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Bayesian characterization of uncertainty in species interaction strengths

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Considerable effort has been devoted to the empirical estimation of species interaction strengths. This effort has focused primarily on statistical significance testing and on obtaining point estimates of parameters that contribute to interaction strength magnitude, leaving characterizations of estimation uncertainty and distinctions between the deterministic and stochastic contributions to variation largely unconsidered. Here we consider a means of quantifying interaction strength uncertainty by formulating an observational method for estimating per capita attack rates as a Bayesian statistical model. This formulation permits the explicit incorporation of multiple sources of uncertainty. In doing so we highlight the informative nature of several so-called non- informative prior choices in modeling the sparse data typical of predator feeding surveys and provide evidence for the superior performance of a new neutral prior choice. A case study application shows that while Bayesian point estimates may be made to correspond with those obtained by frequentist approaches, estimation uncertainty as described by the 95% intervals is more biologically realistic using the Bayesian method in that the lower bounds of the Bayesian posterior intervals for the attack rates do not include zero when the occurrence of a given predator-prey interaction is in fact observed. This contrasts with bootstrap confidence intervals that often do contain zero in such cases. The Bayesian approach provides a straightforward, probabilistic characterization of interaction strength uncertainty. In doing so it provides a framework for considering both the deterministic and stochastic drivers of species interactions and their impact on food web dynamics.
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Bayesian characterization of uncertainty in species interaction
strengths
Considerable effort has been devoted to the empirical estimation of species interaction
strengths. This effort has focused primarily on statistical significance testing and on
obtaining point estimates of parameters that contribute to interaction strength magnitude,
leaving characterizations of estimation uncertainty and distinctions between the
deterministic and stochastic contributions to variation largely unconsidered. Here we
consider a means of quantifying interaction strength uncertainty by formulating an
observational method for estimating per capita attack rates as a Bayesian statistical
model. This formulation permits the explicit incorporation of multiple sources of
uncertainty. In doing so we highlight the informative nature of several so-called non-
informative prior choices in modeling the sparse data typical of predator feeding surveys
and provide evidence for the superior performance of a new neutral prior choice. A case
study application shows that while Bayesian point estimates may be made to correspond
with those obtained by frequentist approaches, estimation uncertainty as described by the
95% intervals is more biologically realistic using the Bayesian method in that the lower
bounds of the Bayesian posterior intervals for the attack rates do not include zero when
the occurrence of a given predator-prey interaction is in fact observed. This contrasts with
bootstrap confidence intervals that often do contain zero in such cases. The Bayesian
approach provides a straightforward, probabilistic characterization of interaction strength
uncertainty. In doing so it provides a framework for considering both the deterministic and
stochastic drivers of species interactions and their impact on food web dynamics.
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Bayesian characterization of uncertainty in species1
interaction strengths2
(Running head: Bayesian Interaction Strengths)3
Christopher Wolf1, Mark Novak2, and Alix I. Gitelman1
4
1Department of Statistics, Oregon State University,5
Corvallis, Oregon 97331, United States6
2Department of Integrative Biology, Oregon State University,7
Corvallis, Oregon 97331, United States8
September 18, 20159
wolfch@science.oregonstate.edu
mark.novak@oregonstate.edu
gitelman@science.oregonstate.edu
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Wolf, Novak, and Gitelman
Abstract10
Considerable effort has been devoted to the empirical estimation of species11
interaction strengths. This effort has focused primarily on statistical significance12
testing and on obtaining point estimates of parameters that contribute to interaction13
strength magnitude, leaving characterizations of estimation uncertainty and14
distinctions between the deterministic and stochastic contributions to variation15
largely unconsidered. Here we consider a means of quantifying interaction strength16
uncertainty by formulating an observational method for estimating per capita attack17
rates as a Bayesian statistical model. This formulation permits the explicit18
incorporation of multiple sources of uncertainty. In doing so we highlight the19
informative nature of several so-called non-informative prior choices in modeling the20
sparse data typical of predator feeding surveys and provide evidence for the superior21
performance of a new neutral prior choice. A case study application shows that while22
Bayesian point estimates may be made to correspond with those obtained by23
frequentist approaches, estimation uncertainty as described by the 95% intervals is24
more biologically realistic using the Bayesian method in that the lower bounds of the25
Bayesian posterior intervals for the attack rates do not include zero when the26
occurrence of a given predator-prey interaction is in fact observed. This contrasts27
with bootstrap confidence intervals that often do contain zero in such cases. The28
Bayesian approach provides a straightforward, probabilistic characterization of29
interaction strength uncertainty. In doing so it provides a framework for considering30
both the deterministic and stochastic drivers of species interactions and their impact31
on food web dynamics.32
Keywords: nonlinear interaction strengths, predator-prey, functional response, JAGS,33
zero-inflated gamma, parameter estimation, non-informative neutral priors.34
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Introduction35
Quantifying the strength of species interactions is an important ecological challenge.36
Estimates can be used to identify keystone species whose impacts are disproportionate to37
their abundance (Power et al., 1996), help explain community structure (Wootton, 1994),38
are key to understanding food web stability (Allesina and Tang, 2012), and underlie39
forecasts of community dynamics (Yodzis, 1988). Estimates made on a per capita basis are40
particularly useful in that they underlie all other measures of interaction strength (Laska41
and Wootton, 1998). In this regard, a fundamental component of predator-prey interaction42
strengths is the nature of the predator’s functional response. For example, with linear43
Holling type I functional responses and linear density dependence in the prey, per capita44
