Ecological Archives E093-197-A4

Joshua M. Rapp, Miles R. Silman, James S. Clark, Cecile A. J. Girardin, Darcy Galiano, and Richard Tito. Year. Intra- and inter-specific tree growth across a long altitudinal gradient in the Peruvian Andes. Ecology 93:2061–2072. http://dx.doi.org/10.1890/11-1725.1

Appendix D. Parameter estimates from the Bayesian analysis of tree growth.


Fig. D1. Markov Chain Monte Carlo chains of eight parameters for the model run of Weinmannia bangii. The parameters shown are the mean diameter growth rate (intercept), the effects of the three covariates (canopy status, dbh, and altitude), individual effects (ind), diameter error (diam), increment error (incr), and process error (proc). The acceptance rate of estimates is also shown. Chains converged quickly (after 1–2000 steps), but the first 10000 steps were removed before calculation of mean effects. Convergence was similar for other species.



Fig. D2. Diameter and growth rates (diameter increment) for five example W. bangii trees with different amounts and combinations of data. Shown are observations (circles - diameter measurement from censuses in left column; diameter increment from dendrometers in right), predictive median (solid lines), and 95% predictive intervals (dashed lines). Each tree is labeled with the altitude of the plot it came from and a unique identification number.



Fig. D3. Magnitude and distribution of five sources of variability included in the Bayesian model (increment and diameter measurement error, process error, and individual variability, and year effect) for W. bangii. Patterns were similar for each species (see Tables D1–D9 for parameter values for each individual species).



Fig. D4. Year effects from the Bayesian model. Points represent the mean posterior parameter value, while lines depict the 95% credible interval of the year effect. Numbers above each line indicate the number of stems from each year included in the analysis.


Table D1. Prior and Posterior parameter estimates for Weinmannia microphylla.

Parameter No.
obs.
Posterior
mean
Bayesian
SE
Credible interval Prior
parameter
values
Prior
mean
2.5% 97.5% par1 par2
Intercept   869 0.335 0.196 -0.0451 0.727 0 10 0
Random
effect
Individual 260 0.0107 0.00124 0.00847 0.0133 2 0.4 0.4
Error
sources
Diameter 776 0.0669 0.00252 0.0623 0.0721 1550 62 0.04
Increment 157 0.00051 0.0000465 0.000427 0.000608 157 0.0624 0.0004
Process 869 0.00308 0.000352 0.00242 0.0038 10 0.09 0.01
Fixed
year
effects
2003 87 -0.00505 0.0122 -0.029 0.0187 NA NA NA
2004 94 -0.00387 0.0118 -0.0271 0.0194 NA NA NA
2005 94 -0.00501 0.0117 -0.0278 0.0178 NA NA NA
2006 146 0.00366 0.0099 -0.0156 0.0231 NA NA NA
2007 227 0.00556 0.00866 -0.0114 0.0226 NA NA NA
2008 221 0.00472 0.00868 -0.0122 0.0217 NA NA NA
Covariates Canopy status 869 0.00279 0.0089 -0.0148 0.0204 0 10 0
DBH 869 -0.00378 0.00519 -0.0139 0.00634 0 10 0
Altitude 869 -0.0578 0.0591 -0.176 0.0571 0 10 0


Table D2. Prior and Posterior parameter estimates for Weinmannia crassifolia.

Parameter No.
obs.
Posterior
mean
Bayesian
SE
Credible interval Prior
parameter
values
Prior
mean
2.5% 97.5% par1 par2
Intercept   3040 0.115 0.0078 0.1 0.131 0 10 0
Random
effect
Individual 811 0.00442 0.000321 0.00383 0.0051 2 0.4 0.4
Error
sources
Diameter 2510 0.0518 0.00112 0.0497 0.054 5020 201 0.04
Increment 119 0.000468 0.0000472 0.00038 0.00057 119 0.0472 0.0004
Process 3040 0.000838 0.0000605 0.00072 0.00096 10 0.09 0.01
Fixed
year
effects
2003 311 -0.00506 0.00395 -0.0128 0.00269 NA NA NA
2004 311 -0.00506 0.00399 -0.0128 0.00273 NA NA NA
2005 311 -0.00506 0.00397 -0.0128 0.00266 NA NA NA
2006 666 0.0052 0.003 -0.0007 0.0111 NA NA NA
2007 731 0.00528 0.00291 -0.0004 0.011 NA NA NA
2008 714 0.0047 0.00294 -0.001 0.0105 NA NA NA
Covariates Canopy status 3040 0.0181 0.00354 0.0109 0.0249 0 10 0
DBH 3040 -0.0286 0.00394 -0.0362 -0.0208 0 10 0


Table D3. Prior and Posterior parameter estimates for Weinmannia bangii.

