# Tutorial 12.11 - Integrated Nested Laplace Approximation (INLA) for (generalized) linear mixed models

24 Feb 2016

## Overview

Tutorial 12.9 introduced the basic INLA framework as well as a fully fleshed out example using a fabricated data set. Tutorial 12.10 provided simple practical examples of the use of INLA for linear and generalized linear models. The current tutorial will focus on (generalized) linear mixed effects (hierarchical) models and using both fabricated data and real worked examples.

In this tutorial we have an opportunity to step up in model complexity and explore the use of INLA for multilevel (hierarchical) models. As with tutorial 12.10, all data sets will be fabricated from set parameters so that we always have a 'truth' from which to compare outcomes and as a point of comparison, each data set will be followed by Frequentist and Bayesian MCMC outcomes.

### Mixed effects model - RCB

set.seed(1)
n.groups <- 6
n.sample <- 10
n <- n.groups * n.sample
block <- gl(n = n.groups, k = n.sample, lab = paste("Block", 1:n.groups, sep = ""))
x <- runif(n, 0, 70)
mn <- mean(x)
sd <- sd(x)
cx <- (x - mn)  #/sd
Xmat <- model.matrix(~block * cx - 1 - cx)  #intercepts and slopes
Xmat <- model.matrix(~-1 + block + x)  #intercepts and slopes
intercept.mean <- 230
intercept.sd <- 20
slope.mean <- 1.5
# slope.sd <- 0.3
intercept.effects <- rnorm(n = n.groups, mean = intercept.mean, sd = intercept.sd)
# slope.effects <- rnorm(n=n.groups, mean=slope.mean, sd=slope.sd) #intercepts
# and slopes
slope.effects <- slope.mean
all.effects <- c(intercept.effects, slope.effects)
lin.pred <- Xmat[, ] %*% all.effects
eps <- rnorm(n = n, mean = 0, sd = 10)
y <- lin.pred + eps
data.hier <- data.frame(y = y, x = cx + mn, block = block)

         y        x  block
1 281.1091 18.58561 Block1
2 295.6535 26.04867 Block1
3 328.3234 40.09974 Block1
4 360.1672 63.57455 Block1
5 276.7050 14.11774 Block1
6 348.9709 62.88728 Block1

library(ggplot2)
ggplot(data.hier, aes(y = y, x = x, fill = block, color = block)) + geom_smooth(method = "lm") +
geom_point() + theme_classic()


$$y_i = \alpha + \beta X_{i} + \sum_{j=1}^J{}\gamma_{ij}\mathbf{Z_i} + \varepsilon_i \hspace{4em}\varepsilon_i\sim{}\mathcal{N}(0, \sigma^2)$$

library(nlme)
data.lme <- lme(y~x, random=~1|block, data=data.hier)
summary(data.lme)

Linear mixed-effects model fit by REML
Data: data.hier
AIC      BIC   logLik
458.9521 467.1938 -225.476

Random effects:
Formula: ~1 | block
(Intercept) Residual
StdDev:    18.10888 8.905485

Fixed effects: y ~ x
Value Std.Error DF  t-value p-value
(Intercept) 232.8193  7.823393 53 29.75937       0
x             1.4591  0.063789 53 22.87392       0
Correlation:
(Intr)
x -0.292

Standardized Within-Group Residuals:
Min          Q1         Med          Q3         Max
-2.09947262 -0.57994305 -0.04874031  0.56685096  2.49464217

Number of Observations: 60
Number of Groups: 6

modelString="
model {
#Likelihood
for (i in 1:n) {
y[i]~dnorm(mean[i],tau)
mean[i] <- inprod(beta[],X[i,]) + inprod(gamma[],Z[i,])
}
#Priors
for (i in 1:nX) {
beta[i] ~ dnorm(0, 1.0E-6) #prior
}
for (i in 1:nZ) {
gamma[i] ~ dnorm(0, tau.block) #prior
}

sigma <- z/sqrt(chSq)    # prior for sigma; cauchy = normal/sqrt(chi^2)
z ~ dnorm(0, 0.0625)I(0,)  # half-cauchy with scale of 4
chSq ~ dgamma(0.5, 0.5)  # chi^2 with 1 d.f.
tau <- pow(sigma, -2)

sigma.block <- z.block/sqrt(chSq.block)    # prior for sigma; cauchy = normal/sqrt(chi^2)
z.block ~ dnorm(0, 0.0625)I(0,)
chSq.block ~ dgamma(0.5, 0.5)  # chi^2 with 1 d.f.
tau.block <- pow(sigma.block, -2)
}
"
X <- model.matrix(~x, data.hier)
Z <- model.matrix(~-1+block, data.hier)
data.list <- with(data.hier,
list(y=y,
X=X, nX=ncol(X),
Z=Z, nZ=ncol(Z),
n=nrow(data.hier)
)
)

data.jags <- jags(data=data.list,
inits=NULL,
parameters.to.save=c('beta','gamma','sigma','sigma.block'),
model.file=textConnection(modelString),
n.chains=3,
n.iter=10000,
n.burnin=2000,
n.thin=100
)

Compiling model graph
Resolving undeclared variables
Allocating nodes
Graph Size: 815

Initializing model

print(data.jags)

Inference for Bugs model at "5", fit using jags,
3 chains, each with 10000 iterations (first 2000 discarded), n.thin = 100
n.sims = 240 iterations saved
mu.vect sd.vect    2.5%     25%     50%     75%   97.5%  Rhat n.eff
beta[1]     232.319   8.624 214.449 227.506 232.405 238.059 249.564 1.016   240
beta[2]       1.453   0.062   1.336   1.406   1.458   1.491   1.567 0.997   240
gamma[1]     26.610   8.904  10.035  21.527  26.274  30.716  45.525 1.034   240
gamma[2]      1.634   8.615 -13.492  -3.522   1.393   6.234  17.306 1.031   240
gamma[3]      7.941   8.647  -8.050   2.688   7.525  12.997  25.231 1.012   240
gamma[4]     -0.764   8.629 -16.457  -6.119  -1.039   3.953  14.586 1.016   240
gamma[5]    -28.293   8.814 -45.139 -32.786 -28.004 -24.376 -11.547 1.016   240
gamma[6]     -3.182   9.096 -19.279  -8.868  -3.169   1.795  14.828 1.017   240
sigma         8.920   0.859   7.311   8.368   8.862   9.480  10.785 1.017   110
sigma.block  19.366   8.242  10.329  14.294  17.387  21.957  42.429 1.056    50
deviance    433.888   4.301 427.870 430.526 433.235 436.252 444.373 1.008   240

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 9.3 and DIC = 443.2
DIC is an estimate of expected predictive error (lower deviance is better).

library(rstan)
modelString="
data {
int<lower=1> n;
int<lower=1> nX;
int<lower=1> nBlock;
vector [n] y;
matrix [n,nX] X;
int Z[n];
}
parameters {
vector[nX] beta;
real<lower=0> sigma;
vector [nBlock] gamma;
real<lower=0> sigmaBlock;
}
transformed parameters {
vector[n] eta;

eta <- X*beta;
for (i in 1:n) {
eta[i] <- eta[i] + gamma[Z[i]];
}
}
model {
#Likelihood
y~normal(eta,sigma);

#Priors
beta ~ normal(0,1000);
sigma~cauchy(0,5);
gamma ~ normal(0,sigmaBlock);
sigmaBlock~cauchy(0,5);
}
generated quantities {
vector[n] log_lik;

for (i in 1:n) {
log_lik[i] <- normal_log(y[i], eta, sigma);
}
}
"

Xmat <- model.matrix(~x,data=data.hier)
data.hier.list <- with(data.hier, list(y = y, X = Xmat, nX=ncol(Xmat),n = nrow(data.hier),
Z=as.numeric(block), nBlock=length(levels(block))))

library(rstan)
data.hier.rstan <- stan(data=data.hier.list,
model_code=modelString,
chains=3,
iter=1000,
warmup=500,
thin=2,
save_dso=TRUE
)

SAMPLING FOR MODEL 'f3aefea794fe54ff9ebd376cb1a2914e' NOW (CHAIN 1).

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#  Elapsed Time: 0.573899 seconds (Warm-up)
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print(data.hier.rstan, pars=c('beta','sigmaBlock','sigma'))

Inference for Stan model: f3aefea794fe54ff9ebd376cb1a2914e.
3 chains, each with iter=1000; warmup=500; thin=2;
post-warmup draws per chain=250, total post-warmup draws=750.

mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
beta[1]    233.59    0.61 8.77 217.32 227.44 233.40 239.27 252.30   208 1.00
beta[2]      1.46    0.00 0.06   1.33   1.42   1.46   1.50   1.58   558 1.00
sigmaBlock  19.20    0.36 6.45  10.41  14.63  18.31  22.08  36.20   320 1.01
sigma        9.05    0.04 0.90   7.47   8.43   9.02   9.63  11.00   527 1.00

Samples were drawn using NUTS(diag_e) at Wed Dec 16 10:43:50 2015.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).

library(brms)
data.brm <- brm(y~x+(1|block), data=data.hier, family='gaussian',
prior=c(set_prior('normal(0,1000)', class='b'),
set_prior('cauchy(0,5)', class='sd')),
n.chains=3, n.iter=2000, warmup=500, n.thin=2
)

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summary(data.brm)

 Family: gaussian (identity)
Formula: y ~ x + (1 | block)
Data: data.hier (Number of observations: 60)
Samples: 3 chains, each with n.iter = 2000; n.warmup = 500; n.thin = 2;
total post-warmup samples = 2250
WAIC: 442.2

Random Effects:
~block (Number of levels: 6)
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sd(Intercept)    19.28      6.92    10.45    37.71        713    1

Fixed Effects:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
Intercept   232.54      7.96   216.25   247.87        498 1.01
x             1.46      0.07     1.34     1.59       1071 1.00

Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sigma(y)     9.13      0.92     7.55     11.1       1450    1

Samples were drawn using NUTS(diag_e). For each parameter, Eff.Sample is a
crude measure of effective sample size, and Rhat is the potential scale
reduction factor on split chains (at convergence, Rhat = 1).

stancode(data.brm)

functions {
}
data {
int<lower=1> N;  # number of observations
vector[N] Y;  # response variable
int<lower=1> K;  # number of fixed effects
matrix[N, K] X;  # FE design matrix
# data for random effects of block
int<lower=1> J_1[N];  # RE levels
int<lower=1> N_1;  # number of levels
int<lower=1> K_1;  # number of REs
real Z_1[N];  # RE design matrix
}
transformed data {
}
parameters {
real b_Intercept;  # fixed effects Intercept
vector[K] b;  # fixed effects
vector[N_1] pre_1;  # unscaled REs
real<lower=0> sd_1;  # RE standard deviation
real<lower=0> sigma;  # residual SD
}
transformed parameters {
vector[N] eta;  # linear predictor
vector[N_1] r_1;  # REs
# compute linear predictor
eta <- X * b + b_Intercept;
r_1 <- sd_1 * (pre_1);  # scale REs
# if available add REs to linear predictor
for (n in 1:N) {
eta[n] <- eta[n] + Z_1[n] * r_1[J_1[n]];
}
}
model {
# prior specifications
b_Intercept ~ normal(0,1000);
b ~ normal(0,1000);
sd_1 ~ cauchy(0,5);
pre_1 ~ normal(0, 1);
sigma ~ cauchy(0, 32);
# likelihood contribution
Y ~ normal(eta, sigma);
}
generated quantities {
}


The INLA implementation will focus on fitting the model and predicting for the purpose of generating a summary figure.