interaction strengths correspond to the predators’ per capita attack rates. For nonlinear45
multi-species functional responses, such as the Holling type II functional response which46
most predators exhibit (Jeschke et al., 2004), the per capita attack rates are parameters47
reflecting the predators’ prey preferences (Chesson, 1983).48
Unfortunately, estimating interaction strengths in natural systems is difficult. In most49
food webs the large number of pairwise interactions—and the large number of weak50
interactions in particular—makes the use of manipulative experiments logistically51
prohibitive. Thus, many have resorted to indirect means of estimation, such as using52
energetic and allometric principles (Wootton and Emmerson, 2005). More effort still has53
been devoted to estimating interaction strength parameters by characterizing predator54
functional responses, largely on a pairwise experimental basis or by using observations of55
predator gut contents and kill rates (Vucetich et al., 2002; Jeschke et al., 2004).56
To date, most of the effort spent on measuring interaction strengths has focused on57
obtaining point estimates of parameters. For example, Paine (1992) used a bootstrapping58
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Wolf, Novak, and Gitelman
procedure only to quantify the uncertainty associated with the mean net strength of59
pairwise species interactions due to variation among experimental replicates. The focus has60
similarly been on point estimates and their statistical significance in the use of functional61
response experiments designed to determine the dependence of feeding rates on prey-62
and/or predator densities. Thus only the “deterministic core” of alternative functional63
response formulations has been of interest (Vucetich et al., 2002; Jeschke et al., 2004).64
More specifically, functional responses have often been fit to data using statistical models65
such as F=cN
1+chN +(the Holling type II response) whereby variation in a predator’s66
feeding rate (F) is assumed to be controlled by a deterministic component governed by67
variation in abundance (N), attack rate (c) and handling time (h), and only a “shell” of68
stochastic variation () is used to describe the variation left unexplained by the69
deterministic core. This is in contrast to considering the variation that is intrinsic to both70
the parameters (cand h) and variables (N) by describing each by a distribution that is71
itself governed by deterministic and stochastic sources of variation.72
The distinction between these two approaches to considering variation in interaction73
strength estimates is important when the uncertainty of estimates itself is of interest. This74
is particularly true when forecasting the dynamics of species rich communities where75
indirect effects can rapidly compound even small amounts of uncertainty (Yodzis, 1988;76
Novak et al., 2011). In such applications, knowledge of the (co-)variation of parameter77
estimates is essential to assessing the sensitivity of predictions under plausible scenarios of78
estimation uncertainty. Of course, estimates of uncertainty are also important in79
comparing the utility and consistency of different interaction strength estimation methods,80
and for the biological interpretation of the estimates themselves. Estimates derived from81
allometric relationships, for example, are typically associated with several orders of82
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Wolf, Novak, and Gitelman
magnitude of variation (Rall et al., 2012).83
Here we extend the observational method for estimating the per capita attack rates of84
predator-prey interactions presented by Novak and Wootton (2008) to characterize85
estimation uncertainty. Our interest in such observational methods stems from their ability86
to more easily accommodate instances of trophic omnivory than experimental and87
time-series methods, and because they retain the species-specific information lost in88
allometric and energetic approaches. Furthermore, with the method of Novak and Wootton89
(2008) attack rates may be estimated for multiple prey simultaneously while accounting for90
an inherent nonlinearity of predator-prey interactions because a multi-species Holling type91
II functional response is considered in the method’s derivation. Our stochastic formulation92
connects a deterministic functional response model with each of the sources of empirical93
data that contribute to the per capita attack rate estimates. The Bayesian framework in94
which we implement our approach thereby permits us to account for both variation due to95
sampling effort and the environment, and thus to explicitly incorporate the deterministic96
and stochastic sources of uncertainty intrinsic to the attack rate estimates.97
To assess the Bayesian method’s utility we apply it to data collected by Novak (2010),98
contrasting these estimates with those obtained by non-parametric and parametric99
bootstrapping procedures. Commensurate with the Bayesian approach we assess the effect100
of alternative prior choices on posterior point estimates and show that these estimates may101
be made consistent with those obtained by bootstrapping approaches by the choice of an102
appropriate non-informative prior. In addition, we show that estimation uncertainty as103
described by 95% intervals is considerably more constrained and biologically realistic when104
estimated within the Bayesian framework. Finally, we provide posterior probability105
distributions on the per capita attack rate estimates that lend themselves to a more useful106
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and descriptive characterization of interaction strength uncertainty.107
Methods108
Model framework109
Novak and Wootton (2008) present a method for estimating a predator’s per capita110
attack rates provided that a multi-species Holling type II functional response,111
Fi=ciNi
1 +
S
P
k=1
ckhkNk
,eqn 1
describes the per predator feeding rate on the ith prey species (i= 1, ..., S). Their112
estimator for the attack rates may be shown to be equivalent to113
ci=Ai
A0
1
hiNi
,eqn 2
where hiand Niare the handling time and abundance of the ith prey species, Aiis the114
number of predators feeding on prey i, and A0is the number of predators not feeding (see115
Appendix S1).116
Estimates of Aiand A0come from one or more predator population “snapshot”117
feeding surveys in which the number of predator individuals feeding on each prey species is118
recorded. Other data sources are used to estimate prey abundances and handling times.119
For example, handling times are more easily measured in laboratory experiments than in120
the field, while abundance estimates may come from independently performed community121
surveys. Even if handling time data are based on field observations, they are unlikely to be122
measured on the predators observed in a feeding survey since the lengths of time those123
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predators have been feeding are unknown due to the snapshot nature of the surveys.124
Both eqn 1 and eqn 2 are implicitly deterministic mathematical models that include125
