Tuesday, July 3, 2012

Faith's Conjecture

Allen’s Rule. Rensch’s Rule. Cope’s Law. Bergmann’s Rule. Cheverud’s Conjecture. Who wouldn’t want one? Hendry’s Rule? Hendry’s Law? Hendry’s Paradox? Faith’s Conjecture? That’s right, Faith’s Conjecture!

Just recently, I spent a week in Bonito, Brazil, for the annual meeting of the bioGENESIS core project for DIVERSITAS. Over dinner, we strayed into rules, laws, paradoxes, and conjectures: which one we wanted to have and what it would be. Dan Faith decided that he wanted a Conjecture – and proceeded to suggest one. He proposed that any fairy tale or fable could be inverted so that its message would change in an interesting and informative way. Take, for example, the tortoise and the hare. Maybe the tortoise had heard the classic fable and so assumed he would win, thus losing owing to arrogance or inattentiveness. Or maybe the hare had heard it and so realized he would lose if he didn’t run steadily. Tortoise loses – hare wins – lesson changes: those who don’t pay attention to history are bound to repeat it. Having succeeded here, we tried a few other fairy tales and fables on the Conjecture and I have to confess it was a bit of a force to make it work. Time to give up on Faith's Conjecture.

A few days later, however, I got to thinking about it a bit more and realized that perhaps Faith’s Conjecture really was valid – but in a different context: ecology and evolution. I hereby restate Faith’s Conjecture as: any correlation from which a causal relationship might be inferred (the thing on the x axis influences the thing on the y axis) can be inverted (the things on the x and y are switched) to lead to a new causal inference. Here are some examples.

Gene flow is often assumed to constrain adaptive divergence. Thus, one would expect that populations experiencing higher levels of gene flow would show lower adaptive divergence. So one goes out in nature, samples a bunch of independent population pairs, and tests whether pairs that are more divergent in adaptive traits (y axis) are also those showing lower gene flow (x axis). If such a correlation is present, one infers that gene flow constrains adaptive divergence. However, it is also true that adaptive divergence might constrain gene flow. This is the hypothesis of ecological speciation, where increasing adaptive divergence causes greater ecologically-based reproductive barriers, which decrease gene flow. So one goes out in nature, samples a bunch of independent population pairs, and tests whether pairs that are more divergent in adaptive traits (now the x axis) are also those that show lower gene flow (now the y axis). If the correlation is present, one infers that adaptive divergence reduces gene flow: ecological speciation! Thus, the exact same correlation can be inverted for two different inferences. Remarkably this example is real: many studies have plotted gene flow on the x axis and adaptive divergence on the y axis and made the first inference while many other studies have plotted adaptive divergence on the y axis and gene flow on the x axis and made the second inference. Faith’s Conjecture in the machine!
A perennial topic in biodiversity science is the idea that increasing biodiversity (e.g., number of species) increases ecosystem productivity. This makes sense because, among several reasons, more species will presumably use a more diverse set of resources and thereby increase the overall productivity of the system. So one goes out in nature, samples a bunch of plots, and shows that plots with higher biodiversity have higher productivity: hypothesis validated - we need to save more biodiversity because it will increase productivity, a key ecosystem service. But, at the same time, it has long been known that increased productivity can increase biodiversity – at least up to a point. After all, higher productivity allows more total individuals in an ecosystem which increases both the width and depth of, and thus the number of species in, the food web. So one goes out in nature … you get the idea.

Many other examples exist – and, importantly, every one of these has been inferred in correlational studies. Smaller species' ranges should obviously lead to less genetic diversity, since smaller population sizes and adaptation to a narrow range of conditions should lead to the depletion of genetic diversity. Yet on the flip side, less genetic diversity should constrain evolutionary potential and thus lead to smaller species ranges. Increased diet diversity within a population leads to increased trophic trait diversity owing to disruptive selection, but increased trophic trait diversity leads to increased diet diversity because individuals with different traits can feed on different foods. Etc. Etc. Ad infinitum.
So, Faith’s Conjecture is upheld. Perhaps it is even a law or a rule but, for my money, I would rather have a conjecture too. It just sounds cooler- and easier! I should point out that Faith’s Conjecture cannot be directly tested simply through logic of the above sort. Instead, one needs to experimentally manipulate each of the axes to see if it really does causally effect the other axis and, interestingly, this has been done – and confirmed – in the above examples. In addition, Faith’s Conjecture does not necessarily apply to situations where one of the axes is not a biological variable. For example, increasing geographical distance decreases gene flow but increasing gene flow does not decrease geographical distance.