For hierarchical models (mixed effects models) in R, predictions can occur at different levels of the hierarchy. For example, we could predict the value of y at a given x and given block, or we could predict the value of y at a given x for the average block. In INLA, the fitted values are equivalent to the former. This is useful for generating residuals. However, to generate partial observations, we also need to predict the value of y at the observed levels of x for the average block.

Hence, in addition to the full prediction sequence, I am going to define two additional versions of our observed data.

• one that reflects fitted values - predicted values at observed x and block (fitted)
• one that reflects predicted values at observed x for the average block (pred)

pred <- fitted <- subset(data.hier, select=c(x,block,y))
fitted$y <- pred$y <- pred$block <- NA newdata <- data.frame(x=seq(min(data.hier$x, na.rm=TRUE),max(data.hier$x, na.rm=TRUE),len=100), block=NA, y=NA) data.pred <- rbind(subset(data.hier, select=c(x,block,y)), fitted, pred ,newdata)  Now lets fit the model. library(INLA) #fit the model data.inla <- inla(y~x + f(block, model='iid'), data=data.pred, control.compute=list(dic=TRUE, cpo=TRUE, waic=TRUE))  Although we would normally perform any model validation routines prior to exploring the model parameters, on this occasion we will quickly explore what is captured in the INLA model. We will start with a summary of the model fit. #examine the regular summary summary(data.inla)  Call: c("inla(formula = y ~ x + f(block, model = \"iid\"), data = data.pred, ", " control.compute = list(dic = TRUE, cpo = TRUE, waic = TRUE))" ) Time used: Pre-processing Running inla Post-processing Total 0.1504 0.2777 0.1751 0.6032 Fixed effects: mean sd 0.025quant 0.5quant 0.975quant mode kld (Intercept) 235.1609 5.2586 224.8011 235.1607 245.5084 235.1608 0 x 1.3938 0.1300 1.1376 1.3938 1.6497 1.3938 0 Random effects: Name Model block IID model Model hyperparameters: mean sd 0.025quant Precision for the Gaussian observations 2.900e-03 5.000e-04 0.002 Precision for block 1.884e+04 1.865e+04 1282.101 0.5quant 0.975quant mode Precision for the Gaussian observations 2.800e-03 4.00e-03 0.0028 Precision for block 1.335e+04 6.83e+04 3496.4786 Expected number of effective parameters(std dev): 2.00(0.00) Number of equivalent replicates : 30.00 Deviance Information Criterion (DIC) ...: 527.38 Effective number of parameters .........: 2.957 Watanabe-Akaike information criterion (WAIC) ...: 527.35 Effective number of parameters .................: 2.794 Marginal log-Likelihood: -281.07 CPO and PIT are computed Posterior marginals for linear predictor and fitted values computed  When fitting the INLA model above, we did not specify any priors. To see what priors were applied, we can use the inla.show.hyperspec() function inla.show.hyperspec(data.inla)  List of 4$ predictor:List of 1
$hyper:List of 1$ theta:List of 8
$name : atomic [1:1] log precision$ short.name: atomic [1:1] prec
$initial : atomic [1:1] 11$ fixed     : atomic [1:1] TRUE
$prior : atomic [1:1] loggamma$ param     : atomic [1:2] 1e+00 1e-05
$family :List of 1$ :List of 3
$label: chr "gaussian"$ hyper:List of 1
$theta:List of 8$ name      : atomic [1:1] log precision
$short.name: atomic [1:1] prec$ initial   : atomic [1:1] 4
$fixed : atomic [1:1] FALSE$ prior     : atomic [1:1] loggamma
$param : atomic [1:2] 1e+00 5e-05$ link :List of 1
$hyper: list()$ fixed    :List of 2
$:List of 3$ label     : chr "(Intercept)"
$prior.mean: num 0$ prior.prec: num 0
$:List of 3$ label     : chr "x"
$prior.mean: num 0$ prior.prec: num 0.001
$random :List of 1$ :List of 2
$label: chr "block"$ hyper:List of 1
$theta:List of 8$ name      : atomic [1:1] log precision
$short.name: atomic [1:1] prec$ prior     : atomic [1:1] loggamma
$param : atomic [1:2] 1e+00 5e-05$ initial   : atomic [1:1] 4
$fixed : atomic [1:1] FALSE  Out of interest, we can compare the prior and posterior for the precision parameter of the residual standard deviation. post <- data.frame(data.inla$marginals.hyperpar[[1]])
post$y <- post$y/10000000
prior <- data.frame(x=post$x, y=dgamma(post$x,1,1.0E-05))
ggplot(post, aes(y=y, x=x)) + geom_line(color='red') +
geom_line(data=prior, color='blue') + theme_classic()


Similarly, we could compare the prior and posterior for the fixed effect of Intercept

data.inla$marginals.fixed[[1]]   x y [1,] 182.5751 1.318072e-19 [2,] 193.0923 2.445666e-13 [3,] 203.6094 1.292026e-08 [4,] 208.8680 1.039060e-06 [5,] 214.1266 4.315473e-05 [6,] 219.3852 9.405064e-04 [7,] 222.7872 4.767681e-03 [8,] 226.5101 1.914578e-02 [9,] 228.4488 3.303948e-02 [10,] 229.7445 4.422887e-02 [11,] 230.7727 5.340365e-02 [12,] 231.6451 6.079235e-02 [13,] 232.4281 6.666708e-02 [14,] 233.1540 7.114612e-02 [15,] 233.8420 7.429546e-02 [16,] 234.5068 7.616540e-02 [17,] 234.8348 7.663098e-02 [18,] 235.1607 7.678562e-02 [19,] 235.4873 7.663057e-02 [20,] 235.8145 7.616599e-02 [21,] 236.4789 7.429836e-02 [22,] 237.1655 7.115775e-02 [23,] 237.8887 6.670071e-02 [24,] 238.6766 6.079170e-02 [25,] 239.5539 5.336113e-02 [26,] 240.5756 4.424251e-02 [27,] 241.8721 3.304592e-02 [28,] 243.8124 1.914173e-02 [29,] 247.5177 4.802801e-03 [30,] 250.9366 9.406587e-04 [31,] 256.1952 4.317105e-05 [32,] 261.4538 1.039687e-06 [33,] 266.7123 1.293071e-08 [34,] 277.2295 2.448550e-13 [35,] 287.7466 1.320028e-19  post <- data.frame(data.inla$marginals.fixed[[1]])
post$y <- post$y/100
prior <- data.frame(x=post$x, y=dnorm(post$x,0,1/0.001))
ggplot(post, aes(y=y, x=x)) + geom_line(color='red') +
geom_line(data=prior, color='blue') + theme_classic()


In this initial fit, I have accepted the default priors (recall that priors are defined in terms of precision)

Now lets fit the model in which we specify priors.

• $log\Gamma(1, 0.1)$ for hyperprior on residual standard deviation
• $N(0,0.001)$ for the slope parameter and $N(0,0.00001)$ for the intercept parameter
• $log\Gamma(0.1,0.1)$ for hyperprior on standard deviation of blocks

#fit the model
data.inla <- inla(y~x + f(block, model='iid', hyper=list(theta=list(prior='loggamma', param=c(0.1,0.1)))),
data=data.pred,
control.family=list(hyper=list(prec=list(prior='loggamma', param=c(0.1,0.1)))),
control.fixed=list(mean=0, prec=0.001, mean.intercept=0, prec.intercept=0.00001),
control.compute=list(dic=TRUE, cpo=TRUE, waic=TRUE))
#examine the regular summary
summary(data.inla)

Call:
c("inla(formula = y ~ x + f(block, model = \"iid\", hyper = list(theta = list(prior = \"loggamma\", ",  "    param = c(0.1, 0.1)))), data = data.pred, control.compute = list(dic = TRUE, ",  "    cpo = TRUE, waic = TRUE), control.family = list(hyper = list(prec = list(prior = \"loggamma\", ",  "    param = c(0.1, 0.1)))), control.fixed = list(mean = 0, prec = 0.001, ",  "    mean.intercept = 0, prec.intercept = 1e-05))")

Time used:
Pre-processing    Running inla Post-processing           Total
0.0798          0.2026          0.0344          0.3167

Fixed effects:
mean     sd 0.025quant 0.5quant 0.975quant     mode kld
(Intercept) 232.6597 8.6553   214.9637 232.6965   250.1279 232.7479   0
x             1.4591 0.0644     1.3320   1.4591     1.5858   1.4592   0

Random effects:
Name	  Model
block   IID model

Model hyperparameters:
mean     sd 0.025quant 0.5quant
Precision for the Gaussian observations 0.0126 0.0025     0.0084   0.0125
Precision for block                     0.0032 0.0021     0.0007   0.0027
0.975quant   mode
Precision for the Gaussian observations     0.0180 0.0121
Precision for block                         0.0085 0.0018

Expected number of effective parameters(std dev): 6.856(0.0819)
Number of equivalent replicates : 8.752

Deviance Information Criterion (DIC) ...: 441.66
Effective number of parameters .........: 7.925

Watanabe-Akaike information criterion (WAIC) ...: 442.24
Effective number of parameters .................: 7.628

