A painless introduction to partitions in ecology and evolution

A partition is a mathematical division of some quantity into component parts. Partitions are incredibly useful in ecology and evolution. But partitions can be tricky to interpret, as illustrated by widespread misunderstanding of some of them. Further, the same interpretive issues crop up over and over with different partitions in different contexts. Hence this little “field guide” to partitions in ecology and evolution.

Can you walk me through an example of a partition?

Sure. Consider an exponentially growing population, dN/dt=rN. And consider a second exponentially growing population, with parameters and state variables denoted by primes: dN’/dt=r’N’. Let’s say we want to understand why dN/dt differs from dN’/dt. Just by looking at the equations, you can tell that any difference between their growth rates must be due to differences in r, differences in N, or both. But to be more precise, you need to do a bit of math. We can express the difference between dN’/dt and dN/dt as

dN’/dt-dN/dt=r’N’-rN

The right hand side of the above equation is a partition of the difference between dN’/dt and dN/dt, but it’s trivially obvious and so not very useful. It doesn’t tell you anything you didn’t already know. But we can rearrange it using algebra into a form that tells us something less obvious. I’m going to show you the steps and encourage you not to skip them even if you’re a math phobe, because they’re quite simple and they’ll help you understand the rest of the post.

dN’/dt-dN/dt=r’N’-rN #the equation we started with

dN’/dt-dN/dt=(r’-r+r)(N’-N+N)-rN #add and subtract r and N. This seems pointless, but it’s got a point, honest. 🙂

dN’/dt-dN/dt=(r+Δr)(N+ΔN)-rN #let Δr denote r’-r and ΔN denote N’-N

dN’/dt-dN/dt=NΔr+rΔN+ΔrΔN #multiply through the parentheses, cancel rN-rN

The final equation above is a three-part additive partition of the difference in population growth rate between the two populations. It decomposes the difference into components attributable to the difference in r values (NΔr), the difference in N values (rΔN), and an “interaction term” depending on the differences in both r and N values (ΔrΔN). Those components have ecological interpretations. You can interpret NΔr as telling you how much population growth rate would change if you took the equation dN/dt=rN and replaced r with r’. That is, it tells you how much of the difference in population growth rate between the two populations is attributable to the change in r, holding everything else constant. The other two terms have analogous interpretations–they isolate the effect of changing N on population growth rate, and isolate the interactive effect of changing r and changing N. That three-part partition is just a rearrangement of the equation we started with, so everything in it is implicit in the first equation. But for reasons discussed below, it’s often really useful to make the implicit explicit.

Hmm. Ok, I guess. But that’s a weird made-up example, it’s like trolley problem ecology. Can you give me some real examples?

Sure! Happy to oblige (although you really ought to give trolley problem ecology a chance…)

• ANOVA. The one partition with which most every ecologist is familiar. Partitions total sums of squares into components attributable to variation among group means, and variation among observations within groups. Thereby allowing a test of the null hypothesis that variation among groups is within the range you’d expect, given the among of variation within groups.
• The Price equation. Originally, a partition of evolutionary change in the mean of some character into components attributable to evolution by natural selection, and transmission bias. See Frank 2012 and references therein for review. Has since been extended and reinterpreted to apply to a much broader range of problems, including ecological problems. Partitioning the effects of individual-level selection from those of group selection (a huge literature, reviewed in Okasha 2006). Partitioning the effects of species richness, species composition, and “context dependence” on ecosystem function (Fox 2006). Partitioning effects of evolution and various ecological factors on community or ecosystem properties (Ellner et al. 2011). Partitioning the determinants of average population growth rate in populations distributed over space (Schauber et al. 2007). Many more. Has also spawned a lot of conceptual debate, since there are different forms of the Price equation that slice the same evolutionary “pie” in different ways, with no universal agreement on which way is correct or best.
• Partitions of the diversity-stability relationship. Several authors have proposed partitions of the temporal variance of total abundance or biomass (an inverse measure of “stability”) into ecologically-interpretable components, with an emphasis on isolating the effect of “diversity” on stability. Lehman and Tilman 2000, Ives and Hughes 2002, Thibault et al. 2013, Loreau and deMazancourt 2013, deMazancourt et al. 2013, others.
• Additive partitions of biodiversity-ecosystem function relationships. Loreau and Hector 2001 proposed a now-standard way to partition the effects of biodiversity on ecosystem function in a particular sort of experiment into effects of “selection” and “complementarity”. In Fox 2005 I proposed a refined “tripartite” version of their partition, arguing that their “selection” effect combines two different effects best kept separate.
• Partitioning gamma diversity into alpha and beta diversity. Proposing and debating alternative partitions of gamma diversity into alpha and beta diversity is almost its own subfield at this point. Lots of debate over whether an additive or multiplicative partition is best, plus there are various additive and multiplicative partitions to choose from. There’s also debate over how to partition beta diversity into subcomponents (e.g., “turnover” vs. “nestedness”). And debate over how or whether to standardize beta diversity by alpha diversity…Perhaps there’s so much debate because nobody can agree on exactly what “beta diversity” is.
• Chesson’s partition of coexistence mechanisms. Peter Chesson has developed a partition (actually, more than one) of the contributions of different classes of coexistence mechanism to species coexistence. Very important work, empirical applications of which are now taking off. See this old post for some entry points into this literature, focusing on those coexistence mechanisms that only operate if species’ abundances fluctuate over time (Chesson’s work also incorporates other coexistence mechanisms, of course).
• Others I’m not thinking of just now.

