Bayesian meta-analysis of regional biomass factors for Quercus mongolica forests in South Korea
Supplementary Material
Tzeng Yih Lam
Xiaodong Li
Rae Hyun Kim
Kyeong Hak Lee
Yeong Mo Son
S1. WinBUGS Codes for Biomass Conversion and Expansion Factor (BCEF)
The followings are WinBUGS codes for modeling regional level BCEF under multilevel Bayesian Hierarchical Model (BHM) framework assuming three Probability Density Functions (PDFs).
S1.1 Gamma PDF
Model{
# Random effects model
for(i in 1:N){
bcef[i]~dgamma(a.hat[i],b.hat[i])
a.hat[i]<-a[study[i]]
b.hat[i]<-b[study[i]]
}
# Prior distributions for Gamma distribution parameters
for(j in 1:J){
a[j]~dgamma(A.a,B.a)
b[j]~dgamma(A.b,B.b)
}
# Prior distribution for hyper-parameters
A.a~dgamma(0.001,0.001)
B.a~dgamma(0.001,0.001)
A.b~dgamma(0.001,0.001)
B.b~dgamma(0.001,0.001)
# Outputs
for(j in 1:J){
bcef.pub[j]<-a[j]/b[j]
bcef.pub.var[j]<-a[j]/pow(b[j],2)
}
post.a<-A.a/B.a
post.b<-A.b/B.b
bcef.mean<-post.a/post.b
bcef.var<-post.a/pow(post.b,2)
}
# Data
list(N=52,J=12,
bcef=c(0.3872671, 0.4534884, 0.5130233, 0.9871638, 1.3583333, 0.8590283,
0.8136239, 1.3797368, 0.6769231, 1.0243902, 0.7144781, 0.7646552,
0.9, 1.3829787, 1.264697, 2.8872222, 1.1297826, 1.2640756, 1.0467337,
0.8224576, 1.255814, 0.8483051, 1.8363636, 1.0052632, 0.9778761,
1.0751773, 0.743314, 0.615, 1.8353933, 1.0866972, 2.216, 1.0290244,
1.1546667, 0.5260759, 1.1733333, 0.9492105, 1.2691782, 0.7871034,
0.8320918, 0.8839029, 1.0295852, 1.1604706, 0.7522128, 0.6790374,
0.1865383, 0.8012882, 0.6543203, 0.7005553, 0.8231579, 0.9083227,
1.0767969, 1.0498184),
study=c(1, 1, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7,
7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12,
12, 12, 12, 12, 12, 12, 12, 12, 12)
)
S1.2 Log-normal PDF
Model{
# Random effects model
for(i in 1:N){
bcef[i]~dlnorm(mu.hat[i],tau)
mu.hat[i]<-mu[study[i]]
}
# Prior distributions for Log-normal distribution parameters
for(j in 1:J){
mu[j]~dnorm(theta,tau.mu)
}
# Prior distribution for hyper-parameters
tau<-pow(sigma,-2)
sigma~dunif(0,100)
theta~dnorm(0,0.000001)
tau.mu<-pow(sigma.mu,-2)
sigma.mu~dunif(0,100)
# Outputs
inv.tau<-pow(tau,-1)
for(j in 1:J){
bcef.pub[j]<-exp(mu[j]+0.5*inv.tau)
bcef.pub.var[j]<-(exp(inv.tau)-1)*exp(2*mu[j]+inv.tau)
}
bcef.mean<-exp(theta+0.5*inv.tau)
bcef.var<-(exp(inv.tau)-1)*exp(2*theta+inv.tau)
}
# Data
list(N=52,J=12,
bcef=c(0.3872671, 0.4534884, 0.5130233, 0.9871638, 1.3583333, 0.8590283,
0.8136239, 1.3797368, 0.6769231, 1.0243902, 0.7144781, 0.7646552,
0.9, 1.3829787, 1.264697, 2.8872222, 1.1297826, 1.2640756, 1.0467337,
0.8224576, 1.255814, 0.8483051, 1.8363636, 1.0052632, 0.9778761,
1.0751773, 0.743314, 0.615, 1.8353933, 1.0866972, 2.216, 1.0290244,
1.1546667, 0.5260759, 1.1733333, 0.9492105, 1.2691782, 0.7871034,
0.8320918, 0.8839029, 1.0295852, 1.1604706, 0.7522128, 0.6790374,
0.1865383, 0.8012882, 0.6543203, 0.7005553, 0.8231579, 0.9083227,
1.0767969, 1.0498184),
study=c(1, 1, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7,
