Janneke Hille Ris Lambers, James S. Clark, and Michael Lavine. 2005. Implications of seed banking for recruitment of southern Appalachian woody species. Ecology 86:85–95.

Appendix A. Estimating seed decay and incorporation into the soil seed bank using a Bayesian statistical approach.

We estimated seed mortality rates (d_j’s), seed incorporation into the soil (v_j’s) from the density of viable seeds in the soil and seed rain inputs into seed traps while accommodating measurement uncertainty in seed rain inputs into sites of soil cores. Because the number of parameters in our model renders it analytically intractable, we used Markov Chain Monte Carlo simulation (Gibbs sampling) to estimate parameters. This Bayesian statistical framework allowed us to incorporate prior information (in the form of probability distributions) into our statistical models to generate parameter estimates (the mean of the posterior distribution).

We developed these prior distributions using data from the National Tree Seed Testing Laboratory (Fig. A1) and from published studies (Fig. A2; Clark and Boyce 1964, Marquis 1975, Wendel 1977, Granstrom and Fries 1985, Granstrom 1987, Houle 1994, Haywood 1994). Sensitivity to priors for each parameter (v_j, d_j) was assessed by comparing parameter estimates with informative priors to those based on non-informative priors; posteriors calculated with non-informative priors (i.e., diffuse or flat distributions) are solely influenced by the data we collected (columns 3 and 6 in Table 3). Both prior information and the data we collected influence posteriors calculated with informative priors (v_j - columns 4 and 5 in Table 3; d_j - column 7 and 8 in Table 3).

Quantifying informed priors for seed viability and seed mortality

Informed priors for our parameter v_j came from maximum likelihood analysis of seed viability estimates from the National Tree Seed Laboratory (representing our prior knowledge of v_j, seed incorporation into the soil). Although seed viability can exceed the proportion of seeds incorporated into the soil after one growing season, which includes losses due to both seed viability and subsequent mortality, we used previous information on seed viability as priors for v_j because we assumed low seed viability would represent the predominant reason for low transition probabilities from dispersed seed to persistent seed banks.

For seed viability, "observations" consisted of the number of viable seeds S_i from a total number of N_i tested in the i^th seed lot from the National Tree Seed Laboratory. We assumed that S_i follows a binomial distribution, but that this probability can vary from seed lot to seed lot as a Beta distribution with parameters
_vand _v. For t seed lots, the marginal beta-binomial sampling distribution has the following likelihood:

(A.1)

Beta distributions from the _vand _v. parameters we estimated using Eq. A.1 correspond well to the observed frequency of seed viability estimates for Acer rubrum, Betula spp., and Liriodendron tulipifera (Fig. A1). We chose to give our informed prior distribution for v_j the same weight as the maximum likelihood seed viability estimates from the National Tree Seed Laboratory. We compared results of model fitting with this informed prior to results with an uniformed (or diffuse) prior to determine how this choice of prior weight affected parameter estimates (as in Clark and Lavine 2001).

The data we used to estimate priors for seed mortality parameters (d_j – estimated for Betula spp. and Liriodendron tulipifera) come from published seed burial studies (Clark and Boyce 1964, Marquis 1975, Wendel 1977, Granstrom and Fries 1985, Granstrom 1987, Houle 1994, Haywood 1994). In these studies, mesh bags of seeds are buried and then removed from the soil after a specified amount of time and seeds tested for viability. The data consist of the number of viable seeds (S_ik) remaining from the total number of seeds buried (N_i) in bag i at time step k. We assumed that the proportion of seeds  remaining viable in bag i at each time step k (in other words, ) depends on two parameters, initial seed viability v and annual seed mortality d:

(A.2)

As in Eq. 1, A.2 assumes a constant rate of seed loss from the seed bank (i.e., exponential), except that seed inputs occur only once (at time of seed bag burial). The binomial is the basis for the likelihood:

(A.3)

The expected loss of seed viability over time (as described by parameters v and d) describes the data observed reasonably well – with the exception of data from one study on Betula spp. (Fig. A2).