I propose the Faith’s Conjecture is commonly – perhaps universally – true, and I challenge the reader to provide putative instances where it succeeds or fails. In the latter case, I intend to either prove that cause and effect really are reversible in the proffered example or that the example does not apply. In short, I hope to redefine Faith’s Conjecture as necessary so that it conveniently excludes examples that do not abide. After all, the next best thing to having one's own conjecture is to be the person to name the conjecture.

Some cool things I conjectured in Brazil.


  1. "increasing geographical distance decreases gene flow but increasing gene flow does not decrease geographical distance" – well, it might. Take two populations separated from each other on a gradient, locally adapted. Now ratchet up the gene flow. They will become maladapted to their present location (presumably), but better adapted to some intermediate part of the gradient, and might then experience a range shift towards the other population. Of course this does not always work. :->

  2. Yes indeed! But it won't always happen. Faith's conjecture is for situations where cause and effect can always be reversed.

  3. Nice post, I'll link to it from Dynamic Ecology presently, but I wanted to reply here. Re: the biodiversity-productivity example, you're right that it seems like the causality can run either way. But this depends on a widespread ambiguity in what people mean by "productivity". When we think of causality as running from diversity to productivity, we generally think of "productivity" as meaning "total biomass" or "net primary productivity" or some other collective attribute of the species doing the producing. But when we think of causality as running from productivity to diversity, we generally think of "productivity" as being "enrichment" or something--that is, an attribute of the environment rather than the species occupying the environment (at least, we *should* think of productivity in this latter case).

    Which isn't to say the arrow of causality couldn't still be reversed, if one were willing to think about the determinants of environmental properties like soil fertility on sufficiently long timescales. For instance, maybe plant biodiversity affects the fraction of nutrients in dead plant matter that get retained in the system as opposed to washed away, thereby affecting soil fertility...

  4. Ok, thought of a clear-cut counterexample: species-area curves. Area causally affects species richness, but not vice versa.

    In general, I suspect correlations between biotic and abiotic variables are good candidate counterexamples.

  5. Ah, yes, I agree on the last point but it falls into that category of "In addition, Faith’s Conjecture does not necessarily apply to situations where one of the axes is not a biological variable. For example, increasing geographical distance decreases gene flow but increasing gene flow does not decrease geographical distance." So the conjecture still holds.

  6. Sorry, missed that limitation on the domain of Faith's Conjecture at the very end of the post...

    There certainly are examples of one-way biotic-biotic causality, some of which might be counterexamples to the conejecture. Think of commensalisms, such as nurse plants. You observe a spatial association (correlation) between nurse plants, and other plants. If it arises because nurse plants causally affect other plants but not vice-versa (e.g., because the other plants are much smaller), then that could be considered a counterexample.

  7. Hmmm. Tricky. You might be right, at least if restated in the context of a correlation as "the abundance of nurse plants (x-axis) determines the abundance of dependent plants (y-axis)" But how specific and narrow is this exception? Think of other possible situations where one species might affect another and vice versa. Obviously the abundance of predators and prey will influence each other, as will the abundance of two mutualists, and two competitors, etc. So none of these would violate Faith's Conjecture. The key then is commensualism: by definition one species influences the other but NOT vice versa. Perhaps that is the exception that proves the rule because the phenomenon obviates the conjecture by definition.

    However, it leaves open the question of just how common and strong is true commensualism. I would argue that it is never strictly true. That is, nurse species will - to at least some extent - be influenced by the plants that occupy them. This effect may be weak but it will almost certainly be present. So the conjecture might technically hold but it can be ignored for all intents and purposes because the effect in one direction is so much stronger than the effect in the opposite direction.