Parameter No.
obs.
Posterior
mean
Bayesian
SE
Credible interval Prior
parameter
values
Prior
mean
2.5% 97.5% par1 par2
Intercept   1780 -0.275 0.0443 -0.362 -0.188 0 10 0
Random
effect
Individual 427 0.0114 0.00104 0.00959 0.0137 2 0.4 0.4
Error
sources
Diameter 1410 0.0517 0.00139 0.0491 0.0546 2820 113 0.04
Increment 310 0.000458 0.0000284 0.000405 0.000517 310 0.124 0.0004
Process 1780 0.00162 0.000137 0.00136 0.0019 10 0.09 0.01
Fixed
year
effects
2003 236 -0.00151 0.00713 -0.0154 0.0124 NA NA NA
2004 242 -0.00461 0.00698 -0.0183 0.00907 NA NA NA
2005 242 -0.00513 0.00705 -0.019 0.0087 NA NA NA
2006 328 0.00522 0.00625 -0.00703 0.0175 NA NA NA
2007 370 0.00461 0.00599 -0.00718 0.0164 NA NA NA
2008 364 0.00142 0.00607 -0.0106 0.0134 NA NA NA
Covariates Canopy status 1780 0.0402 0.00617 0.028 0.0522 0 10 0
DBH 1780 -0.0125 0.0042 -0.0207 -0.00422 0 10 0
Altitude 1780 0.124 0.0138 0.0974 0.151 0 10 0


Table D4. Prior and Posterior parameter estimates for Weinmannia reticulata.

Parameter No.
obs.
Posterior
mean
Bayesian
SE
Credible interval Prior
parameter
values
Prior
mean
2.5% 97.5% par1 par2
Intercept   1700 0.0256 0.0532 -0.0793 0.131 0 10 0
Random
effect
Individual 490 0.00892 0.000758 0.00753 0.0105 2 0.4 0.4
Error
sources
Diameter 1600 0.0491 0.00129 0.0466 0.0516 3200 128 0.04
Increment 277 0.000428 0.0000275 0.000378 0.000485 277 0.11 0.0004
Process 1700 0.00146 0.000126 0.00123 0.00172 10 0.09 0.01
Fixed
year
effects
2003 169 -0.00315 0.00747 -0.0178 0.0115 NA NA NA
2004 170 -0.00491 0.00741 -0.0196 0.00957 NA NA NA
2005 170 -0.00468 0.00745 -0.0192 0.00984 NA NA NA
2006 307 0.00176 0.006 -0.0101 0.0136 NA NA NA
2007 450 0.00506 0.00529 -0.00526 0.0154 NA NA NA
2008 434 0.00593 0.00526 -0.00442 0.0163 NA NA NA
Covariates Canopy status 1700 0.0253 0.00607 0.0134 0.0372 0 10 0
DBH 1700 -0.0177 0.00419 -0.0259 -0.00953 0 10 0
Altitude 1700 0.0373 0.0192 -0.00033 0.0752 0 10 0


Table D5. Prior and Posterior parameter estimates for Weinmannia mariquitae.

Parameter No.
obs.
Posterior
mean
Bayesian
SE
Credible interval Prior
parameter
values
Prior
mean
2.5% 97.5% par1 par2
Intercept   92 0.114 0.0695 -0.0215 0.252 0 10 0
Random
effect
Individual 25 0.0452 0.015 0.0244 0.082 2 0.4 0.4
Error
sources
Diameter 76 0.0609 0.00781 0.0477 0.078 152 6.04 0.04
Increment 23 0.000392 0.0000833 0.00026 0.00059 23 0.0088 0.0004
Process 92 0.00401 0.000781 0.00277 0.00583 10 0.09 0.01
Fixed
year
effects
2003 13 0.000294 0.0567 -0.112 0.112 NA NA NA
2004 13 -0.00403 0.0571 -0.116 0.11 NA NA NA
2005 13 -0.00359 0.0572 -0.116 0.11 NA NA NA
2006 17 0.0143 0.0523 -0.0891 0.118 NA NA NA
2007 18 -0.0074 0.0505 -0.106 0.0928 NA NA NA
2008 18 0.000453 0.0503 -0.0978 0.101 NA NA NA
Covariates Canopy status 92 0.0535 0.0399 -0.0254 0.133 0 10 0
DBH 92 -0.0564 0.0392 -0.135 0.0193 0 10 0


Table D6. Prior and Posterior parameter estimates for Weinmannia multijuga.

