Caughlin T.T., Ferguson J.M., Lichstein J.W. ,Zuidema P.A., Bunyavejchewin S., and Levey D.J.
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Text S3. Additional details of global sensitivity analysis
We used global sensitivity analysis of IBM output to determine the relative importance of
IBM parameters for total population size. Sensitivity analysis addresses the question of how
perturbations in model parameters contribute to variability in model output. Local sensitivity
analyses, such as taking the partial derivative of output with respect to an input parameter, are
often used for analyzing matrix and integral projection models in demographic research (e.g.,
[1], but have the disadvantage of assuming linear, independent relationships between parameters
and output, which may lead to misleading results when relationships between parameters and
output are non-linear [2]. Thus, we used a global sensitivity analysis to perturb all parameters
simultaneously, including input from the full parameter space, non-linear responses, and
interactions between parameters [3,4]. Variance in parameter inputs for a global sensitivity
analysis can be arbitrarily set at a particular scale, determined by expert opinion, or can originate
from variability in the data [4]. Because all of the parameters used in our IBM come from
Bayesian statistical models parameterized with field data, we were able to use samples from the
posterior distribution of each parameter as the input to the IBM.
We used variance-based methods to decompose the variance in IBM output into main
effects of parameters, representing the additive effect of each parameter, and total effects,
representing the total contribution of parameter variance to output variance, including
interactions between parameters [5]. Variance-based methods for global sensitivity analysis
calculate main sensitivities (𝑆𝑖 ) of output Y to parameter 𝑋𝑖 as:
𝑆𝑖 =
𝑉[𝐸(𝑌|𝑋𝑖 )]
𝑉(𝑌)
(1)
Caughlin T.T., Ferguson J.M., Lichstein J.W. ,Zuidema P.A., Bunyavejchewin S., and Levey D.J.
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The main effect sensitivities can be interpreted as a correlation ratio between parameters and
output, and thus sum to one. Total effect sensitivities (𝑆T𝑖 ) decompose output variance 𝑉(𝑌) into
the variance due to all parameters but one, 𝑿~𝑖 :
𝑆T𝑖 = 1 −
𝑉[𝐸(𝑌|𝑿~𝑖 )]
𝑉(𝑌)
(2)
The total sensitivities represent the total contribution of parameters to output variation, and sum
to greater than one.
To conduct the global sensitivity analysis, we drew 2,000 sets of parameters randomly
from the joint posterior parameter distributions. We then applied the winding stairs algorithm [6]
to generate a matrix of 82,000 sets of parameters that were used to calculate main effect and total
sensitivities. We ran the IBM for 10,000 runs using the parameter sets, each time recording total
population size at the 100th time step as output from each run. In order to examine the sampling
properties of the winding staircase algorithm we calculated the main effect and total sensitivities
for independent parameter sets of sample size s, where s was set to 10, 20, 40, 80, 100, 200, 300,
400, and 500. For each sample-size level, s, we calculated 2000/s independent first-order and
total sensitivity values, where 2000 was the number of steps used in the winding staircase
algorithm. We then calculated the mean and standard deviation of the at each sample-size level.
We estimated the bias as a function of sample size by comparing the average estimate at each
sample-size level to the values calculated using all the samples. The trend in bias and standard
deviation as a function of sample size allowed us to bias correct our estimates and estimate the
standard errors, respectively. Thus, our results use these bias-corrected sensitivities (Table S4).
Consistent with theory, we found higher relatively higher standard errors in the main effect
Caughlin T.T., Ferguson J.M., Lichstein J.W. ,Zuidema P.A., Bunyavejchewin S., and Levey D.J.
3
sensitivities than the total sensitivities [6]. In general, the bias correction procedure made the
first-order sensitivities closer to zero and more similar to the total sensitivities.
References
1. Franco, M. & Silvertown, J. 2004 A comparative demography of plants based upon elasticities
of vital rates. Ecology 85, 531–538.
2. Morris, W. F. & Doak, D. F. 2002 Quantitative conservation biology: theory and practice of
population viability analysis. Sinauer Associates.
3. Ellner, S. P. & Fieberg, J. 2003 Using PVA for management despite uncertainty: effects of
habitat, hatcheries, and harvest on salmon. Ecology 84, 1359–1369.
4. Fieberg, J. & Jenkins, K. J. 2005 Assessing uncertainty in ecological systems using global
sensitivity analyses: a case example of simulated wolf reintroduction effects on elk.
Ecological Modelling 187, 259–280.
5. Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M. &
Tarantola, S. 2008 Global sensitivity analysis: the primer. Wiley-Interscience.