7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12,
12, 12, 12, 12, 12, 12, 12, 12, 12)
)
S1.3 Weibull PDF
Model{
# Random effects model
for(i in 1:N){
bcef[i]~dweib(v.hat[i],lambda.hat[i])
v.hat[i]<-v[study[i]]
lambda.hat[i]<-lambda[study[i]]
}
# Prior distributions for Weibull distribution parameters
for(j in 1:J){
v[j]~dgamma(A.v,B.v)
lambda[j]~dgamma(A.lambda,B.lambda)
}
# Prior distribution for hyper-parameters
A.v~dgamma(0.001,0.001)
B.v~dgamma(0.001,0.001)
A.lambda~dgamma(0.001,0.001)
B.lambda~dgamma(0.001,0.001)
# Outputs
for(j in 1:J){
inv.v[j]<-pow(v[j],-1)
bcef.pub[j]<-pow(lambda[j],-inv.v[j])*exp(loggam(1+inv.v[j]))
bcef.pub.var[j]<-(exp(loggam(1+2*inv.v[j]))-pow(exp(loggam(1+inv.v[j])),2))*
pow(lambda[j],-2*inv.v[j])
}
post.lambda<-A.lambda/B.lambda
post.v<-A.v/B.v
inv.post.v<-pow(post.v,-1)
bcef.mean<-pow(post.lambda,-inv.post.v)*exp(loggam(1+inv.post.v))
bcef.var<-(exp(loggam(1+2*inv.post.v))-pow(exp(loggam(1+inv.post.v)),2))*
pow(post.lambda,-2*inv.post.v)
}
# Data
list(N=52,J=12,
bcef=c(0.3872671, 0.4534884, 0.5130233, 0.9871638, 1.3583333, 0.8590283,
0.8136239, 1.3797368, 0.6769231, 1.0243902, 0.7144781, 0.7646552,
0.9, 1.3829787, 1.264697, 2.8872222, 1.1297826, 1.2640756, 1.0467337,
0.8224576, 1.255814, 0.8483051, 1.8363636, 1.0052632, 0.9778761,
1.0751773, 0.743314, 0.615, 1.8353933, 1.0866972, 2.216, 1.0290244,
1.1546667, 0.5260759, 1.1733333, 0.9492105, 1.2691782, 0.7871034,
0.8320918, 0.8839029, 1.0295852, 1.1604706, 0.7522128, 0.6790374,
0.1865383, 0.8012882, 0.6543203, 0.7005553, 0.8231579, 0.9083227,
1.0767969, 1.0498184),
study=c(1, 1, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7,
7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12,
12, 12, 12, 12, 12, 12, 12, 12, 12)
)
S2. WinBUGS Codes for Biomass Expansion Factor (BEF)
The followings are WinBUGS codes for modeling regional level BEF under multilevel Bayesian Hierarchical Model (BHM) framework assuming three Probability Density Functions (PDFs).
S2.1 Gamma PDF
Model{
# Random effects model
for(i in 1:N){
sbef[i]~dgamma(a.hat[i],b.hat[i])
a.hat[i]<-a[study[i]]
b.hat[i]<-b[study[i]]
}
# Prior distributions for Gamma distribution parameters
for(j in 1:J){
a[j]~dgamma(A.a,B.a)
b[j]~dgamma(A.b,B.b)
}
# Prior distribution for hyper-parameters
A.a~dgamma(0.001,0.001)
B.a~dgamma(0.001,0.001)
A.b~dgamma(0.001,0.001)
B.b~dgamma(0.001,0.001)
# Outputs
for(j in 1:J){
sbef.pub[j]<-a[j]/b[j]
sbef.pub.var[j]<-a[j]/pow(b[j],2)
}
post.a<-A.a/B.a
post.b<-A.b/B.b
sbef.mean<-post.a/post.b
sbef.var<-post.a/pow(post.b,2)
}
# Data
list(N=40,J=11,
sbef=c(0.4459647, 0.3270949, 0.7042645, 0.4194682, 0.3892045, 0.3785083,
0.2576757, 0.2809675, 0.3273002, 0.4334471, 0.3797139, 0.258156,
0.4315436, 0.1926606, 0.3566475, 0.3230652, 0.2885656, 0.2915897,
0.292745, 0.246013, 0.2433597, 0.1308844, 0.4380186, 0.7746617,
0.9052, 0.6962705, 0.2389068, 0.5028143, 0.1565736, 0.8610043,
0.5901667, 0.4062117, 0.4206047, 0.9849434, 0.3440617, 1.0287489,
0.8800643, 0.9973532, 0.6743402, 0.7825875),
