interval due to the rarefaction process).

You are working on complex ecological aspects. Good, congratulations, and I wish you all success. I know it is not easy to work on environmental aspects, particularly animal ecology. However, you can deal with various aspects and synchronise in discussions because you deal with living beings controlled by biotic and abiotic factors. Their impact is highly delicate as these are interdependent.

To compare diversity indices between groups while accounting for confidence intervals derived from rarefaction, you can use several approaches. Here are some suitable methods for your analysis:

Confidence Interval Overlap Method

One straightforward approach is to use the confidence interval overlap method. This method is particularly useful when dealing with rarefied diversity estimates.

• Use 84% confidence intervals instead of 95% confidence intervals. Non-overlapping 84% confidence intervals approximate a statistical test with α = 0.05 .
• Compare the 84% confidence intervals between groups. If they do not overlap, you can infer a statistically significant difference at p < 0.05.

This method is simple to implement and interpret, making it a popular choice among ecologists.

Hutcheson t-test

The Hutcheson t-test is specifically designed to compare Shannon diversity indices between two samples.

• Calculate the variance of the Shannon index for each sample.
• Use the modified t-test formula provided by Hutcheson.
• Determine the degrees of freedom using the provided formula.
• Assess statistical significance using the calculated t-value and degrees of freedom.

While this test is primarily for Shannon diversity, it can be adapted for other indices as well.

Bootstrapping Approach

For a more robust comparison that accounts for the uncertainty in your rarefied estimates:

• Generate bootstrap replicates of your diversity indices for each community.
• Calculate the mean difference between groups for each bootstrap replicate.
• Construct a 95% confidence interval for the difference between groups.
• If the confidence interval does not include zero, conclude a significant difference.

This method is particularly useful when dealing with non-normally distributed data or when parametric assumptions are not met.

Permutation Tests

Permutation tests offer a flexible, distribution-free approach to comparing groups:

• Calculate the observed difference in mean diversity between groups.
• Randomly reassign communities to groups and recalculate the difference.
• Repeat this process many times (e.g., 10,000 permutations).
• Calculate the p-value as the proportion of permuted differences as extreme as or more extreme than the observed difference.

This method is robust to violations of normality and can be applied to various diversity indices.

Multiple Comparisons

Since you're comparing multiple groups, consider adjusting for multiple comparisons:

• Use methods like Bonferroni correction or false discovery rate (FDR) control.
• These adjustments help maintain the overall Type I error rate when making multiple pairwise comparisons.

Visualization

To complement your statistical analysis:

• Create box plots or violin plots of diversity indices for each group, including confidence intervals.
• Use a point-range plot to show mean diversity with confidence intervals for each group .

These visualizations can provide intuitive understanding of the differences between groups and the associated uncertainty.

Remember to choose the method that best fits your data structure and research questions. It's also advisable to use multiple approaches to ensure robustness of your conclusions. If you're unsure about the most appropriate method, consulting with a statistician familiar with ecological data analysis can be beneficial.

To calculate the variance for each sample in the Hutcheson t-test, you need to use a specific formula designed for the Shannon diversity index. Here's how to do it:

To calculate this variance:

• Use a spreadsheet or statistical software to perform these calculations, as they can be complex for large datasets.
• Double-check your calculations, especially the logarithms and summations.
• Ensure you're using the correct base for logarithms (natural logarithm, ln).

Importance in Hutcheson t-test

This variance calculation is crucial for the Hutcheson t-test because:

• It's used in the denominator of the t-statistic formula.
• It accounts for the uncertainty in the Shannon diversity estimate due to sampling.
• It allows for a more accurate comparison between two samples by considering the variability within each sample.

By calculating this variance for each of your samples, you'll be able to proceed with the Hutcheson t-test to compare the Shannon diversity indices between your groups, taking into account the uncertainty in each estimate .

Yes, PERMANOVA (Permutational Multivariate Analysis of Variance) can be applied to analyze the diversity indices between your groups. It's particularly well-suited for your situation for several reasons:

1. Non-parametric: PERMANOVA doesn't assume normality, which is beneficial when dealing with diversity indices that may not be normally distributed .

2. Multivariate: It can handle multiple response variables simultaneously, allowing you to compare all your diversity indices at once .

3. Flexible: PERMANOVA can be used with various distance measures, including those appropriate for ecological data .

4. Powerful: It's generally more powerful at detecting changes in community structure compared to some other methods like ANOSIM .

5. Handles complex designs: PERMANOVA can be applied to complex experimental designs, including multiple factors and interactions .

To apply PERMANOVA to your data:

1. Prepare your data: Organize your diversity indices for each community into a matrix.

2. Choose a distance measure: Select an appropriate distance measure for your data type.

3. Run the analysis: Use software like R (with the vegan package) to perform PERMANOVA .

4. Interpret results: PERMANOVA provides a pseudo F-statistic and a p-value. A significant p-value (typically