** Comparing
the diversities of two communities**

Before
reading this, make sure you understand **how
to measure the diversity of a single communit**y.

Suppose you want to compare the epiphyte diversity of a primary
forest to the epiphyte diversity of a disturbed forest. You
take big samples of each of these two communities. Now you want
to answer the question "Are the diversities different?"

There are two parts to the answer. We must measure the **magnitude**
of the difference, which enables us to judge its biological
importance, and we must measure the statistical significance
of the difference, to see whether it is arose by simple sampling
variability or because of a real difference in diversity. These
are completely different kinds of questions. Many published
studies apply some statistical test and find statistically significant
results, and then go no further. But mere statistical significance
is not in itself interesting. Any tiny difference can be made
statistically significant if sample size is large enough. Given
that the difference is statistically significant, the more interesting
question is: How big is the difference?

Biologists have to get past their fixation on statistical significance
and concern themselves more with this question of the size of
the difference. The use of raw diversity indices made it difficult
to answer that question in the past. If the Gini-Simpson indices
of the communities are .99 and .97, is that an important difference
or a small one? If the Shannon entropies (Shannon-Wiener indices)
of the two communities are 4.5 and 4.1, is that a big difference
or a little one? It is hard to say based on these numbers. At
first glance the difference in Gini-Simpson indices of .99 and
.97 looks small. But this index is highly nonlinear. Converting
to effective number of species (which is the true diversity,
as explained in the other parts of this site) proves that the
difference is in fact huge: the community with a Gini-Simposn
index of 0.99 has the same diversity as a community with 100
equally-common species, while the community with a Gini-Simposn
index of 0.97 has the same diversity as a community with 33
equally-common species. The difference between a community with
33 equally common species and one with 100 equally common species
is enormous. The same holds for the Shannon entropy values just
mentioned: 4.5 converts to 90 effective species while 4.1 converts
to 60 effective species. The second community is only 2/3 as
diverse as the first community, according to this measure. If
the diversity drops that much between undisturbed forest and
disturbed forest, that is a serious and biologically significant
drop. It is hard to realize that the drop is so dramaric if
one looks only at the raw indices.

Suppose you calculate the true diversities (effective numbers
of species) and find a big drop. That is when it is time to
ask the other question, the question of statistical significance:
could such a drop be due to random sampling effects? If the
samples are large, the t-test suggested by Hutcheson (1970)
can answer this question for diversity of order 1 (Shannon measures).
Randomization tests are also available. On this websit I do
not deal with this side of the question because the statistical
tests in the literature are mostly correct. Where the literature
fails is in its treatment (or its lack of treatment) of the
other (and more important) side of the question, the magnitude
or biological significance of the difference.

So, to compare the magnitudes of two diversities, calculate
the effective numbers of species (the exponential of the Shannon
entropy, for example) of the two communities so that you can
compare them on a linear scale and get an intuitive feel for
the difference. (Download my Excel sheet**
Indices to Diversities** which does this conversion
for you.)You can divide the smaller diversity by the larger
one and come up with a meaningful fractional drop in diversity
(something you can't do with raw diversity indices because they
are nonlinear with increasing diversity). Then check to see
if that drop could be due to chance, by calculating the t-test
of Hutcheson (for Shannon measures) or other tests. If the difference
in true diversities (effective numbers of species) is both large
in magnitude *and* statistically significant, then you
have found something important. Congratulations!

Many researchers prefer to use presence/absence measures like
species richness and its similarity-index relatives, Sorensen
and Jaccard, even when frequency data is available. Often they
do this because presence/absence measures are simple to interpret;
many researchers have a lingering distrust of frequency-based
diversity measures. The raw frequency-based indices were abused
so much in the literature that this mistrust of the usual techniques
was justified. However, conversion of these indices to effective
number of species gives these measures many of the same intuitive
properties that species richness has, and frequency-based measures
will detect community differences more easily than presence-absence
measures.

Here are some examples from the literature that illustrate
the above points:

**Example 1, illustrating the increased power of frequency
measures over presence-absence measures: **

An interdisciplinary team of scientists, including my friends
Nigel Pitman and Mark Thurber, found an odd forest type on the
western edge of Amazonia. Although the rest of western Amazonia
is covered with the most diverse forests on earth, this one
was peculiarly poor in species. The authors chose to express
the diversities of each community using species richness, which
is the diversity of order zero. They found that a plot in the
odd forest had a species richness of 102 species while similar
plots in the rest of the region had an average species richness
of 239. Thus, according to species richness, the average forest
plot is 2.3 times more diverse than the species-poor plot. That's
quite a difference.

