box 1: seedling growth and light availabilityecology.cs.duke.edu/ecology/stats1.doc · web...

Last update: 5/22/2023

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Seedling growth and light availabilityJames S. Clark, Duke University

Seedling growth responses to light availability incorporate individual differences in growth response and stochasticity in light availability (Clark et al. 2002). The model is hierarchical Bayes. Here we describe the model and compare results with those obtained with a classical analysis. The classical analysis finds differences between species that are artifacts of unrealistic assumptions.

PROCESS MODEL

The Monod function is a standard model for growth response to resources,

, 1

for xj resource level at resource level j, having asymptote G = 0 + 1, minimum resource for positive growth , and “half saturation” , the resource level at which growth is half way between 0 and 1. An alternative parameterization is

. 2

Both 1 and 2 are useful. We fit parameters using eqn 1, because we can write more efficient algorithms that take advantage of the linear parameters 0 and 1. Parameters of eqn 2 are more ‘biological’, so we interpret the reparameterized eqn 2.

DATA

Growth data derive from annual measurements of seedling heights on 1 m2 plots from the Duke Forest, Orange County, North Carolina. We include several data sets, including one for Acer rubrum seedlings growing beneath a closed canopy and larger data sets for Acer rubrum and Liriodendron tulipifera seedlings that derive from a range of canopy conditions. The growth of seedling i on plot j is defined as the increase in height averaged over two years (three censuses). This is yij. There are nj seedlings per plot, m total plots, and total seedlings.

To estimate understory light we obtained hemispherical canopy photographs during uniform sky conditions (early morning/late afternoon) in mid-summer at a height of 1.15 m above each seedling plot. Images were obtained on 400-speed color slide film using a Nikon FM2 camera with a Sigma 8mm 180-degree fish-eye lens and leveling tripod (Delta-T Devices Ltd.). Scanned images were analyzed using HemiView Canopy Analysis Software (Version 2.1, Delta-T Devices Ltd.). Photo analysis involves an user-defined threshold intensity for each photo that determines whether pixels are classified as


open (sky) or obscured (canopy). The Global Site Factor (GSF) represents the proportion of full sunlight penetrating the forest canopy. The GSF combines direct radiation, by calculating the annual solar track, and diffuse radiation, based on a uniform overcast sky model (Rich 1989). It does not account for backscatter within the canopy.

CLASSICAL APPROACH

To motivate our approach, we begin by describing classical methods (Fig. 1). Let yij be the ith measurement of growth at resource level xj . A simple model says

3The first term is the deterministic model (eqn 1, 2). The only uncertainty we associate with it is that conferred by our degree of confidence in the estimates of the parameters b = [0 1]T and (“estimation error in Figure 1). All variability is contained in the second

term, often assumed to follow a normal distribution, . This term contains

an additional parameter that must be estimated, and it is the source of variability in the model. In this instance, we might think of this as ‘process error’, ‘observation error’, or both (Fig. 1). To estimate parameters we write a likelihood for the data under the assumption that the mean response is j with error, described by a fitted parameter ,

. The likelihood,

, 4

is basis for maximum likelihood (ML) or Bayesian estimation, the latter requiring a prior for parameters.

Fig. 1. Structure of classical maximum likelihood model. The upper box indicates ‘data’, i.e., entities that are observed. Sources of stochasticity include ‘process/obs error’ and parameter uncertainty (estimation error).

The classical method results in low parameter uncertainties (parameter error distributions in Fig. 2a), in large part due to the fact that standard errors are proportional to 1/M. Error propagation to the predicted response (dashed lines in Fig. 2a) can give a misleading impression, in that is estimated with great confidence, a result of assumptions that light is known precisely and that all individuals have this identical response to light—in other words, stochasticity is ‘error’ or ‘noise’. Tight confidence intervals on belie the true scatter–far more than 95% of the data fall outside the 95% propagated parameter error. The distribution of residual variation accounts for most of the scatter (right side of Figure 2a). Regardless of cause, the analysis assigns it to , because we have provided no other place for it. Of course, we could propagate the stochasticity associated with (and

Seedling growth: Saturating function Data:

Parameters:

Light data

G, x0, , 2 Estimation error

Process/obs error


uncertainty in the estimate 2) but the confidence interval for data, with negative values at the low light levels where light limitation matters (Fig. 2b), would be of questionable use. Moreover, the assumption that scatter is ‘error’ (the impetus for this classical model) justifies ignoring it and focus on the CI for .