Marginal log-Likelihood:  -243.04
CPO and PIT are computed

Posterior marginals for linear predictor and fitted values computed


Now compare the priors and posteriors for the hyperpriors of the two levels of standard deviation (blocks and residuals), to see what it was we defined when altering the priors.

post <- data.frame(data.inla$marginals.hyperpar[[1]]) post$y <- post$y prior <- data.frame(x=post$x, y=dgamma(post$x,0.1,0.1)) ggplot(post, aes(y=y, x=x)) + geom_line(color='red') + geom_line(data=prior, color='blue') + theme_classic()  post <- data.frame(data.inla$marginals.hyperpar[[2]])
post$y <- post$y
prior <- data.frame(x=post$x, y=dgamma(post$x,0.1,0.1))
ggplot(post, aes(y=y, x=x)) + geom_line(color='red') + geom_line(data=prior, color='blue') + theme_classic()


The model outputs are spread between a large number of objects captured in the list returned by inla().

names(data.inla)

 [1] "names.fixed"                 "summary.fixed"
[3] "marginals.fixed"             "summary.lincomb"
[5] "marginals.lincomb"           "size.lincomb"
[7] "summary.lincomb.derived"     "marginals.lincomb.derived"
[9] "size.lincomb.derived"        "mlik"
[11] "cpo"                         "po"
[13] "waic"                        "model.random"
[15] "summary.random"              "marginals.random"
[17] "size.random"                 "summary.linear.predictor"
[19] "marginals.linear.predictor"  "summary.fitted.values"
[21] "marginals.fitted.values"     "size.linear.predictor"
[23] "summary.hyperpar"            "marginals.hyperpar"
[25] "internal.summary.hyperpar"   "internal.marginals.hyperpar"
[27] "offset.linear.predictor"     "model.spde2.blc"
[29] "summary.spde2.blc"           "marginals.spde2.blc"
[31] "size.spde2.blc"              "model.spde3.blc"
[33] "summary.spde3.blc"           "marginals.spde3.blc"
[35] "size.spde3.blc"              "logfile"
[37] "misc"                        "dic"
[39] "mode"                        "neffp"
[41] "joint.hyper"                 "nhyper"
[43] "version"                     "Q"
[45] "graph"                       "ok"
[47] "cpu.used"                    "all.hyper"
[49] ".args"                       "call"
[51] "model.matrix"


Returning our focus to the summary output above, we might like to explore the standard deviations (variance components) of the random effects so as to get a sense for the scales of variation within the hierarchy. However, recall that the hyper-parameters are on a precision scale. Lets derive the standard deviations.

s <- inla.contrib.sd(data.inla, nsamples=1000)
s$hyper   mean sd 2.5% 97.5% sd for the Gaussian observations 8.992904 0.8715473 7.456642 10.77948 sd for block 20.357307 6.8055129 10.747708 37.04383  So there is substantially more variation at the level of block than in the sampling units. #### Predictions Recall that in preparation of data to model with INLA, we appended a sequence of 100 new x values and associated NA values for y to the data. INLA imputes (predicts) missing values and thus the last 100 values in the INLA linear predictor will be the predicted values (since we also indicated NA values for block, predictions are above the level of block - that is, they are for an average block.. newdata <- cbind(newdata, data.inla$summary.linear.predictor[(nrow(data.hier)+nrow(fitted)+nrow(pred)+1):nrow(data.pred),])

                      x block  y     mean       sd 0.025quant 0.5quant
predictor.181 0.9373233    NA NA 234.0362 8.648585   216.3513 234.0686
predictor.182 1.6292032    NA NA 235.0455 8.636937   217.3803 235.0779
predictor.183 2.3210830    NA NA 236.0549 8.625505   218.4090 236.0873
predictor.184 3.0129628    NA NA 237.0643 8.614290   219.4373 237.0967
predictor.185 3.7048426    NA NA 238.0737 8.603293   220.4652 238.1060
predictor.186 4.3967225    NA NA 239.0831 8.592515   221.4928 239.1154
0.975quant     mode          kld
predictor.181   251.5321 234.1152 1.697732e-06
predictor.182   252.5217 235.1244 1.647773e-06
predictor.183   253.5116 236.1336 1.598649e-06
predictor.184   254.5019 237.1428 1.550397e-06
predictor.185   255.4926 238.1520 1.503109e-06
predictor.186   256.4837 239.1612 1.456826e-06

newdata <- reshape:::rename(newdata, c("0.025quant"="lower", "0.975quant"="upper"))


Of course, we can use these predictions to reconstruct the fitted line and credibility bounds on a figure.

fitted <- cbind(fitted,
data.inla$summary.linear.predictor[(nrow(data.hier)+1):(nrow(data.hier)+nrow(fitted)),]) fitted <- reshape:::rename(fitted, c("0.025quant"="lower", "0.975quant"="upper")) pred <- cbind(pred, data.inla$summary.linear.predictor[(nrow(data.hier)+nrow(fitted)+1):
(nrow(data.hier)+nrow(fitted)+nrow(pred)),])
pred <- reshape:::rename(pred, c("0.025quant"="lower", "0.975quant"="upper"))

ndata <- data.hier
ndata$fit <- fitted$mean
ndata$pred <- pred$mean
ndata$Res <- (ndata$fit - data.hier$y) ndata$Pobs <- ndata$pred + ndata$Res

         y        x  block      fit     pred        Res     Pobs
1 281.1091 18.58561 Block1 285.9574 259.7838   4.848297 264.6320
2 295.6535 26.04867 Block1 296.8467 270.6724   1.193259 271.8656
3 328.3234 40.09974 Block1 317.3486 291.1738 -10.974876 280.1990
4 360.1672 63.57455 Block1 351.6006 325.4280  -8.566593 316.8614
5 276.7050 14.11774 Block1 279.4383 253.2653   2.733390 255.9987
6 348.9709 62.88728 Block1 350.5978 324.4251   1.626893 326.0520

ggplot(newdata, aes(y=mean, x=x)) +
geom_point(data=ndata, aes(y=Pobs)) +
geom_ribbon(aes(ymin=lower, ymax=upper), fill='blue', alpha=0.2) +
geom_line() + theme_classic()


Alternatively, predictions can also be performed by generating a model matrix and incorporating these into INLA as linear combinations. This is particularly useful if we want to derive specific comparisons etc. For example, we may want to indicate the change in the response over the range of x.

Of course, this would typically be a decision we would make prior to running the analysis, and indeed must be defined prior to fitting the inla model. I have previously illustrated the use of linear combinations for exploring contrasts amongst categorical variable levels. In this example, we want to contrast different regions of a continuous. Such a comparison cannot easily be achieved via a model matrix. However, R-INLA's inla.make.lincomb function allows us to contrast predictions associated with the input data.

Recall that when we developed the input for the INLA model at the start of this example, we appended the raw data with three additional data sets (each containing NA values for the response). The last of these data frames represented 100 new x values ranging from the minimum observed x to the maximum observed x. Hence, what we want to do is indicate that we wish to contrast the predictions associated with the last (max) and first (min) of these data. To do so, we just need to find the index of these two rows in the total input data frame and use them to define the linear combinations from the Predictor.

##Define linear combinations
## We want to to compare the predicted value of the first prediction
## to the predicted value for the last predition
## the first prediction starts at the following index
idx = nrow(data.hier) + nrow(fitted)+ nrow(pred) +1
## and the length of the prediction data
nr=nrow(newdata)
##create the linear combinations by defining the values to compare using their indices
## to indicate which value in the data set to refer to
lincomb=inla.make.lincomb(Predictor=c(rep(NA,idx-1),-1, rep(NA,nr-2),1))

