Sometimes we can measure beta-diversity as a difference in alpha-diversity.

And most of these indexes use presence/absence data that are, of course,

incidence data.

For instance, one of them is the beta-diversity of Whittaker.

That's simply the number of species, S, the total number of species of the area.

That means, for instance, gamma-diversity.

Divided by alpha, that is, the mean diversity of the samples.

But there are some condition to calculate this index.

For instance, we need that samples have the same dimension,

that diversity is measured as a species richness, and then one is the maximum

similarity that we can get from this index and two is the minimum similarity.

If we have samples of different dimension, we can calculate another index that is

just the same Whittaker that is subtracted by 1 divided by N-1 multiplied by 100.

That is N, just the number of samples along the transect.

So if we have a transect, we measure the number of samples and

then we divide this number by the Whittaker index.

In this case, we have 0 as in no turnover,

that is no diversity, and 100 that is minimum similarity.

That is that the samples have very low similarity means that beta-diversity

is high.

If we want to understand when species are lost and others are gained,

there's an index that is not influenced by trends of species richness and is the same

of previous one, but is not divided by alpha min but is divided by alpha max.

That is the maximum richness within the taxon, for example, so this is useful

to study the turnover variation of different taxa in relation to a disturb.

Another measure is the Cody measure or Cody index that sums the number

of species gained along a transect to the number of species lost.

So we have the number g, that is number of species gained,

plus l that's the number of species lost, divided by 2.

This is also another index to measure differences in alpha-diversity.

Then there are three indexes that are called Routledge

indexes that are based on the decomposition of alpha and beta-diversity.

One of them is just the square value of total species richness

divided by (2r+S)-1, where r is just the number of

couples of species with overlapping distributions.

The second of the three Routledge indexes is for incidence data and

needs samples of the same dimension.

You will see the formula in the picture, where e is just the number of samples

where the species i is present, S j is the species richness of the sample j.

There is another version, that is the exponential form of the Routledge index,

and it's just the exponential of B I, so

this is the second index that I show you in the exponential form.

The last measure of differences in alpha-diversity

that I present you is the Wilson and Shmida's.

This index is just the sum of gained species plus the number of lost

species divided by 2 multiplied by S j that is the same of Whittaker index.

In general, Whittaker, I mean beta W, index is just the best performing measure.

If we want to measure beta-diversity in space,

so we want to understand complementary and similarity, and in this

case we want to know if there is a low spatial turnover or high spatial turnover,

it's interesting to understand how this similarity can be calculated.

For instance, if we want to measure, to understand which of

the samples that we have or which of the areas that we have in a specific space,

is the best to reach the highest number of species.

So we have area one, as an example in this table,

that the Site 1 has Species a, Species b, Species c, but not d,

e, and f, and Site 5 that has Species c, Species d, Species e, and f.

In this case, the best combination of these sites is Site 1 and Site 5, and

this way we get a highest number of species, if we want to set, for instance,

a protected area.

So all these indexes has the same three variables that are a, b, and

c, where a is the number of shared species,

b is the first site exclusive species, and c is the second site exclusive species.

The Marczewski-Steinhaus index is just

calculated as 1-a divided by (a +b+c).

It is just the complemental form of the famous Jaccard index that is

the same instead 1 minus.

So just remove 1 minus, you get the Jaccard index.

That's very useful and very easy to calculate.

When you have more than two samples, you can just use a different formula that

divided the sum of U jk divided by n.

Where U jk is just the couple of samples that is summed with the V

jk that's the common numbers species between the two samples analyzed, j and

k, and S j and S k is the number of species of sample j and

sample k, and n, the total number of samples.

Another very famous index is Sorensen's index.

This index divide 2 multiplied by a, that as I told you is the variable

of shared species, divided by 2 multiplied by a+b+c.

It's one of the most efficient of the measures for complementarity and

similarity.

But if the differences of S, so the total species number,

is i, the results are always i.

A new version of this index has been provided to reduce this effect.

This index is just 1-(a divided by a+min(b,c)).

There are some other indexes that are based on abundances.

One of them is the Bray-Curtis index, that modified version of Sorensen index.

In this case we have that the index is just 2 multiplied by j multiplied

by N divided by (N a + N b), where N a is the total abundance of the site a,

N b is the total abundance of the site b, and

2 multiplied j N is the sum of the minimum abundance values of both sites,

so it's the min(n a) and the min(n b).

All the previous measures are influenced by the richness and

the examples I mentioned, but there is one that is not so affected, and is

the Morisita-Horn index, that is not but influenced by the most abundance pieces.

When we want to give less weight to abundant species,

we use a modified version of the Morisita-Horn index.

That is this one provided in the formula, where N a is the total

abundance of site a, N b is the total abundance of the site b, and

a i is the total abundance of individuals of species in the site a,

b i is the total abundance of the speciesism of the site b,

and d a and d b are calculated as the formulas that are provided here.

Another interesting index of similarity is the percentage similarity index.

In this case we subtract by 100-0.5 and the sum of P ai and

P bi, where P ai and P bi are the percentual abundances of the species i

in the samples a and b, and S is the total number of species.