What good are partitions?

Plenty! You can of course follow up the specific examples above to see what good each of them is. But here are a couple of general, overarching uses of all partitions. There are others, but I’ll stick with two because this post is already long.

• They reveal if you’re missing anything. If you want to quantify the relative strength or importance of the different factors that determine the value of some variable of interest, well, you’d better not miss any factors! That’s where partitions come in: taken together, everything in your partition (i.e. everything on the right hand side of your equation) has to equal the thing you’re partitioning (i.e. the quantity on the left hand side of your equation). For instance, Rankin et al. 2015 used an extension of the Price equation to partition the contributions of three macroevolutionary forces (species selection, within-lineage evolution, and immigration) to evolution of mean mammalian body size across a major climate change (the Paleocene-Eocene Thermal Maximum). Previous work had completely missed the possibility of species selection, treating within-lineage evolution and immigration as the only two possibilities. The Price equation shows that the three forces Rankin et al. considered are exhaustive. They’re the only three, it’s mathematically impossible that there are any others out there waiting to be discovered. Similarly, Peter Chesson’s work proves that the classes of coexistence mechanism he partitions are the only ones possible under his (very broad) assumptions.
• They tell you exactly how to measure stuff. For instance, the Price equation says that the effect of evolution by natural selection on the mean phenotype of a population is given by the covariance between parental relative fitness and parental phenotype. Not the correlation. Not twice the covariance. Not the square root of the covariance. Not the covariance between parental fecundity and parental phenotype. Etc. Ok, in practice maybe you won’t always be able to measure exactly what the relevant partition tells you to measure. But you at least need to know what you’re trying to measure, before you can decide to go with an index or whatever. The Price equation also tells you that total evolutionary change is the sum of the effects of natural selection and transmission bias. Not their product, or their ratio, or etc.

There’s often more than one way to partition the same quantity, isn’t there? How do you identify the correct partition? Or the best one, if there isn’t a single correct one?

Good question. As noted above, there are indeed often alternative partitions of the same quantity.

Sometimes, two partitions differ because one subdivides terms that another lumps together, as with the additive partition and the tripartite partition. In such cases, the more finely resolved partition (here, the tripartite partition) should always be looked at first. If it turns out that you can lump together some terms without much loss of information or insight, you can always do so later. But if you never look at the more finely resolved partition, you don’t know what information you might be throwing away.

Sometimes, partitions might differ because they’re based on different assumptions. For instance, you partition sums of squares in ANOVA differently depending on whether you have a nested or factorial experimental design, and on whether your treatment effects are fixed or random effects. In such cases, you should choose whichever partition makes the most nearly-true assumptions. As another example, two different forms of the Price equation can be interpreted as making different assumptions about ordering of events: selection and then transmission, or vice-versa. You need to pick whichever version best matches the ordering of events in your system, or else look at both and use them to bracket the truth.

Sometimes, it’s unclear which alternative partition is best, or there is no “best” partition at all. It’s not that we have nothing to go on–usually, there are various agreed criteria that any partition ought to satisfy. For instance, any sensible partition of alpha and beta diversity should define beta diversity to be zero when all sites have exactly the same species at exactly the same abundances. But it’s unclear which partition is best if those criteria either conflict with one another and so can’t all be satisfied by a single partition, or else are satisfied by more than one partition. Those are signs that we don’t know exactly what we’re trying to partition. The debate over how to partition alpha and beta diversity seems like a good example. As another example, Okasha 2004 shows that evolutionary biologists have conflicting intuitions about what constitutes “group selection”, leading to different partitions of individual and group selection. Because those intuitions conflict, it’s mathematically impossible for a single partition to satisfy all of them. Until we all agree on exactly what we mean by “beta diversity” or “group selection”, there is no “correct” or “best” partition.

Can partitions be interpreted causally?

Depends on the partition, and on what you mean by “cause”. Sometimes, a causal interpretation will seem perfectly natural. For instance, if you use Chesson’s approach to partition the effects of fluctuation-dependent and fluctuation-independent mechanisms of coexistence, and find that only the former are positive, you’d say that the species in your system are coexisting because of fluctuation-dependent mechanisms. But in other cases, it’s not so clear. I can talk about this further in the comments if anyone’s interested.