#fit the model
data.inla <- inla(y~x + f(block, model='iid', hyper=list(theta=list(prior='loggamma', param=c(0.1,0.1)))),
data=data.pred,
lincomb=lincomb,
control.inla=list(lincomb.derived.only=FALSE),
control.family=list(hyper=list(prec=list(prior='loggamma', param=c(0.1,0.1)))),
control.fixed=list(mean=0, prec=0.001, mean.intercept=0, prec.intercept=0.00001),
control.compute=list(dic=TRUE, cpo=TRUE, waic=TRUE))
#examine the regular summary
data.inla$summary.lincomb   ID mean sd 0.025quant 0.5quant 0.975quant mode kld lc 1 99.94242 4.428648 91.20684 99.94429 108.6562 99.94849 6.955833e-14  So the response increases by 99.94 units over the range of x. Another very flexible, yet not purely deterministic (and thus yielding slightly different outcomes each time it is run) alternative is to draw random values from the posteriors associated with the relevant predicted values. This method is analogous to performing contrasts on MCMC samples. Once we have a large number of samples of predicted values for the minimum and maximum x, we just need to summarize the paired differences. Again, we just need to work out what the indices are for the relevant predictions. library(coda) idx = nrow(data.hier) + nrow(fitted)+ nrow(pred) +1 ## and the length of the prediction data nr=nrow(newdata) rmin=inla.rmarginal(10000,data.inla$marginals.fitted.values[[idx]])
rmax=inla.rmarginal(10000,data.inla$marginals.fitted.values[[idx+nr-1]]) change=(rmax-rmin) (change=data.frame(Mean=mean(change), Median=median(change), HPDinterval(as.mcmc(change))))   Mean Median lower upper var1 99.86877 99.84981 75.73204 124.6698  ## or as a percentage change (increase) pchange=100*(rmax-rmin)/rmin (pchange=data.frame(Mean=mean(pchange), Median=median(pchange), HPDinterval(as.mcmc(pchange))))   Mean Median lower upper var1 42.88382 42.68041 30.36272 56.36062  ### Split-plot design Random data incorporating the following properties • the number of between block treatments (A) = 3 • the number of blocks = 35 • the number of within block treatments (C) = 3 • the mean of the treatments = 40, 70 and 80 respectively • the variability (standard deviation) between blocks of the same treatment = 12 • the variability (standard deviation) between treatments withing blocks = 5 library(plyr) set.seed(1) nA <- 3 nC <- 3 nBlock <- 36 sigma <- 5 sigma.block <- 12 n <- nBlock * nC Block <- gl(nBlock, k = 1) C <- gl(nC, k = 1) ## Specify the cell means AC.means <- (rbind(c(40, 70, 80), c(35, 50, 70), c(35, 40, 45))) ## Convert these to effects X <- model.matrix(~A * C, data = expand.grid(A = gl(3, k = 1), C = gl(3, k = 1))) AC <- as.vector(AC.means) AC.effects <- solve(X, AC) A <- gl(nA, nBlock, n) dt <- expand.grid(C = C, Block = Block) dt <- data.frame(dt, A) Xmat <- cbind(model.matrix(~-1 + Block, data = dt), model.matrix(~A * C, data = dt)) block.effects <- rnorm(n = nBlock, mean = 0, sd = sigma.block) all.effects <- c(block.effects, AC.effects) lin.pred <- Xmat %*% all.effects ## the quadrat observations (within sites) are drawn from normal distributions ## with means according to the site means and standard deviations of 5 y <- rnorm(n, lin.pred, sigma) data.splt <- data.frame(y = y, A = A, dt) head(data.splt) #print out the first six rows of the data set   y A C Block A.1 1 30.51110 1 1 1 1 2 62.18599 1 2 1 1 3 77.98268 1 3 1 1 4 46.01960 1 1 2 1 5 71.38110 1 2 2 1 6 80.93691 1 3 2 1  tapply(data.splt$y, data.splt$A, mean)   1 2 3 67.73243 52.25684 37.79359  tapply(data.splt$y, data.splt$C, mean)   1 2 3 37.57486 55.33468 64.87331  replications(y ~ A * C + Error(Block), data.splt)   A C A:C 36 36 12  library(ggplot2) ggplot(data.splt, aes(y = y, x = C, linetype = A, group = A)) + geom_line(stat = "summary", fun.y = mean)  ggplot(data.splt, aes(y = y, x = C, color = A)) + geom_point() + facet_wrap(~Block)  $$y_i = \alpha + \beta X_{i} + \sum_{j=1}^J{}\gamma_{ij}\mathbf{Z_i} + \varepsilon_i \hspace{4em}\varepsilon_i\sim{}\mathcal{N}(0, \sigma^2)$$ library(nlme) #Assuming sphericity # random intercepts model data.splt.lme <- lme(y~A*C, random=~1|Block, data=data.splt, method='REML') # random intercept and slopes model - note this is not an identifiable model data.splt.lme1 <- lme(y~A*C, random=~C|Block, data=data.splt, method='REML') AIC(data.splt.lme, data.splt.lme1)   df AIC data.splt.lme 11 714.3519 data.splt.lme1 16 719.7929  # random intercepts model 'best' summary(data.splt.lme)  Linear mixed-effects model fit by REML Data: data.splt AIC BIC logLik 714.3519 742.8982 -346.1759 Random effects: Formula: ~1 | Block (Intercept) Residual StdDev: 11.00689 4.309761 Fixed effects: y ~ A * C Value Std.Error DF t-value p-value (Intercept) 44.39871 3.412303 66 13.011362 0.0000 A2 -8.86091 4.825726 33 -1.836182 0.0754 A3 -11.61064 4.825726 33 -2.405988 0.0219 C2 31.17007 1.759453 66 17.715775 0.0000 C3 38.83108 1.759453 66 22.069979 0.0000 A2:C2 -15.33891 2.488242 66 -6.164556 0.0000 A3:C2 -24.89184 2.488242 66 -10.003787 0.0000 A2:C3 -4.50515 2.488242 66 -1.810574 0.0748 A3:C3 -30.09278 2.488242 66 -12.093993 0.0000 Correlation: (Intr) A2 A3 C2 C3 A2:C2 A3:C2 A2:C3 A2 -0.707 A3 -0.707 0.500 C2 -0.258 0.182 0.182 C3 -0.258 0.182 0.182 0.500 A2:C2 0.182 -0.258 -0.129 -0.707 -0.354 A3:C2 0.182 -0.129 -0.258 -0.707 -0.354 0.500 A2:C3 0.182 -0.258 -0.129 -0.354 -0.707 0.500 0.250 A3:C3 0.182 -0.129 -0.258 -0.354 -0.707 0.250 0.500 0.500 Standardized Within-Group Residuals: Min Q1 Med Q3 Max -1.908000844 -0.541899250 0.003782048 0.542865052 1.810720228 Number of Observations: 108 Number of Groups: 36  modelString=" model { #Likelihood for (i in 1:n) { y[i]~dnorm(mu[i],tau.res) mu[i] <- inprod(beta[],X[i,]) + inprod(gamma[],Z[i,]) y.err[i] <- y[i] - mu[1] } #Priors and derivatives for (i in 1:nZ) { gamma[i] ~ dnorm(0,tau.block) } for (i in 1:nX) { beta[i] ~ dnorm(0,1.0E-06) } tau.res <- pow(sigma.res,-2) sigma.res <- z/sqrt(chSq) z ~ dnorm(0, .0016)I(0,) chSq ~ dgamma(0.5, 0.5) tau.block <- pow(sigma.block,-2) sigma.block <- z.block/sqrt(chSq.block) z.block ~ dnorm(0, .0016)I(0,) chSq.block ~ dgamma(0.5, 0.5) } " X <- model.matrix(~A*C, data.splt) Z <- model.matrix(~-1+Block, data.splt) data.list <- with(data.splt, list(y=y, X=X, nX=ncol(X), Z=Z, nZ=ncol(Z), n=nrow(data.splt) ) ) data.jags <- jags(data=data.list, inits=NULL, parameters.to.save=c('beta','gamma','sigma.res','sigma.block'), model.file=textConnection(modelString), n.chains=3, n.iter=10000, n.burnin=2000, n.thin=100 )  Compiling model graph Resolving undeclared variables Allocating nodes Graph Size: 5511 Initializing model  print(data.jags)  Inference for Bugs model at "5", fit using jags, 3 chains, each with 10000 iterations (first 2000 discarded), n.thin = 100 n.sims = 240 iterations saved mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat n.eff beta[1] 44.289 3.868 36.293 41.869 44.381 46.984 51.076 1.039 63 beta[2] -9.084 5.164 -19.226 -12.751 -8.834 -5.699 0.619 1.008 220 beta[3] -11.439 4.969 -21.170 -14.938 -11.202 -8.346 -0.663 1.012 160 beta[4] 31.230 1.843 27.619 29.999 31.258 32.500 34.618 1.003 240 beta[5] 38.795 1.869 35.358 37.372 38.834 40.159 42.560 1.004 240 beta[6] -15.538 2.607 -20.671 -17.352 -15.328 -13.912 -10.329 1.014 240 beta[7] -24.761 2.664 -30.154 -26.623 -24.783 -22.720 -19.911 0.997 240 beta[8] -4.530 2.685 -9.967 -6.253 -4.514 -2.693 0.838 1.005 240 beta[9] -29.918 2.610 -34.625 -31.695 -29.986 -28.152 -24.709 1.007 170 gamma[1] -10.354 4.145 -18.338 -12.981 -10.459 -7.666 -2.105 1.035 58 gamma[2] -1.555 4.489 -10.421 -4.500 -1.683 1.462 6.996 1.018 160 gamma[3] -12.533 4.043 -20.494 -15.352 -12.652 -9.676 -4.752 1.021 92 gamma[4] 14.979 4.264 6.636 12.128 14.962 17.826 23.128 1.040 51 gamma[5] 1.506 4.085 -7.280 -1.032 1.649 4.191 8.957 1.025 75 gamma[6] -15.567 4.252 -24.584 -18.191 -15.377 -12.802 -7.336 1.012 130 gamma[7] 6.073 4.237 -0.768 3.146 5.635 8.653 15.191 1.008 210 gamma[8] 3.561 4.226 -4.098 0.841 3.373 6.571 11.067 1.031 70 gamma[9] 7.116 4.272 -0.409 4.075 7.127 9.767 15.772 1.046 50 gamma[10] -8.572 4.557 -15.873 -11.605 -8.906 -5.813 0.557 1.015 120 gamma[11] 13.154 3.953 4.703 10.488 13.021 16.373 19.557 1.033 92 gamma[12] 3.229 4.058 -4.251 0.810 3.486 5.686 11.227 1.052 51 gamma[13] -10.042 4.023 -18.292 -12.658 -9.958 -7.041 -3.116 0.996 240 gamma[14] -25.716 3.634 -32.214 -28.050 -25.803 -23.263 -18.656 1.006 240 gamma[15] 10.945 3.979 3.108 8.289 10.864 13.681 18.596 1.045 240 gamma[16] -1.576 3.918 -9.405 -3.977 -1.559 1.133 5.360 0.997 240 gamma[17] 2.662 4.049 -5.193 0.321 2.811 5.307 11.232 1.001 240 gamma[18] 11.232 3.863 3.445 8.656 11.181 13.784 19.571 1.024 98 gamma[19] 12.064 3.899 5.349 9.442 11.912 14.854 19.142 0.998 240 gamma[20] 10.868 3.806 4.005 8.199 10.914 13.538 18.130 1.007 240 gamma[21] 5.422 3.943 -2.621 3.048 5.378 7.714 13.414 0.997 240 gamma[22] 7.083 4.144 -0.541 4.113 7.396 9.356 15.841 0.999 240 gamma[23] -1.507 3.811 -9.257 -3.930 -1.445 1.128 5.511 1.012 180 gamma[24] -17.275 3.806 -24.474 -20.057 -17.313 -14.482 -10.362 0.999 240 gamma[25] 11.314 4.062 2.820 9.022 11.183 13.790 19.509 1.003 240 gamma[26] 1.584 3.991 -5.260 -1.315 1.658 4.359 9.635 1.012 120 gamma[27] -1.402 3.847 -8.902 -3.858 -1.285 1.133 5.091 1.000 240 gamma[28] -14.559 4.204 -22.387 -17.148 -14.455 -11.640 -6.699 1.005 200 gamma[29] -2.553 3.910 -9.497 -5.440 -2.464 0.183 5.127 0.998 240 gamma[30] 7.490 3.837 -0.275 5.116 7.411 9.964 15.956 1.000 240 gamma[31] 16.192 3.878 8.842 13.650 16.257 18.632 23.669 1.019 180 gamma[32] -0.800 4.007 -9.479 -3.350 -0.723 1.860 7.501 1.009 240 gamma[33] 5.111 4.036 -2.594 2.381 5.173 7.690 13.132 1.011 200 gamma[34] -2.519 4.072 -11.135 -5.072 -2.611 -0.049 5.238 1.015 130 gamma[35] -17.969 3.987 -26.347 -20.657 -17.664 -15.212 -11.042 1.007 180 gamma[36] -4.408 3.843 -12.305 -6.819 -4.419 -1.697 2.703 1.000 240 sigma.block 11.256 1.325 8.763 10.384 11.135 12.057 14.049 1.005 240 sigma.res 4.348 0.399 3.693 4.077 4.322 4.630 5.119 1.000 240 deviance 623.502 11.438 602.496 615.229 623.356 631.607 644.954 0.995 240 For each parameter, n.eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor (at convergence, Rhat=1). DIC info (using the rule, pD = var(deviance)/2) pD = 65.9 and DIC = 689.4 DIC is an estimate of expected predictive error (lower deviance is better).  