R squared irrelevant?

In addition to problems in experimental design, the key results were felt to be weak.  In Fig. 2, the most significant result is the relationship between physiological drought tolerance and abundance for uplands, which has an R2 of 0.12.  The data presented are not strong enough to support the main points in the abstract.


There are two metrics used to judge a particular result: the coefficient of determination (r2) and the P-value, which is the probability that the observed result could have happened by chance. 


For some modelers seeking to replicate observed phenomenon, the coefficient of determination is the key statistic to assess. Eddy covariance modelers often seek to explain observed patterns in carbon flux and when they can generate a high enough r2 by including different independent variables in their model, the feel they've modeled the system well enough and they move on.  


For hypothesis testing in ecology, r2 is irrelevant. Think about the extremes. Let's say we are attempting to test whether a given trait explains the abundance of species in an ecosystem. Trait A explains 90% of the observed variation in abundance among species. P < 0.001. This would generally be considered highly ecologically significant and important to report. 


Now think about the opposite extreme. Trait A explains 2% of the variation. P = 0.9. Pretty high confidence that you can reject the null and state that trait A is not important. Publishable? Absolutely if there are strong hypotheses relating the two. If we beforehand believe that trait A is likely to predict abundance, then it is likely more important to publish that it didn't explain abundance than if it turned out to explain a high proportion of variance. 


There are some cases where the r2's are important to examine. For example, a model result might be r2 = 0.4, P = 0.1. This happens when there isn't much statistical power. This would be one case where the r2 is a helpful parameter.


What about the above-referenced case of r2 = 0.12, which happened to have a P = 0.008? Is the absolute value of the r2 relevant here? We ran a study that compared physiological drought tolerance to the abundance of 60 species. On the one hand, 12% can be a lot. Let's say there are 8 factors that equally explain abundance. None have ever been identified. you identify one of the 8 with high statistical significance. r2 is only 0.12. That seems important. Or there are only two factors that explain abundance, you've identified one of the two, but there is a lot of measurement error or associated random variation that is inherent to the system. r2 will equal 0.12. [see Shipley's work on relativizing r2 to take account for this.] In short, 12% of the variation is 12% more than we knew before and might be about all we can ever hope to know.


On the other hand, 12% can be considered low if you expected a lot more. In our case we showed for one contrast, drought tolerance explained 12% of the variation in abundance, while in a paired contrast, it explained 0.1% (P = 0.8). If you expect that drought it important and that drought tolerance explains a low proportion of the variation, then that's a scientifically important result that is only strengthened by the low r2. 

r2's in and of themselves are irrelevant. They have to be contextualized to expectations. a high r2 that confirms expectations might be considered less important to publish than a low r2 that contradicts explanation.

In the specific case referenced above, we view an r2 of 0.12 both ways. Our statements in the abstract were 1) In this mesic grassland, physiological drought tolerance appears to increase the abundance of plants in xeric uplands, but does not in the mesic lowlands", and 2) "In all, drought appears to have a limited role in structuring the Konza plant community."

At Konza, drought tolerance explains 12% of the variation in abundance of species in uplands and is responsible for explaining 50-fold variation in abundance. that's a lot, but it happens to against a background of 5 orders of magnitude of abundance across species. At the same time, if one's expectation is that drought is very important in a drought-prone ecosystem where production is highly sensitive to interannual variation in precipitation, that same 12% (when paired with 0%) is actually not a lot. 

The Achilles' Heel of modern scientific publishing is the negative result. It would be silly to only publish papers with positive slopes rather than negative ones, but it seems straightforward to reject papers because r2's are low. Time and time again, it's been shown that not publishing negative results (low r2, low P value) biases our scientific understanding. 


Coefficients of determination are certainly not irrelevant, but they are only relevant within the context of expectations and the statistical significance of a test.

The first of a few notes on temperature sensitivity

Probability (r) curves for the energy of collisions between a substrate and an enzyme at 20 °C and 30 °C. Shown are differences in activation energy (Ea) and Q₁₀ required for reaction for a labile substrate (upper figure) and a recalcitrant substrate (lower figure).

