# Tree growth analysis issues, again, part three

To re-state, the specific issues I have with Stephenson et al. can be placed into three categories:
(1) Sampling issues
(2) Biological implications of stat/math issues
(3) Results presentation issues

I’m actually going to start with issue (2) since it’s the crux of the issue, and try to work issue (1) into the discussion, since the two are intertwined. Issue (3) is a bit of a separate matter and if I get time I’ll address it separately.

I take a theoretical or “bottom up” approach to the issue, examining the authors’ main conclusion(s) in light of the implications of applying their allometry equations to certain tree growth models, to evaluate the degree to which their findings might be artifacts of their methods. There’s math involved, but just algebra and natural log transforms, and I’ll try to explain exactly what I’m doing, and why. This general approach is greatly simplified by the fact that many (56%) of the 403 species they analyzed were tropical “moist” forest species (by their definition), whose above-ground biomass (AGB) is modeled as f(diameter) by a single biomass equation taken from Chave et al. (2005). I’ll therefore concentrate on them, at least for now. A close look at Chave et al. (2005) is required.

First, a reminder of what they are claiming. This is, that in contrast to leaf- or stand-level analyses, which often show decreasing productivity with age and/or size, in individual trees, productivity (measured as tree AGB increase per unit time) actually increases with size: bigger trees get bigger faster than do smaller trees. They claim that 97% of 403 tropical and temperate tree species, around the globe, follow this pattern. It’s a pretty grand claim. Accordingly, they will need to demonstrate that a few critical analytical requirements were considered and followed, such as that they included the full range of possible tree sizes, and assessed that competitive conditions between the different sizes of trees are similar. And there are some others also.

A common problem presents itself right away, in that the raw data needed for full evaluation are not presented, nor are most of them available in public databases. However, even some really important summary information is not presented either; these include the exact equations used to estimate AGB as a function of diameter (and height in 16% of their species), the numbers of trees in their different AGB classes, and the diameter breakpoints corresponding to those classes (they do give the biomass breakpoints, but without the dry density values for each species you can’t convert them to diameter, even if you could navigate the third degree polynomials, in logarithms (base e for the data tables and base 10 for the graphs no less)). Therefore we have to infer what we can from what they present.

The authors evaluate biomass growth rates as a function of tree size not as a function of time. This is likely because they don’t have ages on the vast majority of their trees–rates come in from measurements of diameter (and sometimes height) at two time points, most often 5 to 10 years apart. So, the study’s not tracking individual trees’ growth rates over their lives, but rather measuring shorter-term growth rates on more trees in a narrower time window (i.e. 2-3 decades). That’s understandable, especially in the tropics where you mostly can’t do retrospective studies over long times, using tree rings (because there aren’t any). But it’s an important difference nevertheless, as it greatly affects the ability to test whether trees of different sizes are experiencing similar growth environments or not, a big potential problem. Also, if you’re interested in carbon sequestration rates, the focus there should be the time rate. But again, this is not really a carbon dynamics study.

Because AGB growth rates are presented as a function of diameter, but diameter growth rates as a function of time are not, it is difficult to impossible to get a full picture of the growth dynamics involved. For example, these maximum growth rates they report for the largest trees–we don’t know how close those rates might be to hitting a plateau and/or starting to decline; mass growth rates aren’t going to keep increasing indefinitely, that’s for sure. Given this, we need a way of evaluating some diameter growth scenarios, i.e. D = f(t), constrained by the biomass vs diameter relationship they use, and seeing what these jointly imply when linked together. Therefore, onto the math.

Although they never state exactly which of Chave et al’s AGB equations they use, they mention in the supplement a third degree polynomial for tropical moist forests, which thus must be:

$\ln(M) = -1.499 + 2.148\ln(D) + 0.207\ln(D)^2 - 0.0281\ln(D)^3 + \ln(d)$ (eq. 1)

where M = biomass (AGB in kg, dry weight), D = diameter (cm), d = wood/bark mean density, and ln = logarithms in base e. But I want to work with the native units, not log units; I get there by exponentiating both sides, giving:

$M = de^{-1.499 + 2.148\ln(D) + 0.207\ln(D)^2 - 0.0281\ln(D)^3}$ (eq. 2)
where e is the base of natural logs, ~2.72.