However, species richness pays no attention to frequencies,
and so it is not as good at detecting differences as a frequency-based
measure would be. A virgin forest with fifty equally-common
species is much more diverse than a burned forest with
one abundant fire-tolerant invader species and a couple of
individual survivors of 49 other species. Yet species richness
is the same for both forests. It would be better to use
measures sensitive to frequency, if frequency data existed.
Nigel was kind enough to lend me his raw frequency data so I
could figure out the Shannon or order-one diversities of the
plots. Recall that the Shannon diversity is the fairest diversity
measure, weighing each species exactly by its frequency, not
favoring either rare or common species. (It is also the only
diversity measure which can be decomposed into alpha and beta
components when community weights are unequal.) The diversity
of order one for the species-poor plot is 28.5 effective species
while the average for the other upland forest plots is
134 effective species. In other words, the species-poor forest
has the same Shannon entropy as a forest consisting of 28.5
equally-common species, and the normal forests have the same
Shannon entropy as a forest with 134 equally-common species.
The difference in diversity between the two forest types is
therefore the same as the difference in diversity between a
forest with 28.5 equally-common species and a forest with 134
equally-common species. That's a huge difference! Thus when
frequencies are taken into account, the diversity of normal
forests is actually almost 5 times higher than that of the poor
forest. The difference is much greater than the 2.3 times indicated
by species richness. [The original article containing this data
is: *Catastrophic
natural origin of a species-poor tree community in the world's
richest forest*, by N. Pitman, C. Ceron, C
Reyes, M. Thurber, J Arellano, published in Jour. Trop. Ecology,
2005, 21: 559-568. Download a pdf of this article by clicking
on the title; article courtesy N. Pitman and M. Thurber.]

**Example 2, a study that is typical of current research,
which could be improved by calculating the real magnitudes of
the effects being studied:**

In *Overstory composition and stand structure influence
herbaceous plant diversity in the mixed aspen forests of northern
Minnesota*, **American Midland Naturalist **143:
111-125 (2000), Berger and Puettman measure the Shannon entropy
of the herbaceous understory to see if it is correlated with
overstory variables. The found the following correlations:

[Note that they incorrectly call H (or H') "diversity".
H or H' is the entropy, not the diversity. We will see in the
next example that this is not mere semantics but can have major
practical consequences.] Concentrating on Graph A, which is
the main point of their study, we see that Shannon entropy is
significantly correlated with aspen basal area. But is the difference
between 2.59 (the value of H' when aspen basal area is 0) and
3.19 (the value of H' when aspen basal area is 100% of total
basal area) a big difference or a small one? The correlation
coefficient and the p-value do not really answer this question,
but this is all that Berger and Puettman report. Experienced
biologists, who work often with Shannon H, will have an intuitive
feel for the magnitude of the difference between 2.59 and 3.19,
and will recognize it as a substantial difference. But what
is the real magnitude of that difference?

This is where conversion to effective number of species or
true diversities is useful. The difference in herb diversity
as one goes from 0 to 100% aspen basal area is the same
as the difference between a forest with exp(2.59) = 13.3 equally-common
herb species and a forest with exp(3.19) = 24.3 equally-common
herb species. According to Shannon entropy, then, the herb diversity
(the true diversity, not the index H) *doubles *as aspen
basal area increases from 0 to 100%. This shows that Berger
and Puettman have found an effect that is not only statistically
significant but is actually quite large in absolute magnitude.
It is unfortunate that there is no mention of the magnitude
of the effect in their paper, it would have been a very useful
addition to their conclusions. This is a common defect in the
diversity literature, which tends to focus narrowly on statistical
parameters and neglect the more important issue of the real
magnitude of an effect. This neglect has been fostered by a
lack of mathematically-appropriate measures of those magnitudes.
Effective number of species, and the ratios, similarity measures,
and overlap measures derived from it, are the appropriate measures.

**Example 3, showing how ratios of raw diversity indices
can mislead, and how this can be avoided by using effective
numbers of species:**

In *Species diversity in vertical, horizontal, and temporal
dimensions of a fruit-feeding butterfly community in an Ecuadorian
rainforest*, **Biological Journal of the Linnean Society**
62:343-364 (1997), DeVries, Murray, and Lande explore the differences
between tropical butterfly communities along many spatial and
temporal dimensions. This is part of a series of pioneering
papers on the subject, distinguished by their huge sample sizes
and extensive temporal and spatial coverage. A measure of community
similarity has been proposed by one of the authors (Lande 1996)
and is used in this study to analyze similarity among communities
along different ecological dimensions, like canopy vs. understory.
The idea of the measure is as follows: Find the average diversity
of the communities (the alpha diversity), and compare this with
the total diversity of the pooled samples from all the communities
(the gamma diversity). If the communities have few species in
common, the total diversity of the pooled samples will be much
greater than the average diversity of the individual communities.
The ratio of alpha over gamma is his measure of similarity,
a measure of shared diversity. It equals unity when all communities
are identical in species composition.

