Table 1: Poisson Regression Model Output: Summary Statistics of the Posterior Samples

Tables

Table 1: Poisson regression model output: summary statistics of the posterior samples of diversity measurements. The posterior means and medians are used as the estimated diversity measurements and the 95% credible intervals (95% CI) give ranges of the diversity measurements associated with 0.95 inclusion probability.The diversity measurements computed based on the observed counts of different species of butterflies in different locations is also listed to compare with the model estimated diversity measurements. The indicator variable “violation” shows whether the credible interval covers the observed value, 1 stands for failed to cover and 0 means that the observed value is covered. The potential scale reproduction factor, R-hat, measures the convergence of the posterior samples. And R-hat=1.0 indicates that convergence is achieved and the resulting Bayesian estimates are reliable. βA is the additive model of species richness and βM is the multiplicative model.

MEAN / MEDIAN / 95 % CI / OBSERVED / VIOLATION / R-hat
αpatch / 19.43 / 19.42 / (18.12, 20.77) / 16.69 / 1 / 1.0
αcluster / 33.56 / 33.50 / (31.17, 36.00) / 31.67 / 0 / 1.0
γ / 47.13 / 47.00 / (44.00, 49.00) / 49.00 / 0 / 1.0
βApatch / 14.14 / 14.13 / (12.35, 15.96) / 14.93 / 0 / 1.0
βAcluster / 13.57 / 13.67 / (10.83, 16.17) / 17.33 / 1 / 1.0
βMpatch / 1.73 / 1.73 / (1.63, 1.83) / 1.90 / 1 / 1.0
βMcluster / 1.41 / 1.41 / (1.31, 1.51) / 1.55 / 1 / 1.0
DIC = 2840.750

Table 2: ZIP model (species-specific priors for the mixture proportion) output: summary statistics of the posterior samples of diversity measurements. The posterior means and medians are both considered as the estimated diversity measurements and the 95% credible intervals (95% CI) give ranges of the diversity measurements associated with 0.95 inclusion probability.The diversity measurements computed based on the observed counts of different species of butterflies in different locations is also listed to compare with the model estimated diversity measurements. The indicator variable “violation” shows whether the credible interval covers the observed value, 1 stands for failed to cover and 0 means that the observed value is covered. The potential scale reproduction factor, R-hat, measures the convergence of the posterior samples. And R-hat=1.0 indicates that convergence is achieved and the resulting Bayesian estimates are reliable. βA is the additive model of species richness and βM is the multiplicative model.

MEAN / MEDIAN / 2.5% / 97.5% / OBSERVED / VIOLATION / R-hat
αcluster / 31.59 / 31.67 / 29.50 / 33.83 / 31.67 / 0 / 1.0
αpatch / 16.76 / 16.77 / 15.69 / 17.85 / 16.69 / 0 / 1.0
γ / 46.87 / 47.00 / 44.00 / 49.00 / 49.00 / 0 / 1.0
βApatch / 14.83 / 14.83 / 13.17 / 16.47 / 14.93 / 0 / 1.0
βAcluster / 15.29 / 15.33 / 12.67 / 17.83 / 17.33 / 0 / 1.0
Mpatch / 1.89 / 1.88 / 1.79 / 1.99 / 1.90 / 0 / 1.0
βMcluster / 1.48 / 1.48 / 1.38 / 1.59 / 1.55 / 0 / 1.0
DIC = 2940.5

Table 3: ZIP model with environmental predictors (species-specific priors for the mixture proportion) output: summary statistics of the posterior samples of diversity measurements. The posterior means and medians are both considered as the estimated diversity measurements and the 95% credible intervals (95% CI) give ranges of the diversity measurements associated with 0.95 inclusion probability.The diversity measurements computed based on the observed counts of different species of butterflies in different locations is also listed to compare with the model estimated diversity measurements. The indicator variable “violation” shows whether the credible interval covers the observed value, 1 stands for failed to cover and 0 means that the observed value is covered. The potential scale reproduction factor, R-hat, measures the convergence of the posterior samples. And R-hat=1.0 indicates that convergence is achieved and the resulting Bayesian estimates are reliable. βA is the additive model of species richness and βM is the multiplicative model.