no statistical description of their stochastic component. In the next sections of this paper126
we develop a parameter-based version of eqn 2 that gives a stochastic formulation of the127
attack rate estimator that can incorporate sampling and environmental variation explicitly.128
Bayesian Methods129
Frequentist approaches for combining data from multiple sources to estimate functions130
of parameters generally rely on bootstrap methods or asymptotics like the multivariate131
delta-method. Both these approaches exhibit poor small-sample performance (Efron and132
Tibshirani, 1994; Kilian, 1998). This is relevant when dealing with predator feeding surveys133
as the Aiin eqn 2 are often very small for the rare prey species that typify predator diets134
(Rossberg et al., 2006). Small values of Aican be problematic even when the total number135
of predators surveyed is large (Agresti and Coull, 1998). Ignoring variation in abundance136
and handling time estimates to focus on the variation within the feeding surveys may avoid137
this problem, but will lead to underestimation of the uncertainty in the attack rate138
estimates. The Bayesian framework circumvents this problem.139
The Bayesian machinery is built around Bayes theorem:140
f(θ|data)f(data|θ)·f(θ).eqn 3
Here, f(data|θ) is the likelihood: a function specifying the likelihood of the observed data141
in terms of unknown parameters θ.f(θ) is the prior: a probability density function142
reflecting prior beliefs or uncertainty about the parameters. Together, these inform143
f(θ|data): the posterior distribution of the parameters given the data. Here, we consider144
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only objective (also called non-informative) priors, assuming an absence of prior beliefs or145
information about the parameters in question (Berger, 2006). In other situations,146
informative priors constructed from previously obtained knowledge or data may be useful.147
Note that we use fto represent the density function of an arbitrary distribution for148
convenience only, using the function’s argument(s) to indicate the specific distribution149
being referenced. For example, the density of a random variable Xis indicated by f(x),150
and that of θby f(θ), even though these are not necessarily the same density functions.151
We use bold letters for vectors and bold uppercase letters for matrices of random variables.152
A Bayesian Attack Rate Estimator153
A parametric formulation of the attack rate estimator (eqn 2) is154
ξi=αi
α0
1
νiηi
.eqn 4
Here, for the ith prey species, ξiis the unknown attack rate, νiis the population prey155
abundance, ηiis the population handling time, αiis the population proportion of predators156
feeding, and α0is the population proportion of predators that are not feeding on any prey157
species. In each case, the parameters refer to the broader (statistical) population, rather158
than sampled data only. By framing the attack rates this way, we are able to estimate159
them in the context of the broader population about which inference is desired.160
If data on prey-specific feeding proportions (F), abundances (A), and handling times161
(H) are collected independently, the joint likelihood of these distributions may be written162
as:163
f(α,ν,η|F,A,H) = f(F|α)f(A|ν)f(H|η).eqn 5
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Provided that the corresponding priors are also independent, Bayes theorem implies that164
f(α|F)f(F|α)f(α),eqn 6a
f(ν|A)f(A|ν)f(ν),eqn 6b
f(η|H)f(H|η)f(η).eqn 6c
These may therefore be fit with independent models for each component. That is, the165
posterior distributions of the attack rates in eqn 4 may be estimated using Markov Chain166
Monte Carlo (MCMC) to obtain samples from each of the three posterior distributions in167
eqn 6 and combining these using eqn 4. If the three types of data are not gathered168
independently, then it is necessary to consider likelihood or prior models that account for169
this dependence (see Appendix S3).170
Case study data set171
To provide a concrete explanation of the additional details of the Bayesian approach172
we applied it to a dataset involving the predatory marine intertidal whelk Haustrum173
scobina.Haustrum feeds primarily on barnacles and mussels, often by first drilling through174
the shells of its prey. Handling times, which can be hours to days, are the times needed to175
drill and ingest a prey individual. The dataset contains information from replicate feeding176
surveys, quadrat-based abundance surveys, and laboratory-based handling time177
experiments, which we describe briefly below. Further details may be found in Novak178
(2010, 2013).179
Fifteen feeding surveys were conducted during low tides over two years. In each180
survey, the number of whelks feeding on each prey species was recorded, as was as the181
number not feeding (Table 1). The sizes of the predator individuals (both feeding and not182
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feeding) and of the prey being fed upon were also recorded (±1 mm), along with the183
average temperature of the month in which each survey was conducted. These three184
covariates contribute to the deterministic variation in per capita attack rate estimates.185
Species abundance surveys used 10-15 replicate quadrats randomly distributed along 3186
transects, each repeated 3 times over the same time periods as the feeding surveys. As is187
typical of community abundance surveys, numerous zeros exist in these data as many188
species did not occur in every quadrat (Table 1). The presence of such zeroes reflects both189
deterministic variation associated with real variation in species abundances, as well as190
stochastic variation associated with sampling effort.191
Handling times were estimated in laboratory experiments that manipulated predator192
and prey sizes and temperature. Replicated individuals housed in separate aquaria with193
different prey species were checked hourly to determine handling time durations. As a194
result, handling time measurements are interval censored, equally so for prey species with195
short (hour-long) and long (multi-day) handling times. Such uncertainty, along with196
variation in the number of replicate experiments that were performed for each of the prey197
species (Table 1), reflect additional sources of stochastic uncertainty.198
Treating the abundances, handling times, and feeding surveys data as independent, we199
now specify appropriate likelihood and prior models.200
Model formulation201
Accommodating variation in feeding surveys – Letting Pbe the total number of predators202
surveyed, Xithe number observed feeding on prey i, and X0the number not feeding, we203
modeled the combined feeding survey data using a multinomial likelihood with Dirichlet204
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prior:205
(X0, X1, ..., XS)MultP(α0, α1, ..., αS) eqn 7a
(α0, α1, ..., αS)Dirich(c, c, ..., c).eqn 7b
The resulting posterior distribution is also Dirichlet:206
(α0, α1, ..., αS)|xDirich(c+x0, c +x1, ..., c +xS).eqn 8
We will focus on the posterior medians rather than means as our point estimates of207
interest since they are the more appropriate measure of a skewed distribution’s central208