This would be an intuitive and well-behaved similarity measure
if it used true diversities. But Lande (1996) uses raw indices
instead of true diversities in his similarity measure. This
is not a mere semantic issue, it causes real trouble. The Gini-Simpson
index, which is Lande's recommended index of diversity, is always
close to unity for diverse ecosystems. Thus both the alpha Gini-Simpson
index and the gamma Gini-Simpson index will always be close
to unity for diverse communites. Lande's similarity measure
takes their ratio, which will therefore also be close to unity
no matter how similar or different the communities are in species
composition.

This mathematical artifact, not biology, is responsible for
the high similarity values published in this paper and elsewhere.
The similarity between canopy and understory butterfly communities,
using the Gini-Simpson index, is 0.975, indicating high similarity,
even though (as the authors of the paper note) the communities
are really quite distinct from each other. The similarity between
habitats is also high, 0.935, and similarity between months
is high as well, 0.978. Not only in this paper but in all
published papers using this methodology, the Gini-Simpon similarity
values are always high for diverse communities, between 0.93
and 0.99, regardless of what is being measured.

The problem (and the solution) can be made clearer by idealizing
the example somewhat. Suppose we have two equally-large communities
with 500 equally common species in each. Their Gini-Simpson
indices would each equal 0.998, and so the alpha Gini-Simpson
index would also equal 0.998. Suppose the two communities were
identical in species composition. Then the Gini-Simpson index
of the pooled communities would also be 0.998, and Lande's index
of similarity would be 0.998/0.998 = 1.000, correctly indicating
complete similarity between the communities. Now let's assume
instead that the two communities were completely dissimilar
(i.e. they have no species in common). Then the pooled
communities have a Gini-Simpson index of 0.999. Lande's similarity
measure is therefore 0.998/0.999 = 0.999, indicating virtually
complete similarity, even though the communities are in fact
completely different! Obviously ecologists could easily reach
wrong conclusions if they relied on this measure of similarity.
This measure of "similarity" really does not measure
similarity in any meaningful sense, because it is so strongly
influenced by alpha diversity.

The solution is simple: use true diversities (effective number
of species) in Lande's similarity measure, not the raw indices.
Regardless of the indices on which they are based, true
diversities behave intuitively when made into ratios, because
they have Hill's doubling property (discussed elsewhere on this
site). In the example of the preceding paragraph, when the two
communities are identical, the modified Lande's similarity measure
would be (1/(1-0.998)) / (1/(1-0.998)) = 1.000, as it
should. (Note that we first find the alpha Gini-Simpson index
and then convert this to get the true alpha diversity; the alpha
diversity is NOT the average of the effective numbers of species
of the two communities. Lande showed in his 1996 paper that
the latter is not a concave function and could give paradoxical
results. My **Ecology** paper on the mathematics
of alpha and beta diversity provides the theoretical justification
of the definition of alpha diversity used here.) When the two
communities are completely distinct, the modified Lande similarity
is (1/(1-0.998)) / (1/(1-0.999)) = 0.50. This makes intuitive
sense; if the two communities are completely different, the
total pooled diversity is twice the diversity of any single
community, so the average community contains half the total
pooled diversity. If we had used Shannon entropy or some other
diversity index instead of the Gini-Simpson index, we would
still get 0.50 when the two communities are completely different.
If there had been N distinct communities, we would get 1/N when
the communities are completely distinct.

Sometimes it is useful to have a similarity index that goes
from 0 to 1 instead of 1/N to 1. When the similarity (really
homogeneity) measure proposed here, using the Gini-Simpson effective
number of species, is linearly transformed onto the interval
[0,1], it turns out to be identical to the famous Morisita-Horn
index of similarity. Effective number of species is a powerful
conceptual tool which allows the derivation of many other interesting
results as well. The Jaccard, Sorensen, and Horn indices, and
the proper definitions of alpha and beta, all can be derived
using this concept. This is the subject of my paper, **Partitioning
diversity into independnet alpha andf beta components**,
in **Ecology** Oct 2007..

More examples will follow when time permits.