study=c(1, 1, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 9,
10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11)
)
S2.2 Log-normal PDF
Model{
# Random effects model
for(i in 1:N){
sbef[i]~dlnorm(mu.hat[i],tau)
mu.hat[i]<-mu[study[i]]
}
# Prior distributions for Log-normal distribution parameters
for(j in 1:J){
mu[j]~dnorm(theta,tau.mu)
}
# Prior distribution for hyper-parameters
tau<-pow(sigma,-2)
sigma~dunif(0,100)
theta~dnorm(0,0.000001)
tau.mu<-pow(sigma.mu,-2)
sigma.mu~dunif(0,100)
# Outputs
inv.tau<-pow(tau,-1)
for(j in 1:J){
sbef.pub[j]<-exp(mu[j]+0.5*inv.tau)
sbef.pub.var[j]<-(exp(inv.tau)-1)*exp(2*mu[j]+inv.tau)
}
sbef.mean<-exp(theta+0.5*inv.tau)
sbef.var<-(exp(inv.tau)-1)*exp(2*theta+inv.tau)
}
# Data
list(N=40,J=11,
sbef=c(0.4459647, 0.3270949, 0.7042645, 0.4194682, 0.3892045, 0.3785083,
0.2576757, 0.2809675, 0.3273002, 0.4334471, 0.3797139, 0.258156,
0.4315436, 0.1926606, 0.3566475, 0.3230652, 0.2885656, 0.2915897,
0.292745, 0.246013, 0.2433597, 0.1308844, 0.4380186, 0.7746617,
0.9052, 0.6962705, 0.2389068, 0.5028143, 0.1565736, 0.8610043,
0.5901667, 0.4062117, 0.4206047, 0.9849434, 0.3440617, 1.0287489,
0.8800643, 0.9973532, 0.6743402, 0.7825875),
study=c(1, 1, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 9,
10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11)
)
S2.3 Weibull PDF
Model{
# Random effects model
for(i in 1:N){
sbef[i]~dweib(v.hat[i],lambda.hat[i])
v.hat[i]<-v[study[i]]
lambda.hat[i]<-lambda[study[i]]
}
# Prior distributions for Weibull distribution parameters
for(j in 1:J){
v[j]~dgamma(A.v,B.v)
lambda[j]~dgamma(A.lambda,B.lambda)
}
# Prior distribution for hyper-parameters
A.v~dgamma(0.001,0.001)
B.v~dgamma(0.001,0.001)
A.lambda~dgamma(0.001,0.001)
B.lambda~dgamma(0.001,0.001)
# Outputs
for(j in 1:J){
inv.v[j]<-pow(v[j],-1)
sbef.pub[j]<-pow(lambda[j],-inv.v[j])*exp(loggam(1+inv.v[j]))
sbef.pub.var[j]<-(exp(loggam(1+2*inv.v[j]))-pow(exp(loggam(1+inv.v[j])),2))*
pow(lambda[j],-2*inv.v[j])
}
post.lambda<-A.lambda/B.lambda
post.v<-A.v/B.v
inv.post.v<-pow(post.v,-1)
sbef.mean<-pow(post.lambda,-inv.post.v)*exp(loggam(1+inv.post.v))
sbef.var<-(exp(loggam(1+2*inv.post.v))-pow(exp(loggam(1+inv.post.v)),2))*
pow(post.lambda,-2*inv.post.v)
}
# Data
list(N=40,J=11,
sbef=c(0.4459647, 0.3270949, 0.7042645, 0.4194682, 0.3892045, 0.3785083,
0.2576757, 0.2809675, 0.3273002, 0.4334471, 0.3797139, 0.258156,
0.4315436, 0.1926606, 0.3566475, 0.3230652, 0.2885656, 0.2915897,
0.292745, 0.246013, 0.2433597, 0.1308844, 0.4380186, 0.7746617,
0.9052, 0.6962705, 0.2389068, 0.5028143, 0.1565736, 0.8610043,
0.5901667, 0.4062117, 0.4206047, 0.9849434, 0.3440617, 1.0287489,
0.8800643, 0.9973532, 0.6743402, 0.7825875),
study=c(1, 1, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 9,
10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11)
)
S3. WinBUGS Codes for Root-to-Shoot Ratio (RSR)
The followings are WinBUGS codes for modeling regional level R under multilevel Bayesian Hierarchical Model (BHM) framework assuming two Probability Density Functions (PDFs).