tendency (Gelman et al., 2013). The four most commonly used non-informative priors in209
this setting are Laplace’s prior (c= 1), Jeffreys’ prior (c=1
2), Perks’ prior (c=1
S+1 ), and210
Haldane’s prior (c= 0) (Hutter, 2013). However, these priors all result in posterior211
medians that may differ substantially from the sample proportions, which are the212
maximum likelihood estimates (MLEs), especially when any of the Xiare small. This leads213
to counter-intuitive attack rate point estimates for rarely observed prey species (Fig. 1).214
Kerman et al. (2011) showed that when c=1
3, the multinomial parameter posterior215
medians closely match the MLEs, referring to this prior as the non-informative neutral216
prior. We show that this result applies to the ratios of multinomial parameters as well by217
letting γi=αi
α0and noting that the posterior distribution of γiis the ratio of Dirichlet218
components, which is the ratio of independent gamma random variables. This may be219
written as:220
f(γi|xi, x0) = x0+c
xi+c·gx0+c
xi+c·γi; 2(xi+c),2(x0+c)eqn 9
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where g(y;d1, d2) is an F-distribution probability density function with d1and d2degrees of221
freedom. Using the approximation for the median of an F-distribution222
med(Fm
n)n
3n2
3m2
m(see Appendix S2) and setting it equal to the MLE of αi
α0,xi
x0, yields223
the solution c=1
3. Figure 1, which shows the log differences in posterior median relative to224
the MLE for several values of c, evidences that c=1
3is indeed a reasonable prior to use.225
Accommodating variation in abundance surveys – We used a zero-inflated gamma (ZIG)226
model to account for the numerous zeros in the abundances data. Letting Y1, ..., Yndenote227
the abundance measurements, and by conditioning on whether or not a zero occurs, the228
likelihood density of the ZIG distribution can be written as229
g(y;α, β, ρ) = ρI[y=0] [(1 ρ)f(y, α, β)]I[y>0], y 0,eqn 10
where ρis the probability of a zero, f(y;α, β) is the usual gamma density with shape α,230
rate β, and mean α
β, and I[·] is the indicator function that equals 1 when its argument is231
true and 0 otherwise (Ospina and Ferrari, 2012). The ZIG density is separable in ρand232
(α, β). It follows that the zero-inflation parameter can be treated separately, provided a233
separable prior is used. Thus, for each prey species, we modeled the number of observed234
zeros using a binomial distribution with a uniform prior on ρand took235
log(α)U nif(100,100) and log(β)U nif(100,100) to approximate the independent236
scale-invariant non-informative prior f(α, β) = f(α)f(β)1
α
1
β(Syversveen, 1998).237
Accommodating variation in handling time experiments – We used regression to model the238
relationships between handling times and the predator-size, prey-size and temperature239
covariates of the laboratory experiments. Average handling times for use in the attack rate240
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estimation were obtained by combining these regression coefficients with the same covariate241
information obtained during feeding surveys.242
Specifically, we considered the ith handling time observation for a given prey species to243
be associated with a covariates vector Xiconsisting of 1 followed by temperature, predator244
size, and prey size (all log transformed). We then modeled the likelihood of the ith
245
handling time with a modified-Normal likelihood written as246
HiNli(eXT
iβ, σ2) eqn 11
where the subscript lirefers to the censoring “window” length and indicates that a247
Unif (li
2,li
2) error was added to the normal distribution corresponding to the interval248
censoring with which handling times were observed. The exponential link of eqn 11 avoids249
negative mean handling time estimates.250
Treating the field covariates (predator size, prey size, and temperature) as random to251
account for sampling variability, we modeled the distributions of the (log-transformed)252
covariate observations X1, ..., XN, where Nis the total number of field observations, as253
being independent and identically distributed and drawn from a multivariate normal254
distribution, N(µ,Σ0), with mean vector µand covariance matrix Σ0. Non-informative255
multivariate normal and inverse Wishart priors were used for µand Σ0respectively (Fink,256
1997). Letting Xfollow the posterior predictive distribution (our estimate of the257
distribution of the covariates), the mean handling time may be written as258
E(H) = E[E(H|X)] = E(eβTX).eqn 12
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We estimated this expectation by sampling from the regression parameters’ posterior259
distribution, sampling new covariates from their posterior predictive distribution,260
computing eβTXfor each sample, and averaging across all samples. The weak law of large261
numbers ensures convergence to E(eβTX) as sample size increases (Petrov, 1995).262
Model implementation: Putting the pieces together to estimate per capita attack rates263
Using the likelihoods and priors of the feeding surveys, abundances and handling times264
described above, we drew samples from the parameters’ posterior distributions using265
Markov Chain Monte Carlo (MCMC). MCMC sampling was done using JAGS with the R266
package ‘rjags’ (Plummer and Stukalov, 2014). Parameter samples were then combined267
using eqn 4 to produce samples from each prey’s attack rate posterior distribution. This268
use of eqn 4 treated handling times, H, as being independent of the predator feeding269
surveys, F, even though covariate observations of predator size, prey size and temperature270
from the feeding surveys informing Fwere used to inform Hby combining them with the271
laboratory-based handling time regression coefficients associated with these covariates. We272
established the validity of this assumption by examining the relationship between feeding273
proportions and covariate averages between the individual surveys (see Appendix S3).274
We verified Markov chain convergence using trace plots, removed pre-convergence275
samples, and thinned each chain to ensure independence among the remaining samples.276
Burn-in times and thinning values were selected separately for feeding survey, abundance,277
and handling time models based on trace plots and autocorrelation function plots.278
Inferences were based on 1,000 samples after confirming that independent sets of 1,000279
samples led to the same conclusions.280
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Results281
Comparison of Bayesian and bootstrapping procedures282
To assess the utility and performance of our Bayesian approach we contrasted point283
and 95% interval estimates from (i) the model with Laplace’s prior (c= 1) on the Dirichlet284
feeding proportions, (ii) the model with Haldane’s prior (c= 0), and (iii) the model with285
the neutral prior (c=1
3) to estimates obtained by (iv) non-parametric and (v) parametric286
bootstrapping procedures. In contrast to the Bayesian 95% credible intervals, which reflect287
the range of values within which a parameter will occur with 95% probability, the 95%288
confidence intervals associated with bootstrapping do not have a direct probability-based289
