# Ain’t no ash will burn

A gorgeous traditional tune that Matt Skaggs made me aware of some time back. It’s become a definite favorite; here’s my brief interp.

I have seen snow that fell in May
And I have seen rain on cloudless days
Some things are always bound to change
There ain’t no ash will burn

Love is a precious thing I’m told
It burns just like West Virginia coal
But when the fire dies down it’s cold
There ain’t no ash will burn

You say this life is not your lot
Well I can’t be something that I’m not
We can’t stoke a fire that we ain’t got
There ain’t no ash will burn

In every life there comes a time
Where there are no more tears to cry
We must leave something dear behind
There ain’t no ash will burn

There is one lesson I have learned
There ain’t no ash will burn

# Hardy-Weinberg genetic equilibrium and species composition of the American pre-settlement forest landscape

This post is about how binomial probability models can, and cannot, be applied for inference in a couple of very unrelated biological contexts. The issue has (once again) made popular media headlines recently, been the focus of talk shows, etc., and so I thought it would be a good time to join in the discussions. We should, after all, always focus our attention wherever other large masses of people have focused theirs, particularly on the internet. No need to depart from the herd.

Binomial models give the coefficients/probabilities for all possible outcomes when repeated trials are performed of an event that has two possible outcomes that occur with known probabilities. The classic example is flipping a coin; each flip has two possible outcomes of h = t = 0.5, and if you flip it, say twice (two trials), you get 1:2:1 as binomial coefficients for the three possible outcomes of (1) hh = two heads, (2) ht = one head and one tail, or (3) tt = two tails, which gives corresponding probabilities of {hh, ht, tt} = {0.25, 0.50, 0.25}. These probabilities are given by the three terms of (h + t)^2, where the exponent 2 gives the number of trials. The number of possible outcomes after all trials is always one greater than the number of trials, with the order of the outcomes being irrelevant. Simple and easy to understand. The direct extension of this concept is found in multinomial models, in which more than two possible outcomes for each trial exist; the concept is identical, there are just more total probabilities to compute. Throwing a pair of dice would be a classic example.

The most well-known application of binomial probability in biology is probably Hardy-Weinberg equilibrium (HWeq) analysis in population genetics, due to the fact that chromosome segregation (in diploids) always gives a dichotomous result, each chromosome of each pair having an equal probability of occurrence in the haploid gametes. The binomial coefficients then apply to the expected gamete combinations (i.e. genotypes) in the diploid offspring, under conditions of random mating, no selection acting on the gene (and on closely linked genes), and no migration in or out of the defined population.

# I Know You Rider

My interpretation of that old classic, a very fun one to play.

Manzanar WWII Japanese Internment Camp with Sierra Nevada in back, near Independence CA.

# Tree growth analysis issues, again, part five

Continuing in this series, the point of which is to explore issues in tree growth analysis and their relationship to claims made by Stephenson et al. (2014)

Previously, I derived a function relating tree above-ground biomass (AGB) to time, given an apriori equation from the literature (Chave et al., 2005) relating biomass to diameter that is similar to the one used by Stephenson et al. [I used this similar equation, rather than the one actually used by Stephenson et al. (2014), because doing so allowed for an easier derivation of the function relating diameter to time that would produce a constant mass growth rate.] The point of that exercise was to establish a reference frame of constant mass growth rate, from which I could then investigate what increasing and decreasing mass growth rates would imply for radial growth rates. The analysis showed me that models of that basic structure could only produce continuously increasing or decreasing mass (and radial) growth rates.

But continuously decreasing rates are biologically unreasonable, because they require impossible growth rates when trees are very young, and although continuously increasing rates do support what Stephenson et al. claim, they too are untenable. This is simply because, of course, nothing in nature can grow at an ever-accelerating rate; there has to be a deceleration at some point, at the very least to an asymptote, but more likely to an actual decline. The universe is not over-run by tree mass after all. In fact, one of the things that worries me about their paper is that their data shows very little rate deceleration at all, including in species that reach very large size. So…given the rate accelerations at the maximum tree sizes they report–the main point of their paper–then where are all the trees that are > 1.0X the mass of the largest they report? Shouldn’t they also exist? Did all their tree sizes just happen to fall short of the size threshold where growth rates start to decline? Or do they imagine that tree growth rates never slow down and maximum size is limited by various other forces? If so, what are they? Some things just don’t add up here.

