Always in the case of incidence data, you can use another two incrementals.

One is the Jackknife estimator.

The Jackknife Index is a measure that's number of species observed in

the samples plus the number of samples when you observed only one or two species.

That's very useful, because if you don't have the abundance of species you at least

you can use anyway your data to estimate the effective number of species.

Another measure of the effective number of species is the Bootstrap estimator.

It just eliminates and

randomizes the samples in different order just to have an information so

it sums the number of species observed to this continuous resampling.

So as I told you, richness is just one part of the information.

But if you want to understand how is the diversity here,

how much is the diversity of your samples, of your treatments,

if you are analyzing different impact, for instance, on the ecosystem.

You need to know what is the diversity, and there are two that are the most

commonly used indexes of measure that are the Simpson Index and the Shannon Index.

Shannon Index derives from the interior of information, and

it was developed by Shannon just to understand

the quantity of information in telecommunication, in phone communication.

And then, it was adapted to ecological studies.

For instance, Shannon index uses the sum of the relative abundances of the species.

For instance,

in this case you see that the factor pj is just the number of individuals of this

species (n) divided by the total number of species in the sample N.

So in this case, this index as to usual limits that one

is the minimum level that you can raise your diversity that is almost 1.5 and

the maximum level that you can reach that is maximum 3.5.

So very diverse environment has the kind of reach or

index that is almost approaching to four.

Very poor diverse environment as are any index that is close to one.

Another very similar index is the Simpson index.

Simpson index use the same proportion between a number of individuals of each

species and total number of individuals, and is just the square of this value.

So, the sum of this proportion for every species.

This is very useful in case of finding community-wide

the total number of individuals of course is known.

So, what you can use this index just subtracting -1 to

have a more available information.

Simpson Index is usually expressed as 1- the index so 1 divided the index.

It means that, in this way, you can have an index that goes between 0 and

the maximum value of 1.

The Brillouin index of diversity is very useful when you.

Not all species can be detected,

because of difference in the sensitivity of sampling methods.

The Brillouin index is simply the total abundance factoral

value minus the sum of the logarithm, or

the abundance of each species is always factor divided by the total abundance.

Another way to express the diversities the McIntosh Index.

It considers the community as a sample point in the volume and

dimensional fr which the Euclidean distance Is in the sample from the region

can be useful to estimate the diversity.

It is just the root square of the sum of the Euclidean distance of each species,

or the abundance of each species from, of course, the n dimensional upper volume.

In 1971,

Hulbert noticed that a useful measure of diversity can derive from the probability

that an interaction between two individuals of the same species happened.

It means that we can calculate the probability on an interspecific meeting,

and it's called the Hulbert Index.

Later, provided a meter to summarize all these indexes of diversity,

and they are called and are well-known as Hill's numbers.

If we substitute 0 at alpha, we obtain H 0,

that is just S, such as just the number of species or richness, simple information.

But if we substitute 1 we get exactly the exponential form Shannon index.

When we substitute 2 to this formula,

we will get the inverse formula of Simpson, so the inverse of Simpson index.

And why we substitute plus or minus infinite, we got dominance or

rarity indexes.

So you see that this is very summarizing formula to express

most of the index that I already explained [INAUDIBLE] going to [INAUDIBLE].

So where we have low values of alpha,

this formula gives more importance to various species.

But when we have higher value of alpha,

this formula gives more importance to dominant species.

The second component of the diversity, as I've told you, is the evenness.

So, it's a kind of expression of the homogeneity of the distribution of

abundance is inside the species.

Most of the indexes and the measures that have been provided,

that have been suggested to estimate the evenness are just the original formula

of the diversity for why the information of the richness has been removed.

For instance, the Shannon evenness index is just the Shannon index divided by

the logarithm of the number of species, so the logarithmal richness or

total richness.

It moves from a 0, it means minimum evenness, to 1, that's maximum evenness.

The Simpson index of homogeneity or evenness, has the same, just the Simpson

index, divided by the simple number of species, the total number, so richness.

So, you divided D by S.

A modified version of the Shannon evenness index

with less weight to the species richness so lower values when lower evenness,

is the Heip index, it's just the Shannon index, the exponential form of

Shannon index -1 divided by the total number of species -1.

Another way to measure the evenness is the Brillouin evenness index.

The Brillouin index is very useful in case of no random sampling, for

instance when you are sampling parasites.

The formula seems to be complicated, but you can use, in special cases

it could provide very important and very reliable information.

Another way is the Macintosh index.

It comes from the Macintosh diversity index but

in this case you divide by the total abundance minus the total abundance

divided by the root square of the species' richness.

There is another index, but this is, I'll just tell you,

because of information that is effected by bias, it is the Caramago evenness index.

This index has been suggested as a form to avoid the influence of dominant species,

but as I told you, is affected by bias, so it's not so reliable as the other one.

Smith and Wilson then developed a kind of evenness

that is not affected by the unit measure, and then tried

to summarize which measure of evenness should be utilized in a different case.

So they suggest that if the symmetry between rare and

abundant species, so the importance of the abundant species and

the importance of the rare species should be equal is requested.

They also suggested a method to understand which

measure of evenness should be utilized in different cases.

For instance, if the symmetry between rare and

abundant species is request, you have to use Simpson index so

the inverse index of Simpson if the minimal evenness should be 0.

Or Camargo evenness index if intermediate values of evenness

should correspond to value of the index.

If you don't need the symmetry, such as if the most common species

have more importance than rare species, you can use this log

b of the rank abundance plot that I will explain to you in next lecture.

If you don't need a response to [INAUDIBLE],

otherwise you will use this [INAUDIBLE].

In any case, as also this measure

suggested the Simpson evenness index performs always well.

So it's strongly recommended to use Simpson evenness index

in most of the cases.

A useful and interesting way to analyze the behavior, or richness and

evenness, so it means the behavior or the [INAUDIBLE] diversity is the SHE

analysis that is Shannon-Evenness analysis.

Shannon index can be considered as a combination of the logarithm of

the evenness and the logarithm of the species.

For instance,

the index is just the sum of logarithmic species plus logarithmic evenness.

Why we want to compare two samples of the same sample after the treatment, for

instance, pollution, different station, or others?

We can, in some cases, In three possible scenarios.

One is that the two richness are very different but

the diversity of the two samples are equal.

So where only H keeps constant during the accumulation of the species in

the samples because the richness and evenness compensate each other.

This is a kind of log-series distribution.

There is a case when the richness of the two samples and

the diversity of the two samples is diverse but the evenness is equal.

And at the so the evenness keeps constant but the richness and so diversity change.

This is a kind of distribution called broken stick.

And there is the third case where everything is different between the two

samples.

It means richness is different, diversity is different, and

of course evenness is different.

And this is why diversity changes because evenness and richness do not

compensate each other in this kind of distribution, log-normal distribution.

When you put in a graph all the components of diversity, you will have a kind of

representation like this that helps you to understand the diversity and

the differences in diversity between two treatment, two samples or two areas.

Next time, we will discuss about better diversity, understand how to measure it.

So let's see you next time.

Goodbye.