Parameter No.
obs.
Posterior
mean
Bayesian
SE
Credible interval Prior
parameter
values
Prior
mean
2.5% 97.5% par1 par2
Intercept   315 0.192 0.0389 0.118 0.27 0 10 0
Random
effect
Individual 96 0.0284 0.00537 0.0197 0.0405 2 0.4 0.4
Error
sources
Diameter 304 0.0614 0.00372 0.0547 0.0691 608 24.3 0.04
Increment 63 0.000469 0.0000666 0.000355 0.00062 63 0.0248 0.0004
Process 315 0.00625 0.00123 0.0043 0.00897 10 0.09 0.01
Fixed
year
effects
2003 33 -0.0042 0.0317 -0.0664 0.0579 NA NA NA
2004 33 -0.00772 0.0313 -0.0694 0.0532 NA NA NA
2005 33 -0.00718 0.0312 -0.0686 0.0544 NA NA NA
2006 49 0.00841 0.0269 -0.0442 0.0616 NA NA NA
2007 85 0.0158 0.0223 -0.0278 0.0597 NA NA NA
2008 82 -0.00511 0.0224 -0.0491 0.0392 NA NA NA
Covariates Canopy status 315 0.0411 0.0207 3.32E-05 0.0815 0 10 0
DBH 315 -0.0196 0.0129 -0.045 0.00549 0 10 0


Table D7. Prior and Posterior parameter estimates for Weinmannia ovata.

Parameter No.
obs.
Posterior
mean
Bayesian
SE
Credible interval Prior
parameter
values
Prior
mean
2.5% 97.5% par1 par2
Intercept   626 0.163 0.0169 0.131 0.197 0 10 0
Random
effect
Individual 209 0.00921 0.00115 0.00722 0.0117 2 0.4 0.4
Error
sources
Diameter 685 0.0433 0.00126 0.0409 0.0459 1370 54.8 0.04
Increment 58 0.000436 0.0000616 0.00033 0.00057 58 0.0228 0.0004
Process 0.00201 0.000275 0.00155 0.00262 10 0.09 0.01 0.01
Fixed
year
effects
2003 44 -0.00332 0.0152 -0.0332 0.0266 NA NA NA
2004 44 -0.00274 0.0153 -0.0323 0.0273 NA NA NA
2005 44 -0.00318 0.0152 -0.0329 0.0265 NA NA NA
2006 84 0.00282 0.0121 -0.0211 0.0267 NA NA NA
2007 208 0.00508 0.00924 -0.013 0.0233 NA NA NA
2008 202 0.00133 0.00927 -0.0167 0.0195 NA NA NA
Covariates Canopy status 626 0.0111 0.00902 -0.0066 0.0288 0 10 0
DBH 626 -0.0612 0.0125 -0.0864 -0.0369 0 10 0


Table D8. Prior and Posterior parameter estimates for Weinmannia lechleriana.

Parameter No.
obs.
Posterior
mean
Bayesian
SE
Credible interval Prior
parameter
values
Prior
mean
2.5% 97.5% par1 par2
Intercept   288 0.591 0.235 0.139 1.06 0 10 0
Random
effect
Individual 109 0.0182 0.00314 0.013 0.0251 2 0.4 0.4
Error
sources
Diameter 351 0.0478 0.00238 0.0434 0.0527 702 28 0.04
Increment 24 0.000416 0.0000908 0.000275 0.000624 24 0.0092 0.0004
Process 288 0.00562 0.000969 0.004 0.00777 10 0.09 0.01
Fixed
year
effects
2003 10 -0.000688 0.0466 -0.0919 0.0911 NA NA NA
2004 10 -0.00903 0.0466 -0.101 0.0824 NA NA NA
2005 10 -0.00823 0.0468 -0.102 0.0829 NA NA NA
2006 48 0.0106 0.0273 -0.0431 0.0644 NA NA NA
2007 106 0.0115 0.023 -0.0336 0.0571 NA NA NA
2008 104 -0.00407 0.0227 -0.0487 0.0405 NA NA NA
Covariates Canopy status 288 -0.00293 0.0238 -0.0479 0.0452 0 10 0
DBH 288 -0.00482 0.0248 -0.0536 0.0437 0 10 0
Altitude 288 -0.193 0.122 -0.434 0.0434 0 10 0


Table D9. Prior and Posterior parameter estimates for Weinmannia pinnata.