# I was born that way

I’ve been a fool of a singular cool all by myself
Nobody showed me how, I was born that way
Every day is a solo played on a single string
Nobody shows up, nobody walks away

What do I do when the tune is through?
How’m I gonna get me home?
What would you say if I turned your way and said
“Help me now, I can’t do it alone”?

Because lonesome is, as lonesome does, and I do it
Perfect practice keeps me next to me
Nobody needs to need me, there’s nothing to it
Friends you don’t make always let you be

Where do I go to close this show?
This one man band to the bone
Why does it feel like such a deal to say
“Help me now, I can’t do it alone,
Won’t you help me now, I can’t do it alone”

Help Me Now, Chris Smither

# Tree growth analysis issues, again, part four

Continuing….

Last time, I derived an equation that gives constant tree mass growth rates with time, given an apriori, defined relationship between biomass and diameter. Varying two parameters thereof (q and s) produced continuously increasing or decreasing mass and radial (diameter) growth rates, but only the continuously increasing mass growth rates also had reasonable radial growth rates over the full range of small and large trees. I now want to investigate whether there are radial growth models that can give more complex relationships of mass to time, unimodal responses in particular. But before I get to that I’ll show some graphs illustrating these dynamics, and provide the R code as well (at the end).

I’ve altered two things from the previous post, for convenience. First, I re-defined q so as to make the connection between M = f(D), and D = f(t), easier to follow. Specifically,

$M = de^{-1.864 + 2.608\ln(D)}$ and
$ln(D) = \frac{\ln(st)}{2.608q}$ and therefore
$M = de^{-1.864 + \frac{\ln(st)}{q}}$

where M is tree mass, D is diameter, d is wood density, e is the base of natural logs, t is time in years and s and q are computed parameters that jointly give equal mean growth rates at year t = 200 (as explained in the previous post). In this formulation, any model having q < 1.0 will result in continuously increasing mass growth rates as functions of either time or diameter. Second, I changed the target diameter from 156 cm to 120 cm, at 200 years (i.e. slower mean growth rate). This slower mean growth rate just allows me to extend growth beyond 200 years and not get unrealistically sized trees as quickly, at say 300 years.

But before going on, below are four graphs, each showing the growth dynamics of six models, given what I’ve done so far. All produce increasing mass growth rates as a function of time, result from varying q and s in the above equation, and have q < 1.0. The graphs show: (1) diameter and (2) mass, as functions of time, and then mass growth rates as functions of (3) time and (4) diameter. The values of q and s listed in the legends are for the third equation above $M = de^{-1.864 + \frac{\ln(st)}{q}}$

# 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.

# Phenotypic plasticity and climate adaptation; ecology vs natural history

For those interested in the potentially very important issue of biological adaptation to climate change, you will definitely want to check out the latest issue of Evolutionary Applications, a special issue addressing climate change, adaptation and phenotypic plasticity, all articles open. I’ve not yet been able to read any of the articles, but it looks really good from first glance and I’m certain I will learn a lot from it.

That second phrase there is the topic of Jeremy Fox’s latest post at Dynamic Ecology, and he’s outdone even himself this time; go see, once again, how good blog articles and their discussions can be when the effort is made. I wish I had time to respond to it with anything more than the couple of sentences I stated there, but I do not–whatever extra time I have is devoted to just reading (including the comments) and thinking about it. And those discussions over there give you a lot to think about.

# Tree growth analysis issues, again, part two

This picks up from the previous post on Stephenson et al (2014).  Hoping that you’ve looked at the paper, I’ll try to explain why I have concerns about it. It presents a good opportunity to discuss some important analytical issues, biological and mathematical.

I’ll begin with a criticism that relates more to a general concern with scientific publications: the way statements/claims are made in titles and abstracts.  Many scientists will only ever read the abstract of a given article, or even just the title, so it’s extra important to word things carefully there so as not to create the wrong impression.  And since scientific protocol strongly trains for accuracy and brevity in writing, there’s little room for excuses when failing to do so. Their title reads: “Rate of tree carbon accumulation increases continuously with tree size“. That’s an unambiguous, general statement; they’re basically claiming a new, general finding (law?) regarding tree growth rates, world-wide.  Hmmmm. All I will say to that is, better be able to back something like that up, because a lot of people have been looking at tree growth rates for a long time, and some of them are probably going to look pretty closely at it. Or very closely.