interpretation. Rather, if we repeatedly constructed 95% confidence intervals on ‘new’290
datasets, about 95% of them would contain the ‘true’ value of the parameter.291
Non-parametric bootstrapping was performed by sampling with replacement from each292
of the feeding survey, abundance, and handling-time datasets until the same number of293
samples had been drawn as was present in each dataset (Efron and Tibshirani, 1994). Per294
capita attack rates were calculated for many sets of such resampled data to estimate the295
mean and 95% confidence intervals of their bootstrapped distributions.296
The parametric bootstrap was implemented using the likelihood functions of the Bayes297
method. That is, we used the data to estimate the parameters of the three likelihood298
functions (i.e. eqn 7, eqn 10, eqn 11) by maximum likelihood, used these fit likelihood299
functions to simulate new datasets, combining samples from the three distributions to300
calculate per capita attack rates and estimate medians and 95% confidence intervals of the301
resulting bootstrapped attack rate distributions.302
The comparison of the approaches indicates that the model with the neutral prior303
(c=1
3) on the feeding proportions was indeed both sufficient for, and performed best in,304
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describing the variation inherent in the estimated rates with which Haustrum scobina305
attacked its eight prey species (Fig. 2). It exhibited median point estimates most closely306
matching the point estimates of the two bootstrapping approaches. The two bootstrap307
distributions however, frequently exhibited lower 95% confidence interval end points of308
zero; a nonsensical result given that the consumption of these species was in fact observed.309
As expected (Fig. 1), the models having Laplace’s (c= 1) or Haldane’s (c= 0) prior310
resulted in inflated and depressed attack rate point estimates respectively, particularly on311
prey species that were observed infrequently in the feeding surveys.312
Figure 3 shows the posterior probability distributions of Haustrum scobina’s per capita313
attack rates on each of its prey species as estimated using the neutral prior (c=1
3). The314
distributions are roughly symmetric on the logarithmic scale, indicating right skew and315
justifying the use of the median as the point estimate of their central tendency.316
Discussion317
Effort devoted to estimating the strengths of species interactions has centered on obtaining318
point estimates, leaving the characterization of estimation uncertainty largely unconsidered.319
This shortcoming reflects not only the logistical difficulty of quantifying interaction320
strengths in nature’s species-rich communities, but is also a consequence of the still nascent321
integration of the mathematical and statistical methods available to food web ecologists.322
The fitting of deterministic mathematical models to data requires that they be323
formulated as stochastic statistical models whose constants – like the per capita attack324
rates considered here – be treated as unknown parameters to be estimated. For the325
observational estimator of Novak and Wootton (2008) the unknown attack rate parameters326
of interest are functions of other unknown parameters that must themselves be estimated.327
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Uncertainty in attack rate estimates thus reflects the contributions of both the328
deterministic and stochastic variation of these component parameters. The propagation of329
both such forms of variation is inherent to all other experimental and observational330
approaches as well.331
Advantages of the Bayesian approach332
Unlike frequentist methods, Bayesian methods offer a relatively straightforward way to333
estimate parameters that are functions of other parameters using multiple sources of334
information. This is particularly, though not necessarily, so when the posterior335
distributions of these parameters are independent. Bayesian methods also permit a more336
natural interpretation of the uncertainty that accompanies parameter estimates and337
provide a complete characterization of this uncertainty in the form of posterior probability338
distributions; frequentist methods provide the moments and intervals of distributions339
whose interpretation is arguably less intuitive (Clark, 2005).340
In the context of reticulate food webs and predator-prey interactions, the complete341
probabilistic characterization of uncertainty regarding observational interaction strength342
estimates opens the door for probabilistic predictions of species effects and population343
dynamics (Calder et al., 2003; Yeakel et al., 2011). This stands in contrast to the typical344
use of arbitrarily chosen interaction strength ranges in stochastic simulations and345
numerical sensitivity analyses (Yodzis, 1988; Novak et al., 2011). An alternative choice to346
use bootstrapped (frequentist) confidence intervals to inform predictions could lead to347
additional problems when lower interval bounds extend to zero for prey species that are348
rarely found in a predator’s diet. First, draws of zeros would amount to the outright349
removal of the predator-prey interaction and could lead to biased predictions through the350
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underestimation of food web complexity. Second, as evidenced by Haustrum scobina’s351
feeding on Mytilus galloprovincialis (Fig. 2), prey species whose attack rate confidence352
intervals extend to zero may in fact experience very high per capita attack rates on353
average. Treating these interactions as potentially absent would fail to identify strong354
interactions that are rarely observed only because of strong top-down control of the prey355
populations’ sizes, for example. Such issues do not occur in the Bayesian framework where356
the Dirichlet prior distribution is conjugate for the multinomial likelihood, thereby357
producing a Dirichlet posterior from which MCMC samples of zero cannot occur.358
Considerations and implications359
Bayesian methods offer a powerful tool, but they should not be applied without360
careful consideration of the prior distribution. The choice of objective (‘non-informative’)361
prior is particularly important when data are sparse (Van Dongen, 2006; Boshuizen and362
Van Baal, 2009). It follows that, for rarely observed prey species, different prior363
specifications lead to different point estimates of the per capita attack rates (Figs. 1 and364
2). That is, while priors with concentration parameters c > 1
3(e.g., Laplace’s prior) will365
produce higher attack rate point estimates the less frequently a prey species is observed in366
the predator’s diet, priors with concentration parameters c < 1
3(e.g., Haldane’s prior) will367
produce lower attack rate point estimates the less frequently a prey species is observed in368
the predator’s diet (see also Fig. S2). The biological implication of choosing to use one369
such prior over another is that this choice can alter the relative frequency of weak and370
strong interactions. Thus, the choice of priors can alter inferences of population dynamics371
