## Measuring Biodiversity

7 November, 2011

guest post by Tom Leinster

Even if there weren’t a global biodiversity crisis, we’d want to know how to put a number on biodiversity. As Lord Kelvin famously put it:

When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, your knowledge is of a meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science.

In this post, I’ll talk about what happens when you take a mass of biological data and try to turn it into a single number, intended to measure biodiversity.

There have been more than 50 years of debate about how to measure diversity. While the idea of putting a number on biological diversity goes back to the 1940s at least, the debate really seems to have got going in the wake of pioneering work by the great ecologist Robert Whittaker in the 1960s.

There followed several decades in which progress was made… but there was a lot of talking at cross-purposes. In fact, there was so much confusion that some people gave up on the diversity concept altogether. The mood is summed up by the title of an excellent and much-cited paper of Stuart Hurlbert:

• S. H. Hurlbert, The nonconcept of species diversity: A critique and alternative parameters. Ecology 52:577–586, 1971.

So why all the confusion?

One reason is that the word “diversity” is used by different people in many different ways. We all know that diversity is important: so if you found a quantity that seemed to measure biological variation in a sensible way, you might be tempted to call it “diversity” and publish a paper promoting your quantity over all other quantities that have ever been given that name. There are literally dozens of measures of diversity in the literature. Here are two simple ones:

• Species richness is simply the number of species in the community concerned.
• The Shannon entropy is $-\sum_{i = 1}^S p_i \log(p_i)$, where our community consists of $S$ species in proportions $p_1, \ldots, p_S$.

Which quantity should we call “diversity”? Do all these quantities really measure the same kind of thing? If community A has greater than species richness than community B, but lower Shannon entropy, what does it mean?

Another cause for confusion is a blurring between the questions

Which quantities deserve to be called diversity?

and

Which quantities are we capable of measuring experimentally?

For example, we might all agree that species richness is an important quantity, but that doesn’t mean that species richness is easy to measure in practice. (In fact, it’s not, more on which below.) My own view is that the two questions should be kept separate:

The statistical problem of designing appropriate estimators becomes relevant only after the measure to be estimated is accepted to be meaningful.

(Hans-Rolf Gregorius, Elizabeth M. Gillet, Generalized Simpson-diversity, Ecological Modelling 211:90–96, 2008.)

The problems involved in quantifying diversity are of three types: practical, statistical and conceptual. I’ll say a little about the first two, and rather more about the third.

Practical  Suppose that you’re doing a survey of the vertebrates in a forest. Perhaps one important species is brightly coloured and noisy, while another is silent, shy, and well-camouflaged. How do you prevent the first from being recorded disproportionately?

Or suppose that you’re carrying out a survey, with multiple people doing the fieldwork. Different people have a tendency to spot different things: for example, one person might be short-sighted and another long-sighted. How do you ensure that this doesn’t affect your results?

Statistical  Imagine that you want to know how many distinct species of insect live in a particular area — the “species richness”, in the terminology introduced above. You go out collecting, and you come back with 100 specimens representing 10 species.

But your survey might have missed some species altogether, so you go out and get a bigger sample. This time, you get 200 specimens representing 15 species. Does this help you discover how many species there really are?

Logically, not at all. The only certainty is that there are at least 15 species. Maybe there are thousands of species, but almost all of them are extremely rare. Or maybe there are really only 15. Unless you collect all the insects, you’ll never know for sure exactly how many species there are.

However, it may be that you can make reasonable assumptions about the frequency distribution of the species. People sometimes do exactly this, to try to overcome the difficulty of estimating species richness.

Conceptual  This is what I really want to talk about.

I mentioned earlier that different people mean different things by “diversity”. Here’s an example.

Consider two bird communities. The first looks like this:

It contains four species, one of which is responsible for most of the population, and three of which are quite rare. The second looks like this:

It has only three species, but they’re evenly balanced.