library(rstan) modelString=" data { int<lower=1> n; int<lower=1> nX; int<lower=1> nBlock; vector [n] y; matrix [n,nX] X; int Z[n]; } parameters { vector[nX] beta; real<lower=0> sigma; vector [nBlock] gamma; real<lower=0> sigmaBlock; } transformed parameters { vector[n] eta; eta <- X*beta; for (i in 1:n) { eta[i] <- eta[i] + gamma[Z[i]]; } } model { #Likelihood y~normal(eta,sigma); #Priors beta ~ normal(0,1000); sigma~cauchy(0,5); gamma ~ normal(0,sigmaBlock); sigmaBlock~cauchy(0,5); } generated quantities { vector[n] log_lik; for (i in 1:n) { log_lik[i] <- normal_log(y[i], eta, sigma); } } " Xmat <- model.matrix(~A*C,data=data.splt) data.splt.list <- with(data.splt, list(y = y, X = Xmat, nX=ncol(Xmat),n = nrow(data.splt), Z=as.numeric(Block), nBlock=length(levels(Block)))) library(rstan) data.splt.rstan <- stan(data=data.splt.list, model_code=modelString, chains=3, iter=1000, warmup=500, thin=2, save_dso=TRUE )  SAMPLING FOR MODEL 'f3aefea794fe54ff9ebd376cb1a2914e' NOW (CHAIN 1). Chain 1, Iteration: 1 / 1000 [ 0%] (Warmup) Chain 1, Iteration: 100 / 1000 [ 10%] (Warmup) Chain 1, Iteration: 200 / 1000 [ 20%] (Warmup) Chain 1, Iteration: 300 / 1000 [ 30%] (Warmup) Chain 1, Iteration: 400 / 1000 [ 40%] (Warmup) Chain 1, Iteration: 500 / 1000 [ 50%] (Warmup) Chain 1, Iteration: 501 / 1000 [ 50%] (Sampling) Chain 1, Iteration: 600 / 1000 [ 60%] (Sampling) Chain 1, Iteration: 700 / 1000 [ 70%] (Sampling) Chain 1, Iteration: 800 / 1000 [ 80%] (Sampling) Chain 1, Iteration: 900 / 1000 [ 90%] (Sampling) Chain 1, Iteration: 1000 / 1000 [100%] (Sampling) # Elapsed Time: 0.665549 seconds (Warm-up) # 0.377485 seconds (Sampling) # 1.04303 seconds (Total) SAMPLING FOR MODEL 'f3aefea794fe54ff9ebd376cb1a2914e' NOW (CHAIN 2). Chain 2, Iteration: 1 / 1000 [ 0%] (Warmup) Chain 2, Iteration: 100 / 1000 [ 10%] (Warmup) Chain 2, Iteration: 200 / 1000 [ 20%] (Warmup) Chain 2, Iteration: 300 / 1000 [ 30%] (Warmup) Chain 2, Iteration: 400 / 1000 [ 40%] (Warmup) Chain 2, Iteration: 500 / 1000 [ 50%] (Warmup) Chain 2, Iteration: 501 / 1000 [ 50%] (Sampling) Chain 2, Iteration: 600 / 1000 [ 60%] (Sampling) Chain 2, Iteration: 700 / 1000 [ 70%] (Sampling) Chain 2, Iteration: 800 / 1000 [ 80%] (Sampling) Chain 2, Iteration: 900 / 1000 [ 90%] (Sampling) Chain 2, Iteration: 1000 / 1000 [100%] (Sampling) # Elapsed Time: 0.657476 seconds (Warm-up) # 0.361092 seconds (Sampling) # 1.01857 seconds (Total) SAMPLING FOR MODEL 'f3aefea794fe54ff9ebd376cb1a2914e' NOW (CHAIN 3). Chain 3, Iteration: 1 / 1000 [ 0%] (Warmup) Chain 3, Iteration: 100 / 1000 [ 10%] (Warmup) Chain 3, Iteration: 200 / 1000 [ 20%] (Warmup) Chain 3, Iteration: 300 / 1000 [ 30%] (Warmup) Chain 3, Iteration: 400 / 1000 [ 40%] (Warmup) Chain 3, Iteration: 500 / 1000 [ 50%] (Warmup) Chain 3, Iteration: 501 / 1000 [ 50%] (Sampling) Chain 3, Iteration: 600 / 1000 [ 60%] (Sampling) Chain 3, Iteration: 700 / 1000 [ 70%] (Sampling) Chain 3, Iteration: 800 / 1000 [ 80%] (Sampling) Chain 3, Iteration: 900 / 1000 [ 90%] (Sampling) Chain 3, Iteration: 1000 / 1000 [100%] (Sampling) # Elapsed Time: 0.684195 seconds (Warm-up) # 0.341404 seconds (Sampling) # 1.0256 seconds (Total)  print(data.splt.rstan, pars=c('beta','sigmaBlock','sigma'))  Inference for Stan model: f3aefea794fe54ff9ebd376cb1a2914e. 3 chains, each with iter=1000; warmup=500; thin=2; post-warmup draws per chain=250, total post-warmup draws=750. mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat beta[1] 44.30 0.30 3.26 37.71 42.19 44.34 46.44 50.26 121 1.01 beta[2] -8.69 0.54 4.58 -17.44 -11.86 -9.07 -5.57 0.26 72 1.05 beta[3] -11.93 0.43 4.84 -21.34 -15.31 -11.92 -8.56 -2.23 125 1.00 beta[4] 31.09 0.10 1.82 27.47 29.86 31.24 32.24 34.37 303 1.01 beta[5] 38.71 0.11 1.80 35.07 37.53 38.70 39.91 42.17 246 1.01 beta[6] -15.26 0.18 2.55 -20.02 -17.04 -15.27 -13.53 -10.02 191 1.01 beta[7] -24.71 0.12 2.53 -29.57 -26.41 -24.80 -23.01 -19.80 415 1.01 beta[8] -4.45 0.16 2.52 -9.32 -6.15 -4.51 -2.83 0.56 255 1.03 beta[9] -29.92 0.15 2.59 -34.93 -31.68 -29.95 -28.26 -25.05 303 1.01 sigmaBlock 11.13 0.06 1.49 8.60 10.11 10.92 12.00 14.48 542 1.00 sigma 4.37 0.02 0.39 3.66 4.11 4.33 4.61 5.27 350 1.01 Samples were drawn using NUTS(diag_e) at Wed Dec 16 15:33:26 2015. For each parameter, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat=1).  library(brms) data.splt.brm <- brm(y~A*C+(1|Block), data=data.splt, family='gaussian', prior=c(set_prior('normal(0,1000)', class='b'), set_prior('cauchy(0,5)', class='sd')), n.chains=3, n.iter=2000, warmup=500, n.thin=2 )  SAMPLING FOR MODEL 'gaussian(identity) brms-model' NOW (CHAIN 1). Chain 1, Iteration: 1 / 2000 [ 0%] (Warmup) Chain 1, Iteration: 200 / 2000 [ 10%] (Warmup) Chain 1, Iteration: 400 / 2000 [ 20%] (Warmup) Chain 1, Iteration: 501 / 2000 [ 25%] (Sampling) Chain 1, Iteration: 700 / 2000 [ 35%] (Sampling) Chain 1, Iteration: 900 / 2000 [ 45%] (Sampling) Chain 1, Iteration: 1100 / 2000 [ 55%] (Sampling) Chain 1, Iteration: 1300 / 2000 [ 65%] (Sampling) Chain 1, Iteration: 1500 / 2000 [ 75%] (Sampling) Chain 1, Iteration: 1700 / 2000 [ 85%] (Sampling) Chain 1, Iteration: 1900 / 2000 [ 95%] (Sampling) Chain 1, Iteration: 2000 / 2000 [100%] (Sampling) # Elapsed Time: 0.789865 seconds (Warm-up) # 1.02192 seconds (Sampling) # 1.81179 seconds (Total) SAMPLING FOR MODEL 'gaussian(identity) brms-model' NOW (CHAIN 2). Chain 2, Iteration: 1 / 2000 [ 0%] (Warmup) Chain 2, Iteration: 200 / 2000 [ 10%] (Warmup) Chain 2, Iteration: 400 / 2000 [ 20%] (Warmup) Chain 2, Iteration: 501 / 2000 [ 25%] (Sampling) Chain 2, Iteration: 700 / 2000 [ 35%] (Sampling) Chain 2, Iteration: 900 / 2000 [ 45%] (Sampling) Chain 2, Iteration: 1100 / 2000 [ 55%] (Sampling) Chain 2, Iteration: 1300 / 2000 [ 65%] (Sampling) Chain 2, Iteration: 1500 / 2000 [ 75%] (Sampling) Chain 2, Iteration: 1700 / 2000 [ 85%] (Sampling) Chain 2, Iteration: 1900 / 2000 [ 95%] (Sampling) Chain 2, Iteration: 2000 / 2000 [100%] (Sampling) # Elapsed Time: 0.755736 seconds (Warm-up) # 1.27334 seconds (Sampling) # 2.02907 seconds (Total) SAMPLING FOR MODEL 'gaussian(identity) brms-model' NOW (CHAIN 3). Chain 3, Iteration: 1 / 2000 [ 0%] (Warmup) Chain 3, Iteration: 200 / 2000 [ 10%] (Warmup) Chain 3, Iteration: 400 / 2000 [ 20%] (Warmup) Chain 3, Iteration: 501 / 2000 [ 25%] (Sampling) Chain 3, Iteration: 700 / 2000 [ 35%] (Sampling) Chain 3, Iteration: 900 / 2000 [ 45%] (Sampling) Chain 3, Iteration: 1100 / 2000 [ 55%] (Sampling) Chain 3, Iteration: 1300 / 2000 [ 65%] (Sampling) Chain 3, Iteration: 1500 / 2000 [ 75%] (Sampling) Chain 3, Iteration: 1700 / 2000 [ 85%] (Sampling) Chain 3, Iteration: 1900 / 2000 [ 95%] (Sampling) Chain 3, Iteration: 2000 / 2000 [100%] (Sampling) # Elapsed Time: 0.694082 seconds (Warm-up) # 1.24235 seconds (Sampling) # 1.93643 seconds (Total)  summary(data.splt.brm)   Family: gaussian (identity) Formula: y ~ A * C + (1 | Block) Data: data.splt (Number of observations: 108) Samples: 3 chains, each with n.iter = 2000; n.warmup = 500; n.thin = 2; total post-warmup samples = 2250 WAIC: 666.22 Random Effects: ~Block (Number of levels: 36) Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat sd(Intercept) 11.08 1.41 8.67 14.15 562 1 Fixed Effects: Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat Intercept 44.40 3.44 37.83 51.34 378 1.00 A2 -8.84 4.83 -19.08 -0.08 418 1.00 A3 -11.65 5.01 -21.72 -1.87 374 1.01 C2 31.20 1.77 27.57 34.64 1355 1.00 C3 38.81 1.74 35.37 42.24 1292 1.00 A2:C2 -15.34 2.53 -20.59 -10.38 1366 1.00 A3:C2 -24.89 2.50 -29.79 -20.04 1383 1.00 A2:C3 -4.49 2.51 -9.39 0.42 1362 1.00 A3:C3 -30.07 2.51 -34.96 -25.30 1331 1.00 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat sigma(y) 4.4 0.4 3.73 5.27 966 1 Samples were drawn using NUTS(diag_e). For each parameter, Eff.Sample is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1).  stancode(data.splt.brm)  functions { } data { int<lower=1> N; # number of observations vector[N] Y; # response variable int<lower=1> K; # number of fixed effects matrix[N, K] X; # FE design matrix # data for random effects of Block int<lower=1> J_1[N]; # RE levels int<lower=1> N_1; # number of levels int<lower=1> K_1; # number of REs real Z_1[N]; # RE design matrix } transformed data { } parameters { real b_Intercept; # fixed effects Intercept vector[K] b; # fixed effects vector[N_1] pre_1; # unscaled REs real<lower=0> sd_1; # RE standard deviation real<lower=0> sigma; # residual SD } transformed parameters { vector[N] eta; # linear predictor vector[N_1] r_1; # REs # compute linear predictor eta <- X * b + b_Intercept; r_1 <- sd_1 * (pre_1); # scale REs # if available add REs to linear predictor for (n in 1:N) { eta[n] <- eta[n] + Z_1[n] * r_1[J_1[n]]; } } model { # prior specifications b_Intercept ~ normal(0,1000); b ~ normal(0,1000); sd_1 ~ cauchy(0,5); pre_1 ~ normal(0, 1); sigma ~ cauchy(0, 21); # likelihood contribution Y ~ normal(eta, sigma); } generated quantities { }  In preparation for fitting the INLA model, we should consider what specific questions we wish to address from the analysis and therefore how we will query the posteriors. I suggest the following as a short list (there are obviously others that could also be of interest depending on the ecological context of the data): • Is there are difference between the different A treatments (A1, A2, A3) at level C3 • What is the magnitude of the effect between C1 and C3 for each of the A treatments We will of course define linear prediction frames for the following: • fitted values - called fitted • fitted values (ignoring Block) - called pred • cell means predictions - called newdata To address our additional questions, we will define a series of linear combinations pred <- fitted <- subset(data.splt, select=c(A,C,Block,y)) fitted$y <- pred$y <- pred$Block <- NA
newdata <- expand.grid(A=levels(data.splt$A), C=levels(data.splt$C), Block=NA, y=NA)
data.pred <- rbind(subset(data.splt, select=c(A,C,Block,y)),  fitted, pred ,newdata)