and food web stability when models are parameterized with empirical estimates (Allesina372
and Tang, 2012). These considerations are avoided only when all prey occur frequently in a373
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predator’s diet (see Mytilus galloprovincialis and Chaemaesipho columna in Fig. 2). In such374
cases, the large sample sizes mean that the likelihood overwhelms the prior regardless of its375
information content such that Bayesian and frequentist estimates are similar.376
The use of the neutral prior produces posterior distribution median point estimates377
that are least influenced by the prior and thus most like the point estimates of the378
frequentist bootstrap methods (Figs. 1, 2, S2). We therefore suggest that this be the379
preferred objective prior to use. Tuyl et al. (2008) argue against the use of such sparse380
(c < 1) priors for binomial parameters as they put more weight on extreme outcomes. For381
example, if YBin(n, p) and Y∈ {0, n}, the use of sparse priors leads to inappropriately382
narrow credible intervals. Fortunately, this problem is avoided in our application because383
all considered prey species (and “not feeding”) are observed at least once384
(i.e. Y∈ {1, ..., n 1}). In hierarchical models, to which our Bayesian framework could be385
naturally extended (Cressie et al., 2009), Y∈ {0, n}is more likely for any individual386
survey, but this is not an issue as inference at the survey level is not desired.387
An influence of Bayesian prior choice also occurs in the estimation of prey abundances388
by means of a zero-inflated gamma likelihood model. Here the assumption that a389
zero-inflated gamma is descriptive of the abundance structure of all prey species can lead390
to the inflation of per capita attack rate estimates for species that are ubiquitous. When391
species occur in all but a few sampled quadrats, relatively little data are available to392
estimate the probability of obtaining a count of zero. In such situations the influence of393
even an uninformative uniform prior will be increased, resulting in an inflated estimate of394
the proportion of zeros and thus a reduced estimate of a species’ abundance. Attack rate395
estimates are thereby inflated because a species’ abundance occurs in the denominator of396
the estimator (eqn 2). For our dataset, where many species were present in all sampled397
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quadrats (Table 1), this inflation effect appears to have been weak as seen by comparing398
the results of the Bayesian models to the frequentist bootstrapping procedures for which399
such inflation does not occur (Fig. 2); the probability of obtaining a value of zero during400
bootstrapping is equal to the sample proportion of zeros in the data, which is zero for401
species that are always observed. Arguably, however, this inflation effect of the prior that402
is inherent to the use of the zero-inflated gamma in a Bayesian framework is appropriate403
because observations of species absences at the spatial scale of quadrats are fundamentally404
different from observations of species presences when no prior knowledge about the405
patchiness of species’ abundances is available.406
Issues of prior choice aside, Bayesian methods offer a more complete characterization407
of the estimated uncertainty of parameter estimates in the form of posterior probability408
distributions. Several metrics may be chosen to summarize the shapes of these409
distributions. For example, means, medians and modes are all commonly used as point410
estimates to reflect a distribution’s typical and most likely value. For strongly skewed411
distributions – such as those observed here (Fig. 3) – medians are a more representative412
metric of a distribution’s central tendency. Furthermore, a distribution’s median, unlike its413
mean, will always fall within the equal-tailed interval that is typically used to describe the414
uncertainty or variation surrounding the distribution’s estimated central tendency. Of415
course, point estimates provide little information on a distribution’s shape. Confidence or416
credible intervals provide more such information with which to characterize parameter417
uncertainty and variation. The typical metrics for these intervals are equal-tailed, but for418
posterior distributions the highest posterior density (HPD) interval may also be useful419
(Gelman et al., 2013). While intervals characterized by highest posterior density are more420
resistant to distribution skewness and will always include the distribution’s mode,421
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equal-tailed intervals are invariant under monotone transformations, making them easier to422
interpret after log-transformation. Log-transformation is frequently necessary in the423
context of interaction strengths given the wide range of values that the community-wide424
strengths of species interactions typically exhibit (Wootton and Emmerson, 2005).425
Ultimately, the entire joint posterior distribution should be presented whenever possible426
(Chen and Shao, 1999). When this is not practical, the choice of posterior summaries will427
depend on the goal of the analysis.428
Conclusion429
While many ecological processes can be described in purely mathematical terms,430
mathematical models are often most useful when they are linked with real data (Codling431
and Dumbrell, 2012). Linking models with data is necessary to validate and compare432
models, and to parameterize them for real-world use in predicting future system dynamics433
(Bolker, 2008). This has been a challenging task in the study of species rich food webs, not434
least because of the difficulty of parameter estimation in typical food web models and435
challenges with integrating data collected across multiple spatial and temporal scales.436
Statistical models of predator-prey interactions that consider both deterministic and437
stochastic variation in data are needed to accompany the numerous mathematical models438
that have been proposed. Our work represents a step in this direction.439
Acknowledgments440
This work was supported by NSF DEB-1353827 and DEB-0608178. Data and R scripts are441
available on the Dryad Digital Repository (doi:10.5061/dryad.6k144) and on GitHub442
(https://github.com/wolfch2/Bayesian-Interaction-Strength)443
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1. Tables527
Feeding Abundance Handling-time
Prey Species Observations Zeros Experiments
Chamaesipho columna 265 0 6
Xenostrobus pulex 185 0 52
Austrolittorina antipodum 3 2 0
Austrolittorina cincta 2 0 46
Epopella plicata 2 0 1
Mytilus galloprovincialis 1 24 15
Notoacmea Radialspokes 1 5 66
Risellopsis varia 1 3 68
Not Feeding 1,629
Total Surveyed 2,089
Table 1: Summary statistics for the datasets used to estimate the per capita attack rates
with which the intertidal predator Haustrum scobina feeds on its eight prey species. “Feeding
observations” indicates the total frequency with which predator individuals were observed to
be feeding on each prey species across all feeding surveys. “Abundance zeros” indicates the
number of zeros recorded in the 30 quadrat-based community surveys of species abundances.