Fig. 2. a) Maximum likelihood (Fig. 1) fit for the Liriodendron data set, showing parameter error distributions and the distribution of residuals (with variance determined by the MLE for 2). Dashed lines show 95% confidence intervals for parameters and the model , which is propagated from parameter error distributions using a nonparametric bootstrap. Part b shows the additional effect of error represented by the term (outer confidence intervals). For the kth bootstrap resample in part b, we estimate a growth rate as a random draw from . The marginal distribution is Students’ t due to the sampling distribution of 2.

HIERARCHICAL BAYES

Hierarchical models were developed for responses that are too variable for the assumptions of classical models (Carlin and Louis 2000). In the classical model, we ignore when making inference on , because adds noise, but not insight. Residual variation has nothing to do with light response (although the estimate of 2 does depend on the estimate of ). The ‘error’ enters this model as an addition to the deterministic response , and it applies identically to all individuals. In other words, the classical model assumes that individual differences in y are independent of light availability and independent of the light response .

If individuals vary in their response to light, say ij, then the response itself is stochastic (Fig. 3). Now has a mean, and its variance is not asymptotically zero. To determine whether one species has a different response from another we construct credibility intervals that combine variability in with the uncertainty in parameter estimates (see below).


Hierarchical structures allow for context, including variability among individuals within populations. Our implementation is Bayesian: parameters have prior distributions that might be informed by assumptions (the asymptote G should be finite, the intercept should be negative (0 < 0 or, equivalently, x0 > 0), the half saturation should be positive, and so on). We do not impose strong restrictions here (our priors provide flexibility). However, we note that restrictive priors need be no more subjective than the model selection itself–if we did not believe that growth requires at least some resource or that growth eventually saturates, we would not have chosen model 1-2. We demonstrate a single model structure that admits reasonable assumptions about stochasticity, including variability and uncertainty in resources and variable responses among individuals (Fig. 3).

Fig. 3. The hierarchical model of seedling growth responses to variable resources. In contrast with the classical model (Fig. 1), we now admit the stochasticity most ecologists believe actually operates in the real world (sources are in brown text). Light is uncertain and variable; it is now more like a ‘parameter’, than data, and thus, must be estimated. Parameters are not fixed constants, but rather, vary among individuals. The ‘individual effects’ (middle row) now have context as variables within a heterogeneous population. Thus, estimation involves a lower stage with ‘hyperparameters’.

Process, data, and parameter model

We treat the resource-growth relationship (eqn 1) as a nonlinear regression, but take advantage of linear parameters. Define the transformed variable for light at plot j to be

. A design matrix consists of a column of ones and a column with each zj

repeated nj times. For the first plot, we have . The full design matrix is

Seedling growth: Saturating functionData:

Light dataProcess/obs error

‘actual’ lightParameters:

Hyperparameters:

Population effects:distribution for individual responses

Individual effects: individual responses to light

Estimation error

Heterogeneity in response Variability/estimation error


Further define parameter vector bT = [0 1]. Together with the beta

distribution for light observations, we have the likelihood

5

Our method accommodates individual differences in light response in the form of a parameter set that describes the response for the ith individual on the jth plot, ij. Individual parameters are linked to the full population by distributions, defined by hyperparameters (Fig. 3). These are ‘random effects’ at the individual level. The full model,

(likelihood)

(priors)

(hyperpriors) 6has a vector of growth parameters for each individual bij = [ij0, ij1]T

, which is drawn from the distribution . The parameters have hyperprior density and the Wishart W( ) parameter covariance matrix, a multivariate generalization of the gamma distribution. The hyperprior for regression parameters is made noninformative by applying small prior precision D–1. The prior covariance matrix for hyperparameters D = Diag(800, 1000) is relatively weak (Appendix B). The Wishart hyperparameters consist of weight w ‘degrees of freedom’ and matrix R. A noninformative prior has small w (but not less than the dimension of R, i.e., 2) and R << Vb. R is roughly the prior mean of Vb. We use Carlin and Louis’ (2000) rule of thumb R = Diag((r1/8)2(r2/8)2), where r1 = 20 and r2 = 80 are plausible parameter ranges for bij, and w = M/20. Results here use prior parameter values for h = [-30 100]T.

The hierarchical model estimates that individual differences are large. We have not changed our view that there is a relationship like eqn 2 that applies to all individuals in the population. By contrast, we also do not assume that they are completely independent. Each has its own parameter estimates (‘individual’ G and x0 posteriors in Fig. 4). The distribution for the population can be estimated by averaging over individual posteriors. The hierarchical model provides a summary of the population in the form of estimated hyperparameters. The structure of variability and uncertainty here is more plausible (and useful) than in the previous cases, because it is classified in the way ecologists believe that variability operates (Fig. 3). True light levels are unknown and variable. The parameters for any given individual can be viewed as a combination of the contributions of the data for that individual (light and growth), which are weighted by the residual variance –2, and that for the population as a whole. The latter is the hyperparameter vector b0, which is weighted by the inverse of the parameter covariance Vb. There is thus a ‘balance’ between contributions from the individual and the population, maintained by the degree to which each contributes to the overall fit. The population provides the ‘glue’ that precludes the overfitting of independent parameters for every individual. The 95%


credible interval for y (dashed line in Fig. 4) contains ~ 95% of the data points, because the residual variation is small (right side of Fig. 4). This credible interval for the population prediction is consistent with our view that individuals do not have negative growth rates (e.g., Fig 2b)(although they can lose height for reasons that would typically be unrelated to light).