MEAN / MEDIAN / 2.5% / 97.5% / OBSERVED / VIOLATION / R-hat
αcluster / 31.86 / 31.83 / 29.67 / 34.00 / 31.67 / 0 / 1.0
αpatch / 16.89 / 16.88 / 15.85 / 18.00 / 16.69 / 0 / 1.0
γ / 46.99 / 47.00 / 44.00 / 49.00 / 49.00 / 0 / 1.0
βApatch / 14.97 / 14.97 / 13.29 / 16.62 / 14.93 / 0 / 1.0
βAcluster / 15.13 / 15.17 / 12.50 / 17.67 / 17.33 / 0 / 1.0
Mpatch / 1.89 / 1.89 / 1.79 / 1.99 / 1.90 / 0 / 1.0
βMcluster / 1.48 / 1.48 / 1.38 / 1.58 / 1.55 / 0 / 1.0
DIC = 3078.1

Table 4: ZIP model with environmental predictors (species-specific priors for the mixture proportion) output: summary statistics of the posterior samples of regression coefficients

MEAN / MEDIAN / SD / 2.5% / 97.5%
b1 / 0.18 / 0.18 / 0.04 / 0.10 / 0.26
b2 / 0.17 / 0.17 / 0.08 / 0.01 / 0.32
b3 / -0.05 / -0.04 / 0.07 / -0.20 / 0.08
b4 / 0.02 / 0.02 / 0.01 / 0.01 / 0.04

Code

------

# ZIP model without Environmental information:

library("R2WinBUGS")

########################################################################

#######################data set########################################

# “y” is the count of butterflies, “s” represents the species, “c” represents the cluster, “p”

# represents the patch.

#########################################################################

y <-c(

0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,1,1,2,0,1,1,0,1,5,0,

0,0,0,0,0,0,0,1,1,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,

0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,3,

2,2,1,7,2,2,5,8,4,0,1,0,0,0,0,2,3,0,4,7,1,3,0,1,13,0,

4,3,0,2,1,3,2,4,3,2,4,2,1,2,7,6,1,3,9,4,3,11,0,3,3,1,

3,1,0,5,0,0,5,1,0,1,2,0,0,1,15,3,0,5,10,5,1,6,1,1,2,2,

6,2,3,3,2,0,1,2,2,2,2,0,2,0,7,3,1,0,0,1,0,2,1,4,1,1,

7,0,0,5,1,0,4,0,5,3,1,0,1,0,2,0,0,0,0,0,0,3,0,2,0,0,

0,0,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,

0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,

1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,

0,0,0,0,0,0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,2,0,0,0,3,

1,0,2,1,0,1,1,0,5,0,2,0,1,0,1,1,0,0,0,0,0,4,0,1,0,0,

0,6,2,2,3,0,3,0,12,1,1,2,8,1,5,2,12,7,4,2,12,24,3,5,0,15,

0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,4,1,1,1,3,0,0,0,1,0,

0,4,0,0,0,0,6,1,4,2,1,0,0,2,4,1,1,1,2,1,4,3,0,2,1,1,

0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,3,0,5,0,1,

0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,

0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,

0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,

0,1,2,0,0,0,1,1,1,1,2,0,0,0,0,0,2,1,0,1,1,0,0,1,0,2,

23,4,6,10,9,8,15,6,15,2,23,8,1,17,13,12,11,9,3,3,9,22,13,13,2,12,

2,0,2,1,1,1,10,3,0,1,2,0,1,6,11,16,3,7,13,14,6,16,5,5,14,22,

0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,4,0,0,0,

0,0,0,1,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,

0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,

2,1,0,1,1,2,3,0,1,0,0,0,0,1,0,0,0,0,1,4,0,1,0,0,0,2,

1,1,0,0,0,0,2,0,1,0,0,0,0,0,1,0,1,0,0,2,0,1,0,0,0,0,

4,1,1,4,0,0,1,3,1,2,0,0,2,0,1,0,1,0,2,2,1,7,1,3,0,0,

0,1,3,4,1,0,0,0,0,0,0,0,0,0,2,0,3,2,1,2,3,3,0,3,3,0,

0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,

0,1,0,0,2,0,1,0,1,0,0,0,1,1,6,0,0,0,3,0,0,0,0,1,0,0,

0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

0,0,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,

1,0,1,0,0,0,2,2,0,0,0,0,0,0,0,1,1,1,0,0,0,1,1,1,2,0,

0,0,0,0,0,0,0,1,1,0,2,0,1,1,0,0,1,0,0,0,0,0,1,0,0,0,

0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

2,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,

4,0,1,0,0,0,0,0,3,2,0,0,0,0,0,2,0,2,2,0,0,1,0,1,1,2,

4,0,0,2,0,0,0,3,1,1,1,0,1,2,1,0,0,0,0,0,0,0,0,3,1,0,

0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,

0,0,1,0,0,0,0,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,

0,0,1,0,3,0,0,0,2,0,2,2,4,4,3,1,2,1,2,1,1,3,2,2,0,2,

3,1,0,0,1,1,0,0,0,0,1,1,0,2,0,0,0,0,3,0,0,2,0,6,0,7,

0,0,0,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,

0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,3,0,0,1,0,4,1,0)