## linear combinations
Xmat = model.matrix(~A*C, data=newdata)
## to compare A1,A2,A3 at C3, we need to generate a model matrix that reflects the associated
## differences in their Xmat rows.
## The rows we are interested in are 7-9 as evident in the following
newdata

  A C Block  y
1 1 1    NA NA
2 2 1    NA NA
3 3 1    NA NA
4 1 2    NA NA
5 2 2    NA NA
6 3 2    NA NA
7 1 3    NA NA
8 2 3    NA NA
9 3 3    NA NA

## Question 1: Is there are difference between the different A treatments (A1, A2, A3) at level C3
Xmat1=rbind('A1 - A2'=Xmat[7,]-Xmat[8,],
'A1 - A3'=Xmat[7,]-Xmat[9,],
'A2 - A3'=Xmat[8,]-Xmat[9,])
## Question 2: What is the magnitude of the effect between C1 and C3 for each of the A treatments
Xmat1=rbind(Xmat1,
'C3-C1|A1'=Xmat[7,]-Xmat[1,],
'C3-C1|A2'=Xmat[8,]-Xmat[2,],
'C3-C1|A3'=Xmat[9,]-Xmat[3,]
)

lincomb=inla.make.lincombs(as.data.frame(Xmat1))


Now lets fit the model.

#fit the model
data.inla <- inla(y~A*C + f(Block, model='iid'),
data=data.pred,
lincomb=lincomb,
control.inla=list(lincomb.derived.only=FALSE),
control.compute=list(dic=TRUE, cpo=TRUE, waic=TRUE))
#examine the regular summary
summary(data.inla)

Call:
c("inla(formula = y ~ A * C + f(Block, model = \"iid\"), data = data.pred, ",  "    lincomb = lincomb, control.compute = list(dic = TRUE, cpo = TRUE, ",  "        waic = TRUE), control.inla = list(lincomb.derived.only = FALSE))" )

Time used:
Pre-processing    Running inla Post-processing           Total
0.1871          0.3012          0.0732          0.5615

Fixed effects:
mean     sd 0.025quant 0.5quant 0.975quant     mode kld
(Intercept)  44.3882 3.3265    37.8109  44.3930    50.9295  44.4021   0
A2           -8.7807 4.6904   -18.0132  -8.7879     0.4786  -8.8007   0
A3          -11.5528 4.6905   -20.7835 -11.5606    -2.2915 -11.5746   0
C2           30.8737 1.7433    27.4281  30.8779    34.2920  30.8861   0
C3           38.5318 1.7433    35.0861  38.5360    41.9500  38.5443   0
A2:C2       -15.0035 2.4645   -19.8432 -15.0083   -10.1430 -15.0174   0
A3:C2       -24.4923 2.4648   -29.3299 -24.4981   -19.6289 -24.5090   0
A2:C3        -4.1834 2.4645    -9.0237  -4.1880     0.6764  -4.1968   0
A3:C3       -29.6824 2.4649   -34.5196 -29.6883   -24.8184 -29.6995   0

Linear combinations (derived):
ID    mean     sd 0.025quant 0.5quant 0.975quant    mode kld
lc1  1 12.9642 4.7082     3.6537  12.9715    22.2205 12.9857   0
lc2  2 41.2352 4.7083    31.9223  41.2433    50.4896 41.2589   0
lc3  3 28.2710 4.7396    18.9168  28.2716    37.6081 28.2732   0
lc4  4 38.5318 1.7433    35.0861  38.5360    41.9500 38.5443   0
lc5  5 34.3484 1.7500    30.9025  34.3480    37.7919 34.3475   0
lc6  6  8.8494 1.7501     5.4071   8.8478    12.2965  8.8448   0

Linear combinations:
ID    mean     sd 0.025quant 0.5quant 0.975quant    mode kld
lc1  1 12.9642 4.7082     3.6537  12.9715    22.2205 12.9857   0
lc2  2 41.2352 4.7083    31.9223  41.2433    50.4896 41.2589   0
lc3  3 28.2710 4.7396    18.9168  28.2716    37.6081 28.2732   0
lc4  4 38.5318 1.7433    35.0861  38.5360    41.9500 38.5443   0
lc5  5 34.3484 1.7500    30.9025  34.3480    37.7919 34.3475   0
lc6  6  8.8494 1.7501     5.4071   8.8478    12.2965  8.8448   0

Random effects:
Name	  Model
Block   IID model

Model hyperparameters:
mean     sd 0.025quant 0.5quant
Precision for the Gaussian observations 0.0552 0.0095     0.0387   0.0545
Precision for Block                     0.0089 0.0022     0.0051   0.0087
0.975quant   mode
Precision for the Gaussian observations     0.0760 0.0532
Precision for Block                         0.0139 0.0083

Expected number of effective parameters(std dev): 40.23(0.5059)
Number of equivalent replicates : 2.685

Deviance Information Criterion (DIC) ...: 663.72
Effective number of parameters .........: 41.28

Watanabe-Akaike information criterion (WAIC) ...: 665.73
Effective number of parameters .................: 34.52

Marginal log-Likelihood:  -468.85
CPO and PIT are computed

Posterior marginals for linear predictor and fitted values computed

s <- inla.contrib.sd(data.inla, nsamples=1000)
s$hyper   mean sd 2.5% 97.5% sd for the Gaussian observations 4.292209 0.367951 3.628069 5.030097 sd for Block 10.817808 1.352594 8.433952 13.741639  newdata <- cbind(newdata, data.inla$summary.linear.predictor[(nrow(data.splt)+nrow(fitted)+nrow(pred)+1):nrow(data.pred),])