Fig. 4. The hierachical analysis (Fig. 3), which assumes that each individual may respond differently to light (“individual” parameters), with individual distributions summarized by hyperparameters. Bayesian posteriors are shown for all parameters except , which is represented instead by the distibution for together with the predictive distribution for the model. The two hierarchical parameters are shown as individual posteriors, marginalized over all individuals, and hyperparameters.

Comparing trophic responses

A comparison of two species shows that a more realistic structure blurs species differences at all light levels. A classical fit predicts large differences in between Liriodendron and Acer rubrum (Fig. 5b). Sample sizes are large and, thus, confidence intervals are narrow. So narrow that each estimate assigns essentially zero probability to the other response. The models for the two species are statistically different.

The uncertainties and individual variability result in broadly overlapping confidence intervals (Fig 5a). Whereas the classical model predicts distinct differences, a fuller accounting of stochasticity predicts trophic near equivalence. A full accounting of the uncertainty in the classical fit (i.e., propagating the effect of 2) would also produce


broad overlap (e.g., Fig. 2b), provided we were willing to accept the scatter as something other than ‘error’. If we did accept this, we could not justify the classical model.

Figure 5. A comparison of hierarchical (a) and classical (b) models for two species. Data are for Acer rubrum (data for Liriodendron are shown in previous figures). Ninety-five percent confidence intervals are shown as they are typically interpreted. For the classical model, they are for the species-specific response . For the hierarchical Bayesian model, they are credible intervals on the population response.

Additional background

This analysis is abstracted from Clark et al. (2002). Gelfand and Smith (1990) introduce the Gibbs sampler methodology. Several references provide further background on this model. Linear regression with known covariance is discussed by Carlin and Louis (2000). Our hierarchical structure is similar to Gelfand et al. (1990), who used a linear model to examine multiple observations per individual. Our ‘linearization’ is recommended when efficiency is an issue (as it is here). Gamerman (1997) mentions application in a Bayesian context. Gelman et al. (1995) and Carlin and Louis (2000) discuss the Wishart as conjugate for the regression covariance matrix.

APPENDIX A

Parameters for beta distributions of light availability on a given plot were obtained by moment matching. The mean light availability for plot j is equated to the mean of a beta


density for parameters aj and bj. We likewise equate variances and obtain the

beta parameters and . Beta parameters define a

distribution of light availability for each plot.

APPENDIX B

The joint posterior distribution was determined by Gibbs sampling (Gelfand and Smith 1990). A Gibbs sampler is a Markov chain Monte Carlo technique that executes a random walk with a limiting distribution equal to the parameter distribution of interest. Because random samples cannot be drawn directly from this complex distribution, we factor the posterior into conditional distributions given here. We used 10,000 iterations starting from different initial values. A ‘burn-in’ of 500 steps was discarded. Ninety-five percent Bayesian credible intervals are posterior quantiles. Propagation to the response was determined using quantiles of drawn from the 9500 MCMC samples for all parameters.

Regression parametersThe conditional posterior for regression parameters contains contributions from the likelihood and prior. The conditional posterior for the variance was sampled from the inverse gamma distribution

Regression parameters come from , where ,

, , and . Hyperparameters are drawn

from , where and . The

parameter covariance matrix is drawn from the Wishart conditional posterior

(e.g. Gelman et al. 1995). Although regression parameters are given in matrix notation (for compactness), matrix inversions are too slow. All parameters are solved without resort to matrix inversion or (for the hierarchical model) loops over individuals.


Half saturation constantParameter depends on the likelihood and on its prior Beta density. The conditional

posterior is

We use a Metropolis-Hastings step with a beta proposal density centered on the current value of .

Latent (“true”) light availabilitiesLight availability on plot j enters the likelihood for the nj seedlings on plot j, and it has a density describing measurement uncertainty. It is conditionally independent of other plots,

Sampling was executed with a Metropolis-Hastings step based on a Beta proposal density.

box 1: seedling growth and light availabilityecology.cs.duke.edu/ecology/stats1.doc · web...

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