6. Chan, K., Saltelli, A. & Tarantola, S. 2000 Winding stairs: a sampling tool to compute
sensitivity indices. Statistics and Computing 10, 187–196.
Tables
Table S3. Main effect sensitivities from global sensitivity analysis.
Table S4. Total effect sensitivities from global sensitivity analysis.
These tables presents the bias-corrected estimates of main and total effects of parameter
uncertainty on total population size after 100 years from simulated populations. Main effects of
parameters represent the additive effect of each parameter on population size and total effects
represent the total contribution of parameter variance to output variance, including interactions
between parameters.
Caughlin T.T., Ferguson J.M., Lichstein J.W. ,Zuidema P.A., Bunyavejchewin S., and Levey D.J.
Table S3. Main effect sensitivities from global sensitivity analysis. This table presents the
bias-corrected estimates of main and total effects of parameter uncertainty on total population
size after 100 years from simulated populations. Main effects of parameters represent the
additive effect of each parameter on population size.
Submodel
Parameter
Initial distribution
Initial conditions
of trees
Seed dispersal
𝑢. 𝑑𝑖𝑠𝑝𝑒𝑟𝑠𝑎𝑙
𝑓𝑟
Seed production
𝛽0
𝑓𝑟
Seed production
𝛽1
𝑓𝑟
Seed production
𝛽2
Seed dispersal
𝐵𝑎𝑡ℎ
𝑔𝑒𝑟
Germination
𝜇0
Germination
𝛽 𝑔𝑒𝑟
Germination
𝛼 𝑔𝑒𝑟
Germination
𝑑𝑖𝑠. 𝑔𝑒𝑟
Germination
μ𝑖𝑛.ℎ𝑡
Germination
𝑠𝑐𝑎𝑙𝑒 𝑖𝑛.ℎ𝑡
Germination
𝑠ℎ𝑎𝑝𝑒 𝑖𝑛.ℎ𝑡
Seedling survival
𝜇0𝑠𝑠
Seedling survival
𝛽 𝑠𝑠
Seedling survival
𝑠𝑖𝑧𝑒 𝑠𝑠
𝑠𝑔
Seedling growth
𝜇0
Main effects
Standard Error
Main effects
0.008
0.029
0.001
<0.001
<0.001
<0.001
<0.001
0.009
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
0.016
<0.001
0.02
0.033
0.032
0.032
0.031
0.034
0.032
0.028
0.033
0.034
0.035
0.044
0.038
0.034
0.036
0.039
0.029
0.033
4
Caughlin T.T., Ferguson J.M., Lichstein J.W. ,Zuidema P.A., Bunyavejchewin S., and Levey D.J.
Table S3. Main effects and total effect sensitivities from global sensitivity analysis, cont.
Submodel
Seedling growth
Seedling growth
Seedling growth
Seedling growth
Seedling growth
Seedling growth
Seedling heightDBH allometry
Seedling heightDBH allometry
Seedling heightDBH allometry
Seedling heightDBH allometry
Seedling heightDBH allometry
Tree survival
Tree growth
Tree survival
Tree growth
Tree growth
Tree growth
Tree survival
Tree survival
Tree growth
Tree growth
Tree growth
Tree growth
𝛼 𝑠𝑔
𝑑𝑖𝑠. 𝑠𝑔
𝑠𝑖𝑧𝑒 𝑠𝑔
𝛽 𝑠𝑔
𝑠𝑐𝑎𝑙𝑒 𝑠𝑔
𝑠ℎ𝑎𝑝𝑒 𝑠𝑔
0.048
0.052
0.033
0.055
0.019
0.038
Standard Error
Main effects
0.029
0.033
0.034
0.034
0.028
0.024
𝑎ℎ𝑡
0.073
0.023
𝑏 ℎ𝑡
0.054
0.028
𝜎 ℎ𝑡
0.044
0.028
𝑎 𝑠𝑤𝑖𝑡𝑐ℎ
0.046
0.035
𝑏 𝑠𝑤𝑖𝑡𝑐ℎ
0.056
0.034
𝛼 𝑎𝑠
𝛼 𝑎𝑔
𝑑𝑖𝑠. 𝑎𝑠
𝑑𝑖𝑠. 𝑎𝑔
𝐺. 𝑔
𝑃. 𝑔
𝐺. 𝑠
𝑃. 𝑠
𝑎𝑠ℎ𝑎𝑝𝑒
𝑎𝑠𝑐𝑎𝑙𝑒
𝑏𝑠ℎ𝑎𝑝𝑒
𝑏𝑠𝑐𝑎𝑙𝑒
0.029
0.017
0.038
0.02
0.057
0.037
0.716
<0.001
<0.001
<0.001
<0.001
<0.001
0.035
0.031
0.029
0.031
0.034
0.032
0.016
0.038
0.034
0.037
0.032
0.031
Parameter
Main effects
5
Caughlin T.T., Ferguson J.M., Lichstein J.W. ,Zuidema P.A., Bunyavejchewin S., and Levey D.J.