Globally, soils contain about twice as much carbon as found in the atmosphere and three times as much found in vegetation. The fate of organic carbon stored in the terrestrial biosphere depends in large part depends on the temperature sensitivity of microbial decomposition. Currently, there is still debate over the relative sensitivity of different carbon pools to increases in temperature. Modelers have largely punted on the issue, assuming that respiration doubles with every 10°C increase in temperature (Q10 = 2).

Decomposition is complex, but proximally decomposition is an enzymatic process. As such, at least short-term responses to temperature changes should be governed by chemical laws. The degree to which they actually do is still up in the air. The Arrhenius equation describes the relationship between the rate of reaction (k), the activation energy of a reaction (Ea) and temperature (T)

k=A*e^(-Ea/RT)

where R is the gas constant and A is the frequency factor that is specific to each reaction and represents how many collisions between reactants have the correct orientation for reaction. The Arrhenius equation can be used to determine the temperature sensitivity of reactions as well as the fundamental chemical principle that the temperature sensitivity of any given reaction will be proportional to the net activation energy of the reaction.

Mathematically, from the Arrhenius equation, Q10 increases with increasing Ea. For example, at an Ea of 51 kJ mol-1, the Q10 of a reaction between 20 and 30 °C is 2, while at 81 kJ mol-1 the Q10 is 3. The reason for this is derived from molecular collision theory. In brief, at a given temperature, only a small number of the collisions between an enzyme and a substrate will be energetic enough for a reaction to occur, as described by the Maxwell-Boltzmann distribution (see figure above). As temperature increases, the number of collisions increases negligibly, but the fraction of collisions with sufficient energy increases significantly, leading to an increase in reaction rate. The ratio of the number of collisions of sufficient energy at two temperatures is the temperature sensitivity of the reaction. And based on teh Maxwell-Boltzman distributions, we can see that this ratio is much higher for higher Ea's.

The Arrhenius equation is a foundational principle for understanding temperature sensitivity. Later, I'll show recent work just now being published in Nature Geoscience that tests whether the temperature sensitivity of microbial decomposition to short-term increases in temperature follows the Arrhenius equation well or whether other factors might be more important. If it does, predicting temperature responses and the fate of terrestrial carbon pools just got a lot easier.

Grassland Climate Change 3.0

Critical climate periods for ANPP, flowering of three grasses, weight gain of calves, yearlings, and adults, as well as calving rates the following year for Konza. Gray bars indicate a negative effect of precipitation on the process, black positive.

If you look at the development of climate change research in grasslands, there have been two main stages. Climate Change 1.0 was trying to understand the importance of changes in growing season precipitation on ecosystem dynamics. Wet years are compared to dry years. Experiments that test climate change in 1.0 modify total precipitation.

We're still largely using Climate Change 1.0. Climate Change 2.0 examines effective precipitation during the growing season. Effective precipitation calculations largely take into account event size and distribution. Light rain events might lower effective precipitation as they are intercepted by canopies. Heavy rain events might lower effective due to greater flow through or runoff. Too light or too heavy and plants might not ever get a chance to use all the rain, hence lower effective precipitation. Some early-adopters are investigating Climate Change 2.0, but it's not mainstream yet. Certainly the projections and climate change models are not built to forecast in a manner that promotes 2.0.

One of my goals has been to push Climate Change 3.0. With 3.0, it's not just how much rain falls during the growing season, nor how much effective rain falls during the growing season. but when the rain falls. If you look at the critical climate periods for aboveground net primary productivity (ANPP), they largely show that 1.0 works--the more precipitation in the growing season, the more ANPP. For flowering of the major grasses, it's largely 1.0. Growing season precipitation largely determines flowering, with some differences among the species in their sensitivity to rainfall.

For Konza bison, there is just no relationship between growing season precipitation and weight gain for any sex or age class. But factor in the timing of precipitation, and you can explain up to 80% of the variation among years in weight gain. Why? It's because mid-season precipitation suppresses weight gain, while late-season precipitation promotes it. The climate-nutrition-performance cascade hits bison hard. Most likely, the same thing applies to cattle, although it hasn't been shown.

Climate Change 3.0 is nothing new conceptually. But in practice, 3.0 is. Training our models to predict when precipitation falls can be more important than how much falls for humid grasslands. Training ecologists to start to examine this will be probably be harder.