s=c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,

3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,

5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,

6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,

7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,

8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,

9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,

10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,

11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,

12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,

13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,

14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,

15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,

16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,

17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,

18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,

19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,

20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,

21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,

22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,

23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,

24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,

25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,

26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,

27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,

28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,

29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,

30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,

31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,

32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,

33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,33,

34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,34,

35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,

36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,36,

37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,

38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,38,

39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,

40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,

41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,

42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,

43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,43,

44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,

45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,

46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,

47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,

48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,

49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49)

c=c(

1,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,

1,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6)

p=c(

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)

###################Load the saved workspace file#################################

setwd("C:/Jing_miami/PJHou/paper_draft_Crist/code_07032012/")

modfile <- "bugs_code.txt"

bugsdir <- "C:/WinBUGS14"

############Write the Winbugs model file directly in R#############################

mod <- function()

{

mu ~ dnorm(0.0,1.0)

for(i in 1:49) {

species[i] ~ dnorm(0.0,tau.s)

}

for(i in 1:6) {

cluster[i] ~ dnorm(0.0,tau.c)

}

for(i in 1:26) {

patch[i] ~ dnorm(0.0,tau.p)

}

sigma.s ~ dunif(0,5)

sigma.c ~ dunif(0,5)

sigma.p ~ dunif(0,5)

tau.s <- pow(sigma.s,-2)

tau.c <- pow(sigma.c,-2)

tau.p <- pow(sigma.p,-2)

for(i in 1:1274) {

y[i] ~ dpois(m[i])

m[i]<-(1-eta[i])*lambda[i]

eta[i] ~dbern(p0[s[i]])

log(lambda[i]) <- mu+species[s[i]]+cluster[c[i]]+patch[p[i]]

fd[i]<- exp(-lambda[i]+y[i]*log(lambda[i])-loggam(y[i]+1))

l[i] <- log(p0[s[i]]*equals(y[i],0)+(1-p0[s[i]])*fd[i])

y.pred[i]~dpois(m[i])

}

for (j in 1:49)

{

p0[j]~dbeta(1,1)

}

# compute alpha and beta(cluster) and beta(patch) here

for(i in 1:26){

for(j in 1:49){

yp_matrix[i,j]<-y.pred[(j-1)*26+i]

}

totabun<-sum(yp_matrix[,])

for(i in 1:49){

s_sum[i]<-sum(yp_matrix[,i])

ind1[i]<-step(s_sum[i]-0.5)

}

gamma<-sum(ind1[1:49])

# calculate richness by patch and cluster

for(i in 1:26){ for(j in 1:49){

ind2[i,j]<-step(yp_matrix[i,j]-0.5)

}

patch.rich[i]<-sum(ind2[i,])

}

for(j in 1:49){

cluster.mat[1,j]<-sum(yp_matrix[1:4,j])

cluster.mat[2,j]<-sum(yp_matrix[5:7,j])

cluster.mat[3,j]<-sum(yp_matrix[8:10,j])

cluster.mat[4,j]<-sum(yp_matrix[11:13,j])

cluster.mat[5,j]<-sum(yp_matrix[14:21,j])

cluster.mat[6,j]<-sum(yp_matrix[22:26,j])

}

for(i in 1:6){ for(j in 1:49){

ind3[i,j]<-step(cluster.mat[i,j]-0.5)

}

crich[i]<-sum(ind3[i,])

cabun[i]<-sum(cluster.mat[i,])

}

# calculate additive alphas and betas for patch and cluster

alpha.patch<-sum(patch.rich[1:26])/26

alpha.cluster<-sum(crich[1:6])/6

beta.patch<-alpha.cluster-alpha.patch

beta.cluster<-gamma-alpha.cluster

# calculate multiplicative betas

beta.patch.mult<-alpha.cluster/alpha.patch

beta.cluster.mult<-gamma/alpha.cluster

}

#############################################################################

write.model(mod,modfile)