S3.1 Gamma PDF
Model{
# Random effects model
for(i in 1:N){
r[i]~dgamma(a.hat[i],b.hat[i])
a.hat[i]<-a[study[i]]
b.hat[i]<-b[study[i]]
}
# Prior distributions for Gamma distribution parameters
for(j in 1:J){
a[j]~dgamma(A.a,B.a)
b[j]~dgamma(A.b,B.b)
}
# Prior distribution for hyper-parameters
A.a~dgamma(0.001,0.001)
B.a~dgamma(0.001,0.001)
A.b~dgamma(0.001,0.001)
B.b~dgamma(0.001,0.001)
# Outputs
for(j in 1:J){
r.pub[j]<-a[j]/b[j]
r.pub.var[j]<-a[j]/pow(b[j],2)
}
post.a<-A.a/B.a
post.b<-A.b/B.b
r.mean<-post.a/post.b
r.var<-post.a/pow(post.b,2)
}
# Data
list(N=47,J=9,
r=c(0.2147239, 0.1919597, 0.196933, 0.2633352, 0.2022727, 0.1922619,
0.1918002, 0.1967306, 0.2034693, 0.1917949, 0.151831, 0.1636521,
0.181547, 0.2110022, 0.193951, 0.1849562, 0.1958333, 0.1998002,
0.1764176, 0.1874346, 0.2633484, 0.2631926, 0.1521314, 0.1645146,
0.1665136, 0.2401857, 0.1407942, 0.19586, 0.2048829, 0.3267565,
0.3140092, 0.1421093, 0.2064128, 0.0953246, 0.3292643, 0.325117,
1.1195047, 0.5079447, 0.1699677, 0.1456458, 0.9726821, 0.3164511,
1.0467703, 0.4526007, 0.7343642, 0.155599, 0.1425218),
study=c(1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5,
5, 5, 5, 6, 6, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9,
9)
)
S3.2 Log-normal PDF
Model{
# Random effects model
for(i in 1:N){
r[i]~dlnorm(mu.hat[i],tau)
mu.hat[i]<-mu[study[i]]
}
# Prior distributions for Log-normal distribution parameters
for(j in 1:J){
mu[j]~dnorm(theta,tau.mu)
}
# Prior distribution for hyper-parameters
tau<-pow(sigma,-2)
sigma~dunif(0,100)
theta~dnorm(0,0.000001)
tau.mu<-pow(sigma.mu,-2)
sigma.mu~dunif(0,100)
# Outputs
inv.tau<-pow(tau,-1)
for(j in 1:J){
r.pub[j]<-exp(mu[j]+0.5*inv.tau)
r.pub.var[j]<-(exp(inv.tau)-1)*exp(2*mu[j]+inv.tau)
}
r.mean<-exp(theta+0.5*inv.tau)
r.var<-(exp(inv.tau)-1)*exp(2*theta+inv.tau)
}
# Data
list(N=47,J=9,
r=c(0.2147239, 0.1919597, 0.196933, 0.2633352, 0.2022727, 0.1922619,
0.1918002, 0.1967306, 0.2034693, 0.1917949, 0.151831, 0.1636521,
0.181547, 0.2110022, 0.193951, 0.1849562, 0.1958333, 0.1998002,
0.1764176, 0.1874346, 0.2633484, 0.2631926, 0.1521314, 0.1645146,
0.1665136, 0.2401857, 0.1407942, 0.19586, 0.2048829, 0.3267565,
0.3140092, 0.1421093, 0.2064128, 0.0953246, 0.3292643, 0.325117,
1.1195047, 0.5079447, 0.1699677, 0.1456458, 0.9726821, 0.3164511,
1.0467703, 0.4526007, 0.7343642, 0.155599, 0.1425218),
study=c(1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5,
5, 5, 5, 6, 6, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9,
9)
)
Table
Table S1 Stem and above-ground biomass (ton/ha) for individual study site of each publication used for meta-analysis of BCEF, BEF and RSR.