Comparing phenology curves

Packera plattensis, which was found first flowering on April 13 in 2010. 

The timing of flowering is a critical component of the ecology of plants. Flowering during environmentally stressful times or when other plants that utilize the same pollinators can lower a plant’s fecundity if not lead to its extirpation from an ecosystem. As such, the timing of flowering should be under strong selection pressure and be an important component of community assembly. 


Over the past two years, Gene Towne and I (mostly Gene) collected first flowering date (FFD) data on 430 Konza herbaceous species. The last species found to start flowering (a gentian) was found in early October, 189 days after the first herbaceous species--Holosteum umbellatum--was found in late March.


The patterns at Konza are interesting. More on those later. The unexpected find was comparing the patterns with two other predominantly grassland flora. The first was from Chinnor, Oxfordshire. The second, Fargo, ND. 

The y-axis is the fraction of each flora flowering on a given day. x-axis is day of year.


Two things pop out. Relative to Konza, the Chinnor flora has an early tail of species, but not a late tail. Is this because species phenology are all shifted earlier, so that the same species would flower ~50 d earlier there? Or is it just a suite of species that flower earlier are found at Chinnor, but  late-flowering species are not?


And relative to Konza, the Fargo phenology is much more compressed. Again, though, why? Does Fargo not have early- and late- flowering species, or are the phenology of individual species compressed.


Turns out we can begin to answer that and the mechanisms that underly the differences between the pairs differ.


Here are the relationships between FFD between the pairs of sites. Dotted line is 1:1.

Species common to Chinnor and Konza flower on roughly the same day. Hence, one would suspect that the differences in curves between the two sites are due to novel types of species at each site. Yet for Konza and Fargo, early flowering species flower later at Fargo, and late-flowering species flower earlier. Phenology gets compressed for individual species.

Theoretically, I'm still getting up to speed, but comparisons between flora just haven't been done like this. Mid-domain theories are prevalent to test, but each site would support the idea of a mid-domain peak. What's more interesting is why sites differ. Right now, hypotheses about functional novelty/plugging of holes in niche space vs. functional stretching/compression are pretty interesting ones to test here. Flowering is interesting to think about, but the really interesting comparisons (at least for me) will come with comparing functional traits associated with resources, not reproduction.

Bison growth curves

Weight of female (lower) and male (upper) bison at Konza Prairie and Ordway with age.

The performance of bison—how much weight they gain, how many calves are produced—is the ultimate expression of the functioning of North American grasslands. If we can compare the performance of bison in different grasslands, we have a window in the functioning of the grassland. Interannual patterns of weight gain show responses to climate variation. Average weights of animals give general indices to the provision of the quantity and quality of grass produced. Yet, no one has ever compared the performance of bison across grasslands in North America. 

We're getting pretty close to doing that. For any one site, we can fit a growth curve to the weights of animals as they age. There are a number of growth curves that are used for these purposes, but a good one is a generalized Michaelis-Menten equation:

where W0 is the birth weight, Wf is the asymptotic weight, K is the age at which animals are half their asymptotic weight, c is a constant describing the shape of the curve, and t is time in years.

If you fit the weights of bison with age with this equation, for each herd you can extract essentially how heavy cows and bulls get, as well as a rate of maturity...or half-maturity as K would represent.

Right now, we have data on weight gain for about six bison herds. There are about 10 herds in the US that have weight data from roundups. 

So far, we see a few basic things about bison. On average, males level off at about 75% greater weights than females (855 vs. 484 kg). It also takes them about 1.5 y longer to reach half their maximal weight. 

We also see that some bison herds are heavier than others. For example, mature bison in Ordway Prairie in South Dakota are 50-100 kg heavier than mature bison from Konza. That's a lot of bison. Is it a fluke? Unlikely. Over 90% of Ordway adult cows produce calves. At Konza, it's only about 60%. 

There must be a big difference in the grass between Konza and Ordway.  Because their bison growth curves are quite different.

Konza flowering phenology and functional groups.

In general, we have little understanding of how communities are assembled and the types of interactions that long-term generate evolutionary pressures, extinctions, and radiations. I'm pretty sure that whole-flora analyses are going to be keys to helping us understand these complex systems and there are precious few datasets on the scale necessary to do this.