#############################################################################

############################################################################

###########R+WINBUGS#########################################################

#############################################################################

library("R2WinBUGS")

butterfly.data <-list("y","c","s","p")

butterfly.parameters <- c("gamma" , "alpha.cluster" , "beta.cluster" , "alpha.patch" , "beta.patch" , "beta.patch.mult" , "beta.cluster.mult" , "y.pred","p0")

inits <- NULL

butterfly.sim <-bugs(butterfly.data,inits,butterfly.parameters,modfile,n.chains=3,n.iter=40000,n.burnin=10000,n.thin=5,bugs.directory="C:/WinBUGS14/")

#############################################################################

alpha.cluster<-butterfly.sim$sims.list$alpha.cluster

gamma<-butterfly.sim$sims.list$gamma

beta.cluster<-butterfly.sim$sims.list$beta.cluster

alpha.patch<-butterfly.sim$sims.list$alpha.patch

beta.patch<-butterfly.sim$sims.list$beta.patch

beta.patch.mult<-butterfly.sim$sims.list$beta.patch.mult

beta.cluster.mult<-butterfly.sim$sims.list$beta.cluster.mult

p0<-butterfly.sim$sims.list$p0

############################################################################

p0m<-matrix(,49,6)

for(i in 1:49){

p0m[i,1] <- mean(p0[,i])

p0m[i,2] <- median(p0[,i])

p0m[i,3]<-quantile(p0[,i],0.025)

p0m[i,4]<-quantile(p0[,i],0.975)

}

plot(p0m[,2],ylim=c(0,1),xlab='species: 1 to 49',ylab='Posterior quantiles of the mixture proportion')

for(i in 1:49){

xx<-c(i-0.05,i+0.05)

yy<-c(p0m[i,3],p0m[i,3])

yy2<-c(p0m[i,4],p0m[i,4])

zz<-c(i-0.05,i-0.05)

zz2<-c(i+0.05,i+0.05)

ss<-c(p0m[i,3],p0m[i,4])

lines(xx,yy,col='blue')

lines(xx,yy2,col='blue')

lines(zz,ss,col='blue')

lines(zz2,ss,col='blue')

}

summary<-matrix(,7,6)

summary[1,1] <- mean(alpha.cluster)

summary[1,2]<-median(alpha.cluster)

summary[1,3]<-quantile(alpha.cluster,0.025)

summary[1,4]<-quantile(alpha.cluster,0.975)

summary[1,5]<-31.67

summary[2,1] <- mean(alpha.patch)

summary[2,2]<-median(alpha.patch)

summary[2,3]<-quantile(alpha.patch,0.025)

summary[2,4]<-quantile(alpha.patch,0.975)

summary[2,5]<-16.69

summary[3,1] <- mean(beta.cluster)

summary[3,2]<-median(beta.cluster)

summary[3,3]<-quantile(beta.cluster,0.025)

summary[3,4]<-quantile(beta.cluster,0.975)

summary[3,5]<-17.33

summary[4,1] <- mean(beta.patch)

summary[4,2]<-median(beta.patch)

summary[4,3]<-quantile(beta.patch,0.025)

summary[4,4]<-quantile(beta.patch,0.975)

summary[4,5]<-14.93

summary[5,1] <- mean(gamma)

summary[5,2]<-median(gamma)

summary[5,3]<-quantile(gamma,0.025)

summary[5,4]<-quantile(gamma,0.975)

summary[5,5]<-49

summary[6,1] <- mean(beta.cluster.mult)

summary[6,2]<-median(beta.cluster.mult)

summary[6,3]<-quantile(beta.cluster.mult,0.025)

summary[6,4]<-quantile(beta.cluster.mult,0.975)

summary[6,5]<-1.55

summary[7,1] <- mean(beta.patch.mult)

summary[7,2]<-median(beta.patch.mult)

summary[7,3]<-quantile(beta.patch.mult,0.025)

summary[7,4]<-quantile(beta.patch.mult,0.975)

summary[7,5]<-1.9

for (i in 1:7)

{

if ((summary[i,5]>summary[i,4])||(summary[i,5]<summary[i,3])) (summary[i,6]<-1)

else (summary[i,6]<-0)

}

rownames(summary) <- c("alpha.cluster","alpha.patch","beta.cluster","beta.patch","gamma","beta.cluster.mult","beta.patch.mult")

colnames(summary) <- c("mean","median","2.5%","97.5","observed","violation")

summary