              A C Block  y     mean       sd 0.025quant 0.5quant 0.975quant
predictor.325 1 1    NA NA 44.38840 3.326560   37.81966 44.39321   50.92952
predictor.326 2 1    NA NA 35.60775 3.346761   29.01963 35.60543   42.20765
predictor.327 3 1    NA NA 32.83571 3.346796   26.24966 32.83273   39.43807
predictor.328 1 2    NA NA 75.26213 3.336744   68.67216 75.26738   81.82216
predictor.329 2 2    NA NA 51.47799 3.351807   44.88009 51.47560   58.08802
predictor.330 3 2    NA NA 39.21712 3.351838   32.62154 39.21398   45.82984
mode          kld
predictor.325 44.40211 1.296486e-09
predictor.326 35.60120 4.667141e-09
predictor.327 32.82730 5.053008e-09
predictor.328 75.27723 1.257169e-09
predictor.329 51.47123 4.692768e-09
predictor.330 39.20821 5.087045e-09

newdata <- reshape:::rename(newdata, c("0.025quant"="lower", "0.975quant"="upper"))

fitted <- cbind(fitted,
data.inla$summary.linear.predictor[(nrow(data.splt)+1):(nrow(data.splt)+nrow(fitted)),]) fitted <- reshape:::rename(fitted, c("0.025quant"="lower", "0.975quant"="upper")) pred <- cbind(pred, data.inla$summary.linear.predictor[(nrow(data.splt)+nrow(fitted)+1):
(nrow(data.splt)+nrow(fitted)+nrow(pred)),])
pred <- reshape:::rename(pred, c("0.025quant"="lower", "0.975quant"="upper"))

ndata <- data.splt
ndata$fit <- fitted$mean
ndata$pred <- pred$mean
ndata$Res <- (ndata$fit - data.splt$y) ndata$Pobs <- ndata$pred + ndata$Res

         y A C Block A.1      fit     pred        Res     Pobs
1 30.51110 1 1     1   1 34.31855 44.38840  3.8074422 48.19584
2 62.18599 1 2     1   1 65.19225 75.26213  3.0062668 78.26840
3 77.98268 1 3     1   1 72.85033 82.92021 -5.1323504 77.78786
4 46.01960 1 1     2   1 43.05227 44.38840 -2.9673325 41.42106
5 71.38110 1 2     2   1 73.92597 75.26213  2.5448720 77.80700
6 80.93691 1 3     2   1 81.58405 82.92021  0.6471391 83.56735

library(grid)
ggplot(newdata, aes(y=mean, x=C, group=A)) +
geom_blank()+
geom_line(aes(linetype=A,x=as.numeric(C)),position=position_dodge(width=0.1)) +
geom_linerange(aes(ymin=lower, ymax=upper),position=position_dodge(width=0.1))+
geom_point(aes(fill=A),position=position_dodge(width=0.1), shape=21, size=3)+
scale_fill_manual('A', breaks=c(1,2,3), labels=c('1','2','3'), values=c('black','grey','white'))+
theme_classic()+
theme(legend.key.width=unit(2,'lines'), legend.position=c(0,1), legend.justification=c(0,1))


And as for our specific comparisons..

data.inla$summary.lincomb   ID mean sd 0.025quant 0.5quant 0.975quant mode lc1 1 12.964174 4.708219 3.653675 12.971501 22.22051 12.985699 lc2 2 41.235159 4.708295 31.922276 41.243254 50.48961 41.258911 lc3 3 28.270985 4.739604 18.916819 28.271621 37.60812 28.273240 lc4 4 38.531814 1.743316 35.086055 38.536013 41.94998 38.544291 lc5 5 34.348382 1.750041 30.902487 34.348030 37.79192 34.347497 lc6 6 8.849444 1.750136 5.407147 8.847806 12.29653 8.844786 kld lc1 3.208567e-13 lc2 3.743741e-13 lc3 1.682744e-13 lc4 6.920369e-13 lc5 3.155241e-13 lc6 3.653648e-13  ### More complex linear mixed effects model Random data incorporating the following properties • the number of Regions = 4 (A,B,C,D) • the number of Blocks within each Region = 5 • the number of Management strategies within each Block = 2 (a, b) • the number of Sites of each Management strategy = 2 • the number of Transects in each Site =3 • the number of years repeatedly sampled • the mean abundance of the response (y) = 25 • the effect of the Regions is -3, -10, -20 (in the first year) • the initial Management effect (in Region A) = 0 • the year and interaction effects provided • the variability (standard deviation) between Blocks of the same Region = 5 • the variability (standard deviation) between Sites within Block/Management = 2 • the variability (standard deviation) between Transect within Sites = 10 • the gaussian noise (standard deviation) from which observations are drawn = 5 set.seed(123) n.region <- 4 n.block <- 5 n.management <- 2 n.site <- 2 n.transect <- 3 n.year <- 15 dat <- expand.grid(Region = 1:4, Block = 1:5, Management = 1:2, Site = 1:2, Transect = 1:3, Year = 2000:2014) n <- n.region * n.block * n.management * n.site * n.transect * n.year Region <- gl(n = n.region, k = n.block * n.management * n.site * n.transect * n.year, length = n, lab = LETTERS[1:n.region]) Block <- gl(n = n.block, k = n.management * n.site * n.transect * n.year, length = n, lab = paste("Block", 1:n.block)) Management <- gl(n = n.management, k = n.site * n.transect * n.year, length = n, lab = letters[1:n.management]) Site <- gl(n = n.site, k = n.transect * n.year, length = n, lab = paste("Site", 1:n.site)) Transect <- gl(n = n.transect, k = n.year, length = n, lab = paste("Transect", 1:n.transect)) Year <- gl(n = n.year, k = 1, length = n, lab = 2000:2014) dat <- data.frame(Region, Block, Management, Site, Transect, Year) dat.x <- data.frame(fRegion = Region, fBlock = interaction(Region, Block), fManagement = Management, fSite = interaction(Region, Block, Site), fTransect = interaction(Region, Block, Site, Transect), Year) Xmat <- model.matrix(~Region + Year + Management + Management:Year + Region:Management + Region:Year + Region:Management:Year, data = dat.x) Zmat1 <- model.matrix(~-1 + fBlock, data = dat.x) Zmat2 <- model.matrix(~-1 + fSite, data = dat.x) Zmat3 <- model.matrix(~-1 + fTransect, data = dat.x) ## Fixed effects intercept.mean <- 25 # mu alpha region.eff <- c(-3, -10, -20) year.eff <- c(1, -2, -6, -9, -12, -7, -5, -5, -3, -1, -3, -6, -3, -8) management.eff <- 0 management.year.eff <- c(0, 0, 0, 2, 8, 12, 12, 12, 12, 10, 8, 7, 10, 15) management.region.eff <- c(0, 0, 0) interaction.eff <- c(-5, 2, 0, 2, 4, 6, 0, 6, 8, 1, 8, 9, 2, 9, 12, -1, 7, 9, 3, 5, 3, 1, 4, 1, 0, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 2) interaction1.eff <- c(0, 0, 0, -1, 0, 0, -2, 0, 0, -1, 0, 0, -3, -5, 0, -1, -5, 0, -3, -5, 0, -2, -5, 0, -2, -5, 0, 0, -5, 0, 0, -5, 0, 0, -5, 0, 0, -5, 0, 0, -5, 0) fixed.effects <- c(intercept.mean, region.eff, year.eff, management.eff, management.year.eff, management.region.eff, interaction.eff, interaction1.eff) # Put them all together ## generate a data frame and model matrix to capture the raw fixed effect cell ## means ndata.fixed <- expand.grid(Region = levels(dat$Region), Year = levels(dat$Year), Management = levels(dat$Management))
ndata.Xmat <- model.matrix(~Region + Year + Management + Management:Year + Region:Management +
Region:Year + Region:Management:Year, data = ndata.fixed)

data.params.f <- cbind(ndata.fixed, y = (ndata.Xmat[, ] %*% fixed.effects))
ggplot(data.params.f, aes(y = y, x = as.numeric(as.character(Year)), color = Management)) +
geom_line() + facet_wrap(~Region)

## random effects
block.eff <- rnorm(n.region * n.block, mean = 0, sd = 5)
site.eff <- rnorm(n.region * n.block * n.site, mean = 0, sd = 2)
transect.eff <- rnorm(n.region * n.block * n.site * n.transect, mean = 0, sd = 10)

lin.pred <- (Xmat[, ] %*% fixed.effects) + (Zmat1[, ] %*% block.eff) + (Zmat2[, ] %*%
site.eff) + (Zmat3[, ] %*% transect.eff)

# lin.pred <- (Xmat[,] %*% fixed.effects)
lin.pred[lin.pred < 0] <- 0

y <- rnorm(n, lin.pred, 5)
data.mlm <- data.frame(dat, y)


  Region   Block Management   Site   Transect Year        y
1      A Block 1          a Site 1 Transect 1 2000 18.54174
2      A Block 1          a Site 1 Transect 1 2001 31.17430
3      A Block 1          a Site 1 Transect 1 2002 20.11012
4      A Block 1          a Site 1 Transect 1 2003 13.53080
5      A Block 1          a Site 1 Transect 1 2004 13.67697
6      A Block 1          a Site 1 Transect 1 2005 10.87249

library(ggplot2)
ggplot(data.mlm, aes(y = y, x = as.numeric(as.character(Year)), color = Management,
group = interaction(Region, Block, Site, Transect))) + geom_line() + geom_point() +
facet_grid(Block * Site ~ Region) + theme_classic()


$$y_i = \alpha + \beta X_{i} + \sum_{j=1}^J{}\gamma_{ij}\mathbf{Z_i} + \varepsilon_i \hspace{4em}\varepsilon_i\sim{}\mathcal{N}(0, \sigma^2)$$

Note, this takes a while to run!
library(nlme)
#Assuming sphericity
# random intercepts model
data.mlm.lme <- lme(y~Region*Management*Year, random=~1|Block/Site/Transect,
data=data.mlm, method='REML')
# random intercept and slopes model - note this is not an identifiable model
# and requires more resources than are available
#data.mlm.lme1 <- lme(y~Region*Management*Year, random=~Year|Block/Site/Year,
#                    data=data.mlm, method='REML')
#AIC(data.mlm.lme, data.mlm.lme1)
# random intercepts model 'best'
summary(data.mlm.lme)

Linear mixed-effects model fit by REML
Data: data.mlm
AIC      BIC    logLik
27085.25 27848.45 -13418.63

Random effects:
Formula: ~1 | Block
(Intercept)
StdDev:   0.7851926

Formula: ~1 | Site %in% Block
(Intercept)
StdDev:    2.870489

Formula: ~1 | Transect %in% Site %in% Block
(Intercept) Residual
StdDev:    2.530732 10.67443