Which community is the more diverse? It’s a matter of opinion. Mostly in the press, and in many scholarly articles too, “biodiversity” is used as a synonym for “species richness”. On this count, the first community is more diverse. But if you’re more concerned with the healthy functioning of the whole community, the presence of rare species might not be particularly important: it’s balance that matters, and the second community has more of that.

Different people using the word “diversity” attach different amounts of significance to rare species. There’s a spectrum of points of view, ranging from those who give rare species the same weight as common ones (as in the definition of species richness) to those who are only interested in the most common species of all. Every point on this spectrum of viewpoints is reasonable. None should have a monopoly on the word “diversity”.

At least, that’s what Christina Cobbold and I argue in our new paper:

• Tom Leinster, Christina A. Cobbold, Measuring diversity: the importance of species similarity, Ecology, in press (doi:10.1890/10-2402.1).

But that’s not actually our main point. As the title suggests, the real purpose of our paper is to show how to measure diversity in a way that reflects the varying differences between species. I’ll explain.

Most of the existing approaches to measuring biodiversity go like this.

We have a “community” of organisms — the fish in a lake, the fungi in a forest, or the bacteria on your skin. This community is divided into $S$ groups, conventionally called species, though they needn’t be species in the ordinary sense.

We assume that we know the relative abundances, or relative frequencies, of the species. Write them as $p_1, \ldots, p_S$. Thus, $p_i$ is the proportion of the total population that belongs to the $i$th species, where “proportion” is measured in any way you think sensible (number of individuals, total mass, etc).

We only care about relative abundances here, not absolute abundances: so $p_1 + \cdots + p_S = 1$. If half of a forest is destroyed, it might be a catastrophe, but on the (unrealistic) assumption that all the flora and fauna in the forest were distributed homogeneously, it won’t actually change the biodiversity. (That’s not a statement about what’s important in life; it’s only a statement about the usage of a word.)

This model is common but crude. It can’t detect the difference between a community of six dramatically different species and a community consisting of six species of barnacle.

So, Christina and I use a refined model, as follows. We assume that we also have a measure of the similarity between each pair of species. This is a real number between 0 and 1, with 0 indicating that the species are as dissimilar as could be, and 1 indicating that they’re identical. Writing the similarity between the $i$th and $j$th species as $Z_{ij}$, this gives an $S \times S$ matrix $\mathbf{Z}$. Our only assumption on $\mathbf{Z}$ is that its diagonal entries are all 1: every species is identical to itself.

There are many ways of measuring inter-species similarity. Probably the most familiar approach is genetic, as in “you share 98% of your DNA with a chimpanzee”. But there are many other possibilities: functional, phylogenetic, morphological, taxonomic, …. Diversity is a measure of the variety of life; having to choose a measure of similarity forces you to get clear exactly what you mean by “variety”.

Christina and I are by no means the first people to incorporate species similarity into the model of an ecological community. The main new thing in our paper is this measure of the community’s diversity:

${}^q D^{\mathbf{Z}}(\mathbf{p}) = ( \sum_i p_i (\mathbf{Z}\mathbf{p})_i^{q - 1} )^{1/(1 - q)}.$

What does this mean?

• ${}^q D^{\mathbf{Z}}(\mathbf{p})$ is what we call the diversity of order $q$ of the community. Here $q$ is a parameter between $0$ and $\infty$, which you get to choose. Different values of $q$ represent different points on the spectrum of viewpoints described above. Small values of $q$ give high importance to rare species; large values of $q$ give high importance to common species.
• $\mathbf{p}$ is shorthand for the relative abundances $p_1, \ldots, p_S$, and $\mathbf{Z}$ is the matrix of similarities.
• $(\mathbf{Z}\mathbf{p})_i$ means $\sum_j Z_{ij} p_j$.