“Handling-time experiments” indicates the number of laboratory experiments that were used
to estimate handling time regression coefficients for each prey species.
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2. Figure Legends528
Figure 1: Comparison of alternative non-informative priors in estimating the ratio of the529
proportions of feeding versus not feeding predator individuals. The x-axis reflects the530
number of predators observed in the process of feeding on a given prey species, with a total531
of 1,629 individuals assumed to have been not feeding, corresponding to the number not532
feeding in our dataset (Table 1). The y-axis shows the difference in logarithms of the533
posterior median using a Dirich(c, ..., c) prior and the maximum likelihood estimate of the534
ratio. From top to bottom in the graph, the values of c are 1 (Laplace), 1
2(Jeffreys’), 1
3
535
(neutral), 1
S+1 =1
9(Perks’), and 0 (Haldane’s). The neutral prior (c=1
3) leads to estimates536
that closely match the maximum likelihood estimates.537
Figure 2: Comparison of the frequentist and Bayesian approaches to estimating the538
per capita attack rates with which Haustrum scobina consumed its 8 prey species.539
Variation in attack rate estimates is illustrated for each procedure by the medians and 95%540
equal-tailed intervals of their distributions. Procedures are organized the same for each541
prey species as, from top to bottom: (i) non-parametric bootstrap, (ii) parametric542
bootstrap, (iii-v) Bayesian procedure with sparsity parameters 0 (Haldane’s prior), 1
3
543
(neutral prior), and 1 (Laplace’s prior) respectively. Unlike the 95% confidence intervals for544
the bootstrap procedures which often span zero (= 107for graphical convenience), the545
95% posterior posterior intervals of the Bayesian method indicate the regions where attack546
rates lie with 95% probability.547
Figure 3: Posterior distributions for Haustrum scobina’s per capita attack rates548
(prey ·predator1·prey1·m2·day1) and its components (ξi=αi
α0·1
νi·1
ηi) using neutral549
(c=1
3) Dirichlet prior on feeding proportions.550
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3. Figures551
−0.2
−0.1
0.0
0.1
0.2
0 10 20 30 40 50
Predators Feeding
log10
θ
^Bayes
XiX0
Sparsity (c)
1
0.5
0.333
0.111
0
Figure 1
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Mytilus
galloprovincialis
Xenostrobus
pulex
Austrolittorina
antipodum
Notoacmea
Radial
Risellopsis
varia
Chamaesipho
columna
Austrolittorina
cincta
Epopella
plicata
10−7 10−6 10−5 10−4 10−3
Attack Rate ( ξi )
Nonparametric Bootstrap Parametric Bootstrap Bayes (c=0)
Bayes (c=0.333) Bayes (c=1)
Figure 2
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Feeding Ratio
αiα0
H. Time
ηi
Abundance
νi
Attack Rate
ξi
10−4 10−3 10−2 10−1 10010110210310410510−8 10−7 10−6 10−5 10−4
Probability
Figure 3
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Appendix552
S1 On Novak and Wootton’s ‘Species x553
This paper expands on the observational method for estimating attack rates presented554
by Novak and Wootton (2008):555
ci=FiAx
(FxAx)hiNi
,
where ciis the attack rate, hiis the handling time, and Niis the abundance, all for the ith
556
prey species. Aiand Fiare the proportions of all predators and feeding predator557
respectively feeding on the ith prey species. xrefers to an arbitrarily chosen prey species558
that is the same for all ci. Here we show that this equation can also be written in a more559
simplified form, showing that the estimates are not dependent on the choice of species x.560
Define A0to be the observed proportion of predators that are not feeding, so that561
A0= 1
S
P
i=1
Ai. Then, the F0
iscan be obtained by normalizing A0
is:Fi=Ai
S
P
j=1
Aj
=Ai
1A0.562
Noting that:563
FxAx=Ax
1A0
Ax=AxAx(1 A0)
1A0
=Ax[1 (1 A0)]
1A0
=AxA0
1A0
.
It follows that564
FiAx
FxAx
=
Ai
1A0·Ax
AxA0
1A0
=AiAx
AxA0
=Ai
A0
.