We assumed that d is Beta distributed. We sampled from the original data set (with replacement) 1000 times and used Eqs. A.2 and A.3 to fit a mortality rate (d_b) at each bootstrap iteration. We then used the mean () and variance () of these 1000 bootstrap estimates to calculate the two parameters of a Beta distribution (_d and _d) describing d, a technique called moment matching:

	(A.4)
and
	(A.5)

We chose to give our informed prior for d_j the same weight as the Beta distribution on d we calculated using Eqs. A.4 and A.5. We compared results of model fitting with this informed prior to results with an uniformed (or diffuse) prior to determine how our choice of prior affected parameter estimates (as in Clark and Lavine 2001).

The MCMC model

We modeled the observed density of seeds in the soil at location i, in plot j and at time k (b_ijk) as a Poisson process, with mean density ():

(A.6)

where (as described in Eq. 1):

(A.7)

Seed densities in seed traps (- at location i, plot j, year k) are Poisson distributed, with mean :

(A.8)

Seed densities falling onto sites of soil coring (s_jk) are proportional to _jk:

(A.9)

where c is the factor by which soil cores are smaller in area than seed traps (0.0356 m²).

Priors on v_j and d_j are Beta distributed:

	(A.10)
and
	(A.11)

We set parameters _v, _v, _d, and _d to 1 when estimating posteriors with uninformative priors, and to maximum likelihood estimates from Eqs. A.1 to A.5 (Table A1) when estimating posteriors with informative priors.

We gave _jk a uniform prior:

(A.12)

with u_j being twice the maximum seed count observed in a single seed trap within a plot over all nine years. The joint posterior is:

(A.13)

Substituting the appropriate probability distributions into expression A.13 leads to an expression that is proportional to the joint posterior distribution of all model parameters. We used Markov Chain Monte Carlo methods to find the posterior distribution by sampling iteratively from this full joint posterior distribution. We implemented MCMC sampling using the statistical package called BUGS (Bayesian inference Using Gibbs Sampling [http://www.mrc-bsu.ca .ac.uk/bugs]). The MCMC algorithm sampled from the following conditional posteriors:

	(A.14)
	(A.15)
	(A.16)

We assessed convergence visually as well as with several diagnostics (calculated with Bayesian Output Analysis program [http://www.public-health.uiowa.edu/boa/Home.html]) – specifically; Heidelberger and Welch’s, Raftery and Lewis, and Geweke’s. For each taxon, we discarded the "burn-in" iterations and thinned chains as suggested by Raftery and Lewis' diagnostic. After thinning and burn-in, all posteriors were based on at least 2000 samples. Diagnostics indicated that there was no evidence against convergence for any parameter (Geweke's diagnostic), and Heidelberger and Welch's half-width test was passed for all parameters. We also made sure posteriors of _jk were not constrained by the upper bound (u_i) of the uniform prior on _jk.

Results

For Acer rubrum and Betula spp., observed seed bank densities agree reasonably well with those predicted from our model (Eq. A.7) and parameter estimates for models with informed as well as with uninformed priors (Fig. A3). With the exception of Liriodendron tulipifera, estimates of seed incorporation into the soil (v_j) from analyses with both uninformed and informed priors are lower than prior seed viability estimates, and had little effect on posterior densities of seed incorporation (Fig. A4). Seed mortality estimates from MCMC simulations using non-informative priors are variable from plot to plot, and different from priors for both species (Fig. A5). Posteriors of d_j are affected by priors for Liriodendron tulipifera, but not for Betula spp. Priors had greatest effects on posterior densities for plots having limited data, and thus, broad posteriors when priors are uninformed (plots 1, 3, and 4 for Liriodendron tulipifera – Fig. A5). 

Posterior estimates of v_j and d_j (using uninformed priors) are correlated for all plots (Fig. A6), which explains why informed priors on one parameter can affect posteriors of the other parameter (Table 3, column 5 vs. 4 and column 7 vs. 8). The strong dependence of these parameters on each suggests that learning about v_j can also give you information on d_j, and vice versa. For example, if we acquired additional information suggesting annual seed mortality (d_j) of Liriodendron tulipifera in plot 4 is between 0.4 and 0.6, our posteriors indicate that annual seed mortality of this species is approximately between 0.2 and 0.4 (Fig. A6).