Table S4. Total effects from global sensitivity analysis. This table presents the bias-corrected
estimates of total effects of parameter uncertainty on total population size after 100 years from
simulated populations. Total effects represent the total contribution of parameter variance to
output variance, including interactions between parameters. See Text S2 for more details.
Submodel
Parameter
Initial distribution
Initial conditions
of trees
Seed dispersal
𝑢. 𝑑𝑖𝑠𝑝𝑒𝑟𝑠𝑎𝑙
𝑓𝑟
Seed production
𝛽0
𝑓𝑟
Seed production
𝛽1
𝑓𝑟
Seed production
𝛽2
Seed dispersal
𝐵𝑎𝑡ℎ
𝑔𝑒𝑟
Germination
𝜇0
Germination
𝛽 𝑔𝑒𝑟
Germination
𝛼 𝑔𝑒𝑟
Germination
𝑑𝑖𝑠. 𝑔𝑒𝑟
Germination
μ𝑖𝑛.ℎ𝑡
Germination
𝑠𝑐𝑎𝑙𝑒 𝑖𝑛.ℎ𝑡
Germination
𝑠ℎ𝑎𝑝𝑒 𝑖𝑛.ℎ𝑡
Seedling survival
𝜇0𝑠𝑠
Seedling survival
𝛽 𝑠𝑠
Seedling survival
𝑠𝑖𝑧𝑒 𝑠𝑠
𝑠𝑔
Seedling growth
𝜇0
Total effects
Standard Error
Total effects
0.025
0.005
0.025
0.023
0.024
0.026
0.038
0.061
0.029
0.044
0.052
0.027
0.027
0.028
0.057
0.025
0.035
0.027
0.004
0.004
0.005
0.004
0.006
0.011
0.005
0.008
0.01
0.005
0.005
0.006
0.01
0.004
0.005
0.005
6
Caughlin T.T., Ferguson J.M., Lichstein J.W. ,Zuidema P.A., Bunyavejchewin S., and Levey D.J.
Table S4. Total effects from global sensitivity analysis, cont.
Submodel
Seedling growth
Seedling growth
Seedling growth
Seedling growth
Seedling growth
Seedling growth
Seedling heightDBH allometry
Seedling heightDBH allometry
Seedling heightDBH allometry
Seedling heightDBH allometry
Seedling heightDBH allometry
Tree survival
Tree growth
Tree survival
Tree growth
Tree growth
Tree growth
Tree survival
Tree survival
Tree growth
Tree growth
Tree growth
Tree growth
𝛼 𝑠𝑔
𝑑𝑖𝑠. 𝑠𝑔
𝑠𝑖𝑧𝑒 𝑠𝑔
𝛽 𝑠𝑔
𝑠𝑐𝑎𝑙𝑒 𝑠𝑔
𝑠ℎ𝑎𝑝𝑒 𝑠𝑔
0.024
0.025
0.03
0.03
0.03
0.034
Standard Error
Total effects
0.004
0.005
0.005
0.007
0.007
0.007
𝑎ℎ𝑡
0.026
0.005
𝑏 ℎ𝑡
0.024
0.004
𝜎 ℎ𝑡
0.024
0.004
𝑎 𝑠𝑤𝑖𝑡𝑐ℎ
0.062
0.01
𝑏 𝑠𝑤𝑖𝑡𝑐ℎ
0.061
0.01
𝛼 𝑎𝑠
𝛼 𝑎𝑔
𝑑𝑖𝑠. 𝑎𝑠
𝑑𝑖𝑠. 𝑎𝑔
𝐺. 𝑔
𝑃. 𝑔
𝐺. 𝑠
𝑃. 𝑠
𝑎𝑠ℎ𝑎𝑝𝑒
𝑎𝑠𝑐𝑎𝑙𝑒
𝑏𝑠ℎ𝑎𝑝𝑒
𝑏𝑠𝑐𝑎𝑙𝑒
0.057
0.039
0.036
0.036
0.094
0.023
0.843
0.03
0.025
0.028
0.029
0.025
0.011
0.007
0.007
0.006
0.02
0.004
0.108
0.008
0.006
0.007
0.007
0.004
Parameter
Total effects
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