[note: I'll get to leaves in a bit, but first...] I remember as a grad student learning about the “density-dependence debate” and the somewhat amazing fact that a series of random positive numbers (which may represent a series of random population sizes) will show (arguably misleading) evidence of density-dependence. Take, for example, this randomly generated population series:

If I calculate the population rate of change as *r* = (*population*(t+1) - *population*(t))/(*population*(t)) and plot it against population size, I get:

Smaller populations tend to increase and larger populations tend to decrease. But isn’t this misleading? The population sizes were generated randomly from one generation to the next! There’s zero correlation between them!

The reason this happens can be viewed two ways: First, mathematically, the denominator of the *y*-axis variable * is* the

*x*-axis variable (

*population*(t)). Although it seemed like a sensible way to calculate and plot these quantities, random variation in

*population*(t) will naturally make large

*x*-axis values small

*y*-axis values and vice versa, i.e. if

*x*is big, then

*c*/

*x*is small and vice versa. Of course the same sort of thing can manifest in other ways whenever the axes of a plot contain functions of the same random variables. And it’s not just plots… if I had calculated a statistic or an additional variable based on the above plot, I’d be in the same boat.

**The upshot is that one must exercise extreme caution when relating quantities that are calculated using some of the same random variables.**

The second explanation takes a more philosophical turn. The assertion is that a population that takes on a random size from generation to generation about some mean (e.g. 1500 in the plot above) *actually is density-dependent* because, on average, larger populations will be followed by smaller populations and vice versa. If instead I let the population randomly walk, where each generation’s population size is increased or decreased by some random amount relative to the previous generation, then large populations are not on average followed by small populations or vice versa:

and the supposed signature of density-dependence goes away

despite the fact that I’m still dividing the *y*-axis variable by the *x*-axis variable!

So what does this have to do with leaf traits? A lot, it turns out. The canonical leaf economics spectrum paper, Wright et al. 2004, includes plots and correlations of various leaf traits on mass- and area-bases. The leaf economics spectrum is a hugely influential framework for understanding leaf traits, and it’s fair to say that people all over the globe have reorganized their thinking and their research agendas based on it. How many talks have I sat in at ESA that made at least a passing reference to the leaf economics spectrum? Umpteen, at least. Likely more.

However, as recent papers by Osnas et al. (2013) and Lloyd et al. (2013) point out, a lot of the apparently strong mass-based correlations may actually reflect tremendous worldwide variation in leaf mass per area… and little more. Both groups of authors make a much more nuanced argument than I will make here, and you should read their papers! I may be biased, coming from the same lab, but I think the Osnas et al. (2013) contribution moves the ball forward to a greater extent by articulating the extent to which various leaf traits are mass- or area-related without falling into “statistical quicksand” (Lloyd et al.’s term for my bolded statement above).

**Those papers are where it’s at, but I’ll add a nice blog-friendly contribution to the conversation… it’s a little exercise you (or your students) can perform in the comfort and safety of your own office, lab, or classroom. It’s easy and vividly illuminates the issue.**

First, get your hands on the Glopnet dataset used in Wright et al. (2004) and Osnas et al. (2013). Wright et al. wisely made it public. You can download it here. Using your favorite statistics program, plot Log(Amass) against Log(Nmass). Amass is the maximum leaf level photosynthetic rate per leaf mass and Nmass is the leaf nitrogen content per leaf mass. Your graph should look something like this:

Sweet correlation, right? Let’s also graph Log(Aarea) against Log(Narea), which are equivalent measures on a leaf area basis:

Not so sweet a correlation. My natural (and as you can anticipate, naive) reaction to the above graphs is that there is a real relationship between *A* and *N* on a mass basis but not on an area basis. However, and this gets back to the population rate of change calculation above, *Nmass* = *Narea/LMA* and *Amass* = *Aarea/LMA*, where *LMA* is the leaf mass per area. So the mass-basis graph divides both axes by *LMA*. All else equal, large values of *LMA* will lead to small values of *Nmass* and *Amass,* and small values of *LMA* will lead to large values of *Nmass* and *Amass*.

The *proper* way to demonstrate that most of the correlation in the mass-based graphs is due to variation in LMA is via the statistical machinery of Osnas et al. (2013) and Lloyd et al. (2013). But you can get a gut feel for it by fabricating fake mass-based data in Glopnet. Simply scramble the *LMA* column and recalculate mass-based values using the *real* *Narea* and *Aarea* values (which are still truly linked) together with the scrambled *LMA* data. Here’s the recipe:

- “Un-Log” the “
*log Aarea*” column:*Aarea*= 10^(*log Aarea)*. - “Un-Log” the “
*log Narea*” column:*Narea =*10*^(log Narea)*. - “Un-Log” the “
*log LMA*” column:*LMA =*10^*(log LMA)*. - Create a new column, “
*LMA scrambled*” that scrambles the*LMA*data (i.e. take each LMA value and randomly associate it with a different row) – it’s up to you to figure out how to do this, but one possibility is to import the column into Microsoft Excel, create another column next to it with random numbers (use the formula “=Rand()”), and then sort*both*columns by the random number column. Bam! The LMA values have been scrambled! - Create fake Amass data:
*log fake Amass*= Log(*Aarea/LMA scrambled*). - Create fake Nmass data:
*log fake Nmass*= Log(*Narea/LMA scrambled*). - Plot the fake data and compare it to the real data!

Sweet correlation, right? But this fake correlation is driven not by the relationship between photosynthesis and nitrogen, but by the the random variation in LMA, which divides both axes. The implication is that the correlation in the real data is similarly driven by random variation in LMA.

Now, there is a lot to be said here, and Westoby et al. (2013) have issued a rejoinder that deserves your careful consideration. With their improved method, Osnas et al. (2013) make the case that some leaf traits (e.g. nitrogen and dark respiration) are to some extent mass-based and to some extent area-based, whereas others (e.g. phosphorus and maximum net photosynthetic rate) are almost entirely area-based. Upshot, there’s a lot of important nuance here that needs to be understood by serious ecophysiologists. The leaf economics spectrum – far from dead – is being improved and refined. Science moves forward!