The expression doesn’t make sense if $q = 1$ or $q = \infty$, but can be made sense of by taking limits. For $q = 1$, this gives

${}^1 D^{\mathbf{Z}}(\mathbf{p}) = 1/(\mathbf{Z p})_1^{p_1} (\mathbf{Z p})_2^{p_2} \cdots (\mathbf{Z p})_S^{p_S} = \exp(-\sum_i p_i \log(\mathbf{Z p})_i)$

If you want to know the value at $q = \infty$, or any of the other mathematical details, you can read this post at the n-Category Café, or of course our paper. In both places, you’ll also find an explanation of what motivates this formula. What’s more, you’ll see that many existing measures of diversity are special cases of ours, obtained by taking particular values for $q$ and/or $\mathbf{Z}$.

But I won’t talk about any of that here. Instead, I’ll tell you how taking species similarity into account can radically alter the assessment of diversity.

I’ll do this using an example: butterflies of subfamily Charaxinae at a site in an Ecuadorian rainforest. The data is from here:

• P. J. DeVries, D. Murray, R. Lande, 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.

They measured the butterfly abundances in both the canopy (top level) and understorey (lower level) at this site, with the following results:

 Species Canopy Understorey Prepona laertes 15 0 Archaeoprepona demophon 14 37 Zaretis itys 25 11 Memphis arachne 89 23 Memphis offa 21 3 Memphis xenocles 32 8

Which is more diverse: canopy or understorey?

We’ve already seen that the answer is going to depend on what exactly we mean by “diverse”.

First let’s answer the question under the (crude!) assumption that different species have nothing whatsoever in common. This means taking our similarity matrix $\mathbf{Z}$ to be the identity matrix: if $i \neq j$ then $Z_{ij} = 0$ (totally dissimilar), and if $i = j$ then $Z_{ii} = 1$ (totally identical).

Now, remember that there’s a spectrum of viewpoints on how much importance to give to rare species when measuring diversity. Rather than choosing a particular viewpoint, we’ll calculate the diversity from all viewpoints, and display it on a graph. In other words, we’ll draw the graph of ${}^q D^{\mathbf{Z}}(\mathbf{p})$ (the diversity of order $q$) against $q$ (the viewpoint). Here’s what we get:

(the horizontal axis should be labelled with a $q$.)

Conclusion: from all viewpoints, the butterfly population in the canopy is at least as diverse as that in the understorey.

Now let’s do it again, but this time taking account of the varying similarities between species of butterflies. We don’t have much to go on: how do we know whether Prepona laertes is very similar to, or very different from, Archaeoprepona demophon? With only the data above, we don’t. So what can we do?

All we have to go on is the taxonomy. Remember your high school biology: for the butterfly Prepona laertes, the genus is Prepona and the species is laertes. We’d expect species in the same genus to have more in common than species in different genera. So let’s define the similarity between two species as follows:

• the similarity is 1 if the species are the same
• the similarity is 0.5 if the species are different but in the same genus
• the similarity is 0 if they are not even in the same genus.

This is still crude, but in the absence of further information, it’s about the best we can do. And it’s better than the first approach, where we ignored the taxonomy entirely. Throwing away biologically relevant information is unlikely to lead to a better assessment of diversity.

Using this taxonomic matrix $\mathbf{Z}$, and the same abundances, the diversity graphs become:

This is more interesting! For $q > 1$, the understorey looks more diverse than the canopy — the opposite conclusion to our first approach.

It’s not hard to see why. Look again at the table of abundances, but paying attention to the genera of the butterflies. In the canopy, nearly three-quarters of the butterflies are of genus Memphis. So when we take into account the fact that species in the same genus tend to be somewhat similar, the canopy looks much less diverse than it did before. In the understorey, however, the species are spread more evenly between genera, so taking similarity into account leaves its diversity relatively unchanged.

Taking account of species similarity opens up a world of uncertainty. How should we measure similarity? There are as many possibilities as there are quantifiable characteristics of living organisms. It’s much more reassuring to stay in the black-and-white world where distinct species are always assigned a similarity of 0, no matter how similar they might actually be. (This is, effectively, what most existing measures do.) But that’s just hiding from reality.

Maybe you disagree! If so, try the the Discussion section of our paper, where we lay out our arguments in more detail. Or let me know by leaving a comment.