That’s a somewhat hairy equation frankly, given that logarithms of diameter are raised to powers and then added to form an exponent to e. That makes what I want to do next, which is to solve an exponential equation, harder. Therefore, for illustration purposes here, I’m going to use their simpler equation that gives similar (but somewhat over-estimated) results:

$M = de^{-1.864 + 2.608\ln(D)}$ (eq. 3)

As stated, the above equation relates mass to diameter only; I need to go two steps further. I need the rate of increase in M, which necessarily involves time. This in turn means I need a relationship that defines diameter (or ring width) with respect to time, i.e. the radial growth rate. This will then give me the relationship between biomass (M) and time, and from that I can derive the rate of mass growth as a function of either tree size or time. Then I can vary the radial growth rate with time, in different ways, and observe the effect on the mass growth rate.

One useful exercise would be to find the radial growth rate that gives a constant mass increase as a function of tree size. [Note that a constant radial growth rate leads to rapidly increasing mass growth rates; the radial growth rate has to decrease with time to get constant mass growth.] Doing so will help define the lower boundary of the possible radial growth relationships, given what Stephenson et al. report for mass growth rates. But how do I determine that relationship, exactly?

In equation (3) above, a critical value is the coefficient of ln(D), 2.608. A non-obvious fact is, that because it’s a logarithmic function, whenever that value is > 1.0, M will increase at an increasing rate per unit of diameter increase (and when < 1.0, M will increase at a decreasing rate). When exactly = 1.0, M will increase only arithmetically (i.e. at a constant rate). We can’t change the value of 2.608–it is what it is. But I can make diameter a log function of time, ln(D) = f(ln(t)), such that the condition of constant mass growth per unit time is achieved. This will give ln(M) as a direct function of ln(t), in which the coefficient of ln(t), q, = 1.0, and which grows a tree to defined, large size in a defined time. [d, density, is assumed constant over time and so is not important.] The latter I set by making the target diameter 156 cm (62 in., the largest trees in Chave et al’s sample), which if reached in 200 years, gives a reasonable mean radial growth rate of 0.75 cm/year. That mean rate is not critical–I can slow down or speed up growth and it doesn’t affect the conclusions.

So, ln(D) as f(ln(t)) must include a factor of (1/2.608), and a time coefficient s that determines the rate of diameter change with time, thus:

$\ln(D) = \frac{\ln(st)}{2.608}$ (eq.4)

This is then solved for s by setting t = t.max (i.e. 200) when D = D.max = 156cm:

$\ln(156) = \frac{\ln(200s)}{2.608}$

and thus, $s = \frac{e^{2.608\ln(156)}}{200} = \frac{e^{2.608\times5.05}}{200} = 2622.04$

Therefore, constant mass(M) growth as a linear function of time (i.e., constant rate) is defined as:

$M = de^{-1.864 + q\ln(2622.04t)}$ (eq. 5)
where q = 1.0

So, I’ve derived the diameter growth rate as a sole function of time that’s needed to give a constant rate of tree mass growth, given the stated relationship between diameter and biomass. Progress!

Unfortunately, we find with this model that although the radial growth rates after year 20 or so are entirely reasonable, those in the first decade or two, especially the first five years, are wildly unrealistic (way too high). And this is not surprising really; it is the direct result of forcing the mass growth to be constant over time, which is biologically implausible in the earliest years. There’s no way a sapling < 130 cm tall is going to be able to add biomass like a large tree can; to do so requires growing to 20 cm dbh in the first year. Not going to happen, ever.

However, that’s not the story’s end. From this equation I can create accelerating, and decelerating, mass growth rates by increasing or decreasing, respectively, the value of q from 1.0, and see what they do to the mass growth rates. Note that I also have to re-solve for the value of s when I do this, in order to still meet the target diameter (156 cm) at 200 years. This keeps the mean radial and mass growth rates over time the same, but varies the time dynamics, thereof, within that period. And that variation’s the critical point, because it’s what determines whether mass growth rates continually increase with tree size or not.

Without showing the results, if I decrease q in equation (5), from 1.0 to say, 0.5, this results in continuously decreasing mass growth rates, as functions of either time or tree mass (opposite of what Stephenson et al. find). However, the radial growth rates in the early years are then even more ludicrous than in the example just given. So that model’s not reasonable either. By contrast however, if I double the value of q, to 2.0, this results in constantly increasing mass growth rates with time (and tree size), and in radial growth rates that are quite plausible. Moreover, the more plausible the radial growth rates become, the more rapid is the resulting acceleration of mass growth rates with time.

So, this represents some good theoretical evidence supporting Stephenson et al.’s claims–these mathematical considerations favor accelerating growth. However, there is more to it than just that, as I hope to get into. In short, there are other theoretical considerations, as well as very important practical ones, related to sampling, that also have to be considered.