LITERATURE CITED

Clark, F. B., and S. G. Boyce. 1964. Yellow-poplar seed remains viable in the forest litter. Journal of Forestry 62:564–567.

Clark, J. S., and M. Lavine. 2001. Bayesian statistics: estimating plant demographic parameters. Pages 327–346 in S. M Scheiner and J. Gurevitch, editors. Design and analysis of ecological experiments, Oxford University Press, New York, New York, USA.

Granstrom A. 1987. Seed viability of fourteen species during five years of storage in a forest soil. Journal of Ecology 75:321–331.

Granstrom, A., and C. Fries. 1985. Depletion of viable seeds of Betula pubescens and Betula verrocosa sown onto some north Swedish forest soils. Canadian Journal of Forest Research 15:1176–1180.

Haywood, J. D. 1994. Seed viability of selected tree, shrub, and vine species stored in the field. New Forests 8:143–154.

Houle, G. 1994. Spatiotemporal patterns in the components of regeneration of 4 sympatric tree species – Acer rubrum, Acer saccharum, Betula alleghaniensis and Fagus grandifolia. Journal of Ecology 82:39–53.

Marquis, D. A. 1975. Seed storage and germination under northern hardwood forests. Canadian Journal of Forest Research 5:478–484.

Wendel, G. W. 1977. Longevity of black cherry, wild grape, and sassafras seed in the forest floor. USDA Forest Service Research Paper NE 375:1–6.

Table A1.  Prior distributions (estimates and parameters describing distributions) of seed viability (v) and annual mortality rates (d).

Species	Priors
	viability (v, v)	mortality (k, k)
Acer rubrum	0.815 (3.86,0.873)	NA
Betula spp.	0.720 (3.146, 1.22)	0.821 (71.4,14.5)
Liriodendron tulipifera	0.309 (0.613,1.371)	0.340 (4.76,9.23)

FIG. A1. Histogram of seed viability estimates used as priors for vj, seed incorporation. Beta distributions (thick lines) fitted to seed viability estimates are superimposed for Acer rubrum (A), Betula spp. (B), and Liriodendron tulipifera (C).

FIG. A2. Proportion of viable seeds remaining over time in seed burial studies, for Betula spp. (A), and Liriodendron tulipifera (B). These data were used to generate informed priors for dj, Symbols represent data from different studies. For Betula spp. (A) open triangles are from Granstrom 1987, circles with crosses from Granstrom and Fries 1985, asterix’s from Marquis 1975; and filled squares from Houle 1992. Liriodendron tulipifera (B) was taken from one study (Clark and Boyce 1965). Thick lines are the expected mortality of seed viability over time from data fit to Eq. A.2. Dashed lines are 95% confidence intervals (from bootstrapping).

FIG. A3. Predicted seed-bank densities (from Eq. 1, fitted parameters vj and dj, and seed rain densities) vs. observed seed bank densities; for Acer rubrum (A,B), Betula spp. (C,D), and Liriodendron tulipifera (E,F). Predictions are based on models with uninformed priors (A,C,E) and informed priors (B,D,F) on vj and dj. Each point represents a plot average in one of five years.

FIG. A4. Density distributions of plot-specific seed incorporation rates (vj) for Acer rubrum, Betula spp., and Liriodendron tulipifera from models with uninformed priors (A,B,C) and models with informed priors (D,E,F). Prior distributions (drawn in thick lines) based on seed viability estimates are drawn alongside estimates based on uninformed priors (A,B,C).

FIG. A5. Distributions of plot-specific seed mortality rates (dj) for Betula spp. (A,B) and Liriodendron tulipifera (C,D). Prior distributions (drawn in thick lines) based on seed mortality estimated from buried bag studies are drawn alongside plot-specific seed mortality estimates based on uniformed priors (A,B).

FIG. A6. Correlations between estimates of plot-specific seed incorporation rates and seed mortality rates for Acer rubrum (A,D,G,J), Betula spp. (B,E,H,K,M) and Liriodendron tulipifera (C,F,I,L) for the 2000+ posterior samples from MCMC models – using uninformative priors. Light shaded areas of plots indicate areas within 95% confidence intervals of prior distributions, darker shaded areas of plots within 95% confidence intervals of both priors (for Betula spp. and Liriodendron tulipifera).