Parameter No.
obs.
Posterior
mean
Bayesian
SE
Credible interval Prior
parameter
values
Prior
mean
2.5% 97.5% par1 par2
Intercept   83 0.16 0.0865 -0.0088 0.331 0 10 0
Random
effect
Individual 36 0.0371 0.0102 0.0219 0.0613 2 0.4 0.4
Error
sources
Diameter 103 0.0499 0.00505 0.0409 0.0607 206 8.2 0.04
Increment 6 0.000416 0.000202 0.00018 0.00094 6 0.002 0.0004
Process 83 0.00655 0.0016 0.00408 0.0103 10 0.09 0.01
Fixed
year
effects
2003 1 -0.0414 0.189 -0.416 0.328 NA NA NA
2004 1 -0.0384 0.189 -0.409 0.336 NA NA NA
2005 1 -0.0411 0.189 -0.415 0.329 NA NA NA
2006 10 0.0322 0.0908 -0.145 0.212 NA NA NA
2007 35 0.0536 0.0749 -0.0927 0.202 NA NA NA
2008 35 0.0352 0.0737 -0.108 0.182 NA NA NA
Covariates Canopy status 83 0.0706 0.0617 -0.0457 0.196 0 10 0
DBH 83 -0.0448 0.0325 -0.108 0.0197 0 10 0


Table D10. Prior and Posterior parameter estimates for across species analysis (excluding W. ovata).

Parameter No.
obs.
Posterior
mean
Bayesian
SE
Credible interval Prior
parameter
values
Prior
mean
2.5% 97.5% par1 par2
Intercept   8170 0.234 0.032 0.171 0.298 0 10 0
Random
effects
Individual 2250 0.0064 0.000285 0.00586 0.00699 2 0.4 0.4
Species 8 0.0802 0.0404 0.0341 0.183 2 0.4 0.4
pinnata 83 -0.000653 0.00498 -0.0104 0.00911 NA NA NA
lechleriana 288 -0.000161 0.00271 -0.00547 0.00515 NA NA NA
multijuga 315 0.000964 0.00255 -0.00403 0.00596 NA NA NA
bangii 1780 0.000497 0.00107 -0.0016 0.00259 NA NA NA
reticulata 1700 0.000204 0.00109 -0.00194 0.00235 NA NA NA
mariquitae 92 -0.00109 0.00469 -0.0103 0.00811 NA NA NA
crassifolia 3040 -0.0001 0.000827 -0.00172 0.00152 NA NA NA
microphylla 869 0.000387 0.00154 -0.00262 0.0034 NA NA NA
Error
sources
Diameter 7130 0.0532 0.000673 0.052 0.0546 14300 570 0.04
Increment 979 0.000483 0.0000174 0.000451 0.00052 979 0.391 0.0004
Process 8170 0.00102 0.0000558 0.000914 0.00113 10 0.09 0.01
Fixed
year
effects
2003 860 -0.0043 0.00909 -0.0224 0.0138 NA NA NA
2004 874 -0.00534 0.00897 -0.0232 0.0124 NA NA NA
2005 874 -0.00557 0.00905 -0.0235 0.0124 NA NA NA
2006 1570 0.00503 0.00717 -0.00904 0.0193 NA NA NA
2007 2020 0.00577 0.00667 -0.00745 0.019 NA NA NA
2008 1970 0.00441 0.00664 -0.00884 0.0174 NA NA NA
Covariates Canopy status 8170 0.0292 0.00544 0.0184 0.0399 0 10 0
DBH 8170 -0.014 0.00405 -0.022 -0.006 0 10 0
Species median altitude 8170 -0.0459 0.0106 -0.067 -0.0251 0 10 0

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