Publication / Site No. / Stem Biomass(ton/ha) / Aboveground Biomass (ton/ha)
Kwon and Lee 2006a / 1 / 132.6 / 176
Kwon and Lee 2006a / 2 / 117.2 / 168
Kwon and Lee 2006a / 3 / 153.8 / 212.2
Kwon and Lee 2006a / 4 / 141 / 177.4
Kwon and Lee 2006a / 5 / 149 / 213.3
Kwon and Lee 2006a / 6 / 163.5 / 195
Kwon and Lee 2006b / 1 / 184.58 / 250.41
Kwon and Lee 2006b / 2 / 157.12 / 207.88
Kwon and Lee 2006b / 3 / 241.99 / 311.82
Kwon and Lee 2006b / 4 / 232.93 / 300.85
Kwon and Lee 2006c / 1 / n/aa / 208.3
Kwon and Lee 2006c / 2 / n/aa / 194.1
Kwon and Lee 2006c / 3 / n/aa / 216
Kwon and Lee 2006c / 4 / n/aa / 200.2
Kwon and Lee 2006c / 5 / n/aa / 222.2
Kwon and Lee 2006c / 6 / n/aa / 191
Kwon and Lee 2006c / 7 / n/aa / 110.5
Kwon and Lee 2006c / 8 / n/aa / 151.6
Kwon and Lee 2006c / 9 / n/aa / 255.7
Kwon and Lee 2006c / 10 / n/aa / 209.1
Kwon and Lee 2006c / 11 / n/aa / 326.7
Kwon and Lee 2006c / 12 / n/aa / 236.9
Lee and Kwon 2006 / 1 / 153.92 / 212.18
Lee and Kwon 2006 / 2 / 141.03 / 177.37
Lee and Kwon 2006 / 3 / 122.79 / 157.29
Lee and Park 1987 / 1 / 78.33 / 112.64
Park 2003 / 1 / 43.12 / 62.35
Park 2003 / 2 / 117.55 / 156
Park 2003 / 3 / 64.72 / 110.3
Park and Moon 1994 / 1 / 70.4 / 97.8
Park et al. 1996 / 1 / 40.65 / 72.14
Park et al. 2005a / 1 / 188.56 / 243.76
Park et al. 2005a / 2 / 101.58 / 126.57
Park et al. 2005a / 3 / 97.51 / 121.24
Park et al. 2005b / 1 / 73.5 / 83.12
Song and Lee 1996 / 1 / 91.02 / 129.2
KFRI 2010 / 1 / 86.44 / 164.68
KFRI 2010 / 2 / 111.91 / 189.83
KFRI 2010 / 3 / 182.18 / 225.7
KFRI 2010 / 4 / 82.97 / 124.69
KFRI 2010 / 5 / 228.79 / 264.61
KFRI 2010 / 6 / 20.48 / 38.11
KFRI 2010 / 7 / 97.6 / 155.2
KFRI 2010 / 8 / 77.27 / 108.66
KFRI 2010 / 9 / 195.15 / 277.24
KFRI 2010 / 10 / 68.63 / 136.22
KFRI 2010 / 11 / 91.82 / 123.41
KFRI 2010 / 12 / 39.9 / 80.94
KFRI 2010 / 13 / 54.16 / 101.82
KFRI 2010 / 14 / 54.78 / 109.41
KFRI 2010 / 15 / 59.38 / 99.43
KFRI 2010 / 16 / 92.14 / 164.25
aStem biomass was not available.
Table S2 Median, 2.5% and 97.5% sample quantiles of hyperparameters for each fitted PDF for Biomass Conversion and Expansion Factor (BCEF).
Median / 2.5% Quantile / 97.5% QuantileGamma
Aα / 61.52 / 6.058 / 1522.0
Bα / 7.798 / 0.694 / 202.5
Aβ / 26.04 / 4.276 / 693.6
Bβ / 3.393 / 0.439 / 97.98
Log-normal
θ / -0.0570 / -0.2954 / 0.1854
φ / 11.18 / 2.731 / 132.7
Weibull
Av / 39.34 / 3.129 / 978.1
Bv / 13.0 / 0.8783 / 348.6
Aλ / 2.423 / 0.5187 / 111.9
Bλ / 2.837 / 0.3 / 144.0
Table S3 Median, 2.5% and 97.5% sample quantiles of hyperparameters for each PDF for Biomass Expansion Factor (BEF).
Median / 2.5% Quantile / 97.5% QuantileGamma
Aα / 93.52 / 7.208 / 1287.0
Bα / 15.45 / 1.006 / 219.4
Aβ / 11.55 / 2.319 / 261.7
Bβ / 0.7132 / 0.106 / 27.35
Log-normal
θ / -0.9846 / -1.29 / -0.6992
φ / 9.382 / 1.773 / 84.97
Weibull
Av / 95.16 / 6.333 / 1185.0
Bv / 28.98 / 1.568 / 379.7
Aλ / 1.064 / 0.3011 / 4.077
Bλ / 0.0393 / 0.0011 / 0.5047
Table S4 Median, 2.5% and 97.5% sample quantiles of hyperparameters for each PDF for Root-to-Shoot Ratio (RSR).
Median / 2.5% Quantile / 97.5% QuantileGamma
Aα / 1.342 / 0.397 / 4.577
Bα / 0.0393 / 0.0040 / 0.3034
Aβ / 1.1 / 0.3458 / 3.244
Bβ / 0.0070 / 0.0009 / 0.0451
Log-normal
θ / -1.484 / -1.736 / -1.242
φ / 24.99 / 3.293 / 2235.0