This can be further simplified by noting that the A0
ishave a common denominator (total565
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number surveyed). This means that the original attack rate equation can be written as566
ci=Ai
A0
·1
hiNi
=#feeding on i
#not feeding ·1
hiNi
.
This shows that the estimate does not involve species x. Moreover, the total number567
surveyed need not be known to estimate a subset of the attack rates.568
S2 F-distribution median569
In general, the median of the F-distribution does not have a closed form. However, we570
can derive an approximation by relating the F-distribution to the beta-distribution.571
Let XFm
n. We can express Xas the ratio of scaled, independent Chi-squared
distributions Cmχ2
mand Cnχ2
n:
X=Cm/m
Cn/n
It follows that we can express Xas the ratio of scaled independent gamma distributions572
Gmgamma(m
2,2) and Gngamma(n
2,2):573
X=Gm/m
Gn/n
=n
m
Gm
Gn
We can then normalize the gamma distributions:574
X=n
m
Gm
Gm+Gn
Gn
Gm+Gn
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Letting (D1, D2)Dir(m
2,n
2) and using the relationship between Dirichlet and gamma575
distributions,576
X=n
m
D1
D2
Using the marginal distribution for Dirichlet components result and the fact that577
D1+D2= 1, we have that578
X=n
m
B
1B
where BBeta(m
2,n
2). Note that this is a monotone transformation of B, so it preserves579
the median. When m > 2 and n > 2, the median of Bis approximately m
21
3
m
2+n
22
3
(Kerman,580
2011). Substituting this result, we have that581
med(X) = n
m
med(B)
1med(B)
=n
m
m
21
3
m
2+n
22
3
n
21
3
m
2+n
22
3
=n
m
m
21
3
n
21
3
=n
m
m
21
3
n
21
3
=n
m
3m2
3n2
=n
2n2
3m2
m
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S3 Accounting for dependence among information sources582
In our dataset, predator feeding surveys included covariate information (predator size,583
prey size, and temperature) that was used to estimate field handling times on the basis of584
regression models for handling times parameterized using laboratory data. In estimating585
attack rates we treated the field covariates as part of the handling times data Hand586
assumed they were independent of the feeding proportions data F. The validity of this587
assumption may be assessed by plotting the regression covariates versus the observed588
feeding proportions, as shown in figure S1. In this figure, every point represents a single589
feeding survey. The x-axes are the averages of the (log-tranformed) covariate and the590
y-axes are the proportions of predators feeding. Only two species had sufficient data to be591
plotted and showed little evidence of a dependence.592
If a lack of independence were evident it would need to be accounted for in the593
covariates distribution model. That is, although our model for the covariates was a594
multivariate normal, feeding survey level information (specifically proportions of predators595
feeding on each prey species) could be added to the model to affects its multivariate mean.596
This way, the mean covariate vector would be a function of the proportion of predators597
feeding on that prey type. Posterior distribution sampling could then be done by first598
sampling from the feeding proportions posterior distributions and then using the sampled599
feeding proportions to obtain samples from the handling times.600
34
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PrePrints
Wolf, Novak, and Gitelman
S4 Supporting Figures601
log10 Predator Size (mm)
log10 Prey Size (mm)
log10 Temperature (°C)
5%
10%
15%
20%
25%
5%
10%
15%
20%
25%
Chamaesipho
columna
Xenostrobus
pulex
1.05 1.10 1.15 1.20 0.00 0.25 0.50 0.75 1.00 1.08 1.12 1.16 1.20
Predators Feeding
Figure S1: Average field covariates versus feeding proportions. Each point corresponds to a
single feeding survey. Only species that appeared in more than three separate feeding surveys
are shown. Of the eight species and three covariates, only Xenostrobus pulex showed any
evidence of a relationship between feeding proportions and feeding covariates (i.e., between
Fand Hin eqn 5).
35
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Wolf, Novak, and Gitelman
10
50
500
1000
10 50 500 1000
Predators not feeding (X0)
Predators feeding (Xi)
−0.01
0.00
0.01
log10
θ
^Bayes
XiX0
10
50
500
1000
10 50 500 1000
Predators not feeding (X0)
Predators feeding (Xi)
0.30
0.31
0.32
0.33
copt
Figure S2: Given the skewed nature of prey-specific per capita attack rate posterior proba-
bility distributions, the distribution median serves as a more appropriate point estimate than
the mean. Fig. 1 illustrates the difference between the posterior median and maximum like-
lihood estimate of the ratio of feeding and non-feeding predators as a function of the number
of feeding individuals, showing how the neutral (c=1
3) prior minimizes this difference. As
a generalization of Fig. 1, in the left panel, we illustrate this difference as a function of both
the number of predators observed feeding and the number observed not feeding. The right
panel shows that the “optimal” value of cthat minimizes this difference (a function of both
feeding and non-feeding individuals) is typically around 1
3. In both cases, the survey data
from our example are shown as black dots.
36
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... Such refocusing on parameters rather than per capita interaction strengths per se will clarify distinctions among differing uses of the term interaction strength (cf. Berlow et al. 2004, Wootton & Emmerson 2005 ), will aid in avoiding the strict equilibrium assumptions implicit in many estimation approaches (Novak & Wootton 2010, Wootton & Emmerson 2005), and will permit the field to move beyond current sensitivity analyses to more probabilistic descriptions of interaction strengths and predictions than are currently possible (Petchey et al. 2015, Wolf et al. 2015). Many additional challenges exist to understanding the dynamics of real ecological systems, such that humility and adaptive strategies will always be necessary (Doak et al. 2008, Petchey et al. 2015). ...
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