With that in mind, here's the latest Konza phenology data by functional group. This is through Sept 1. The x-axis is day of year of first flowering for a species. Based on n = 408 species, which represents about 80% of the herbaceous grassland flora for Konza. We'll probably get another 10-20 species flowering before the year is up.

The y-axis is probability of flowering per day over the year for species of each functional group based on a "smooth" fit of the distribution data. Probabilities are standardized across functional groups. I broke out the Cyperaceae because it was the Carex that flowered early, not any C3 grasses. The C3 grasses that flower late in the year are generally woodland grasses.

This is terribly fascinating, though I'm not sure what the story is yet. For example, why are there C3 forbs that flower in August, but not any C3 grasses? And why are there C4 grasses that begin flowering in March, but not any C4 forbs? 

There certainly is an long-term competitive interactions that sort communities and drive selection. It's almost likely a rock-paper-scissors story. If rock (C4 grasses) then no scissors (C3 grasses), but if paper (grazers) then there are less rocks, so can have knife (C3 forbs).  

The C4 forbs are probably the most interesting story. If high temperatures favor C4 over C3, then why are there so many C3 forbs that are active during the hottest months rather than C4 forbs. Konza's C4 forbs are mostly Chamaesyche (Euphorbiaceae) and Amaranthus. Often they are prostrate forbs and/or weedy species keying in on disturbed areas. The C3 forbs that flower during this time are species like Salvia. Are there C4 forbs that fall into the same niches as these C3 forbs. Is there evolutionary constraint here that allows all the mid- to late-summer C3 forbs to persist? 

As we generate more large-scale trait datasets, more of these patterns should come clear. 

Mycorrhizal fungi and grassland community structure

Relationship between mycorrhizal infection rates and the log-transformed response of species abundance to grazing.

The structuring of plant communities is complex. There are a myriad of proximal and distal factors that can influence the abundance of species. The role of mycorrhizal fungi in structuring grassland communities has always been opaque. In temperate grasslands, many of the species are dependent on arbuscular mycorrhizal fungi, yet many non-mycorrhizal species are found throughout the grasslands. Whether these non-mycorrhizal species tap unique pools or even are facilitated by the mycorrhizal species is really unknown.

Over a decade ago, Wilson and Hartnett (1998) quantified the dependence of ~100 grassland species on mycorrhizal fungi. There had never been a screening study like it. Nor has there been one since. Their work largely compared different functional groups, with the conclusion that C4 grasses are the most dependent on mycorrhizal fungi and legumes the least. The work implied that success at Konza would be dependent on the ability to utilize mycorrhizal fungi, but this was never quantified.

Recently, we've compared the screening data with actual abundances from Konza. It turns out that there is no relationship between abundance and mycorrhizal responsiveness or infection rates. As such, mycorrhizal symbioses are likely not necessary for success.

That said, mycorrhizal symbioses do determine which species perform better under certain conditions. For example, almost 25% of the variation in the response of species abundance to the presence of grazers (bison) was explained by the mycorrhizal infection rate. Grazing promoted non-mycorrhizal species. Similarly, suppression of fire promotes non-mycorrhizal species (data not shown).

In both cases, fire suppression and grazing increase the availability of nutrients relative to other resources. How to think of the role of mycorrhizal under different burning or grazing regimes is still not clear. It's easy to say that fire suppression or grazing increases nutrient availability, which decreases the need for mycorrhizal fungi. But why? Is it because they are too much of a carbon drain? Many of the high-fire, low-grazing species just do not grow at all in the absence of mycorrhizal fungi, so it is unlikely to be associated with competition for nutrients. And why would mycorrhizal responses/infection predict just the responses to grazing/fire, but not abundance overall. In contrast, we see traits like leaf tissue density--which I think of as being associated with low nutrient availability--prediction abundance across Konza, but not the responses to fire and grazing. 

How to proceed on the issue is not easy, but it's a curious pattern to line up with a number of others in understanding how grassland communities are structured.

Wilson, G. W. T. and D. C. Hartnett. 1998. Interspecific variation in plant responses to mycorrhizal colonization in tallgrass prairie. American Journal of Botany 85:1732-1738.