Fixed effects: y ~ Region * Management * Year
Value Std.Error   DF   t-value p-value
(Intercept)                   24.983075  2.226854 3451 11.219002  0.0000
RegionB                       -3.144363  2.756125 3451 -1.140863  0.2540
RegionC                       -6.585873  2.756125 3451 -2.389540  0.0169
RegionD                      -16.786779  2.756125 3451 -6.090716  0.0000
Managementb                   -0.461356  2.756125 3451 -0.167393  0.8671
Year2001                       1.092046  2.756125 3451  0.396225  0.6920
Year2002                      -1.296597  2.756125 3451 -0.470442  0.6381
Year2003                      -7.213107  2.756125 3451 -2.617119  0.0089
Year2004                      -9.110367  2.756125 3451 -3.305498  0.0010
Year2005                     -11.028643  2.756125 3451 -4.001503  0.0001
Year2006                      -6.398429  2.756125 3451 -2.321530  0.0203
Year2007                      -5.101181  2.756125 3451 -1.850852  0.0643
Year2008                      -5.259972  2.756125 3451 -1.908466  0.0564
Year2009                      -3.048462  2.756125 3451 -1.106068  0.2688
Year2010                       0.029576  2.756125 3451  0.010731  0.9914
Year2011                      -3.659707  2.756125 3451 -1.327845  0.1843
Year2012                      -6.387772  2.756125 3451 -2.317664  0.0205
Year2013                      -3.000411  2.756125 3451 -1.088634  0.2764
Year2014                      -6.419585  2.756125 3451 -2.329206  0.0199
RegionB:Managementb           -0.490260  3.897750 3451 -0.125780  0.8999
RegionC:Managementb           -1.506586  3.897750 3451 -0.386527  0.6991
RegionD:Managementb           -1.301994  3.897750 3451 -0.334037  0.7384
RegionB:Year2001              -4.158386  3.897750 3451 -1.066868  0.2861
RegionC:Year2001               2.724414  3.897750 3451  0.698971  0.4846
RegionD:Year2001              -1.243091  3.897750 3451 -0.318925  0.7498
RegionB:Year2002               1.388455  3.897750 3451  0.356220  0.7217
RegionC:Year2002               1.793631  3.897750 3451  0.460171  0.6454
RegionD:Year2002               3.217287  3.897750 3451  0.825422  0.4092
RegionB:Year2003               2.376382  3.897750 3451  0.609681  0.5421
RegionC:Year2003               7.533771  3.897750 3451  1.932851  0.0533
RegionD:Year2003               8.085904  3.897750 3451  2.074506  0.0381
RegionB:Year2004               1.227342  3.897750 3451  0.314885  0.7529
RegionC:Year2004               7.804294  3.897750 3451  2.002256  0.0453
RegionD:Year2004               8.027336  3.897750 3451  2.059480  0.0395
RegionB:Year2005               0.915247  3.897750 3451  0.234814  0.8144
RegionC:Year2005               7.583662  3.897750 3451  1.945651  0.0518
RegionD:Year2005              10.511196  3.897750 3451  2.696734  0.0070
RegionB:Year2006              -1.969627  3.897750 3451 -0.505324  0.6134
RegionC:Year2006               6.324273  3.897750 3451  1.622544  0.1048
RegionD:Year2006               6.576699  3.897750 3451  1.687307  0.0916
RegionB:Year2007               2.492321  3.897750 3451  0.639425  0.5226
RegionC:Year2007               4.241940  3.897750 3451  1.088305  0.2765
RegionD:Year2007               3.030074  3.897750 3451  0.777391  0.4370
RegionB:Year2008               0.819754  3.897750 3451  0.210315  0.8334
RegionC:Year2008               4.957217  3.897750 3451  1.271815  0.2035
RegionD:Year2008               3.217624  3.897750 3451  0.825508  0.4091
RegionB:Year2009              -1.017748  3.897750 3451 -0.261112  0.7940
RegionC:Year2009              -1.278117  3.897750 3451 -0.327911  0.7430
RegionD:Year2009              -0.253945  3.897750 3451 -0.065152  0.9481
RegionB:Year2010              -0.813809  3.897750 3451 -0.208789  0.8346
RegionC:Year2010              -2.963120  3.897750 3451 -0.760213  0.4472
RegionD:Year2010              -1.860916  3.897750 3451 -0.477434  0.6331
RegionB:Year2011              -0.843949  3.897750 3451 -0.216522  0.8286
RegionC:Year2011              -0.388598  3.897750 3451 -0.099698  0.9206
RegionD:Year2011               2.993279  3.897750 3451  0.767951  0.4426
RegionB:Year2012               0.614050  3.897750 3451  0.157540  0.8748
RegionC:Year2012               0.847939  3.897750 3451  0.217546  0.8278
RegionD:Year2012              -0.080905  3.897750 3451 -0.020757  0.9834
RegionB:Year2013               0.844921  3.897750 3451  0.216772  0.8284
RegionC:Year2013               1.232426  3.897750 3451  0.316189  0.7519
RegionD:Year2013               1.325756  3.897750 3451  0.340134  0.7338
RegionB:Year2014              -1.497522  3.897750 3451 -0.384202  0.7009
RegionC:Year2014              -2.710367  3.897750 3451 -0.695367  0.4869
RegionD:Year2014               2.310542  3.897750 3451  0.592789  0.5534
Managementb:Year2001           0.272414  3.897750 3451  0.069890  0.9443
Managementb:Year2002           0.894072  3.897750 3451  0.229382  0.8186
Managementb:Year2003           1.980613  3.897750 3451  0.508143  0.6114
Managementb:Year2004           3.684594  3.897750 3451  0.945313  0.3446
Managementb:Year2005           8.726425  3.897750 3451  2.238836  0.0252
Managementb:Year2006          12.533006  3.897750 3451  3.215446  0.0013
Managementb:Year2007          12.369640  3.897750 3451  3.173533  0.0015
Managementb:Year2008          14.210457  3.897750 3451  3.645810  0.0003
Managementb:Year2009          13.018350  3.897750 3451  3.339965  0.0008
Managementb:Year2010           9.879000  3.897750 3451  2.534539  0.0113
Managementb:Year2011           9.625410  3.897750 3451  2.469479  0.0136
Managementb:Year2012           7.684252  3.897750 3451  1.971458  0.0488
Managementb:Year2013          10.692889  3.897750 3451  2.743349  0.0061
Managementb:Year2014          14.557136  3.897750 3451  3.734754  0.0002
RegionB:Managementb:Year2001   1.218654  5.512251 3451  0.221081  0.8250
RegionC:Managementb:Year2001   0.917516  5.512251 3451  0.166450  0.8678
RegionD:Managementb:Year2001   1.328626  5.512251 3451  0.241032  0.8095
RegionB:Managementb:Year2002  -1.331460  5.512251 3451 -0.241546  0.8091
RegionC:Managementb:Year2002   0.539761  5.512251 3451  0.097920  0.9220
RegionD:Managementb:Year2002   2.136347  5.512251 3451  0.387563  0.6984
RegionB:Managementb:Year2003  -4.724825  5.512251 3451 -0.857150  0.3914
RegionC:Managementb:Year2003  -1.685112  5.512251 3451 -0.305703  0.7598
RegionD:Managementb:Year2003  -0.645305  5.512251 3451 -0.117067  0.9068
RegionB:Managementb:Year2004  -1.909914  5.512251 3451 -0.346485  0.7290
RegionC:Managementb:Year2004   0.923206  5.512251 3451  0.167483  0.8670
RegionD:Managementb:Year2004   1.186634  5.512251 3451  0.215272  0.8296
RegionB:Managementb:Year2005  -1.698437  5.512251 3451 -0.308120  0.7580
RegionC:Managementb:Year2005  -1.188338  5.512251 3451 -0.215581  0.8293
RegionD:Managementb:Year2005   0.566060  5.512251 3451  0.102691  0.9182
RegionB:Managementb:Year2006  -2.251086  5.512251 3451 -0.408379  0.6830
RegionC:Managementb:Year2006  -3.250624  5.512251 3451 -0.589709  0.5554
RegionD:Managementb:Year2006  -0.187909  5.512251 3451 -0.034089  0.9728
RegionB:Managementb:Year2007  -1.280319  5.512251 3451 -0.232268  0.8163
RegionC:Managementb:Year2007  -3.743853  5.512251 3451 -0.679188  0.4971
RegionD:Managementb:Year2007  -1.509483  5.512251 3451 -0.273842  0.7842
RegionB:Managementb:Year2008  -3.154177  5.512251 3451 -0.572212  0.5672
RegionC:Managementb:Year2008  -7.006974  5.512251 3451 -1.271164  0.2038
RegionD:Managementb:Year2008  -4.092433  5.512251 3451 -0.742425  0.4579
RegionB:Managementb:Year2009  -1.170009  5.512251 3451 -0.212256  0.8319
RegionC:Managementb:Year2009  -2.968861  5.512251 3451 -0.538593  0.5902
RegionD:Managementb:Year2009  -4.177561  5.512251 3451 -0.757868  0.4486
RegionB:Managementb:Year2010   1.179948  5.512251 3451  0.214059  0.8305
RegionC:Managementb:Year2010  -1.807891  5.512251 3451 -0.327977  0.7429
RegionD:Managementb:Year2010  -2.325576  5.512251 3451 -0.421892  0.6731
RegionB:Managementb:Year2011   0.171569  5.512251 3451  0.031125  0.9752
RegionC:Managementb:Year2011  -5.218953  5.512251 3451 -0.946792  0.3438
RegionD:Managementb:Year2011  -4.493347  5.512251 3451 -0.815157  0.4150
RegionB:Managementb:Year2012   0.474024  5.512251 3451  0.085995  0.9315
RegionC:Managementb:Year2012  -4.911883  5.512251 3451 -0.891085  0.3729
RegionD:Managementb:Year2012   0.983547  5.512251 3451  0.178429  0.8584
RegionB:Managementb:Year2013  -1.551035  5.512251 3451 -0.281380  0.7784
RegionC:Managementb:Year2013  -5.755470  5.512251 3451 -1.044123  0.2965
RegionD:Managementb:Year2013  -0.969721  5.512251 3451 -0.175921  0.8604
RegionB:Managementb:Year2014   0.958792  5.512251 3451  0.173938  0.8619
RegionC:Managementb:Year2014  -2.388451  5.512251 3451 -0.433299  0.6648
RegionD:Managementb:Year2014  -1.893498  5.512251 3451 -0.343507  0.7312
Correlation:
(Intr) ReginB ReginC ReginD Mngmnt Yr2001 Yr2002
RegionB                      -0.619
RegionC                      -0.619  0.500
RegionD                      -0.619  0.500  0.500
Managementb                  -0.619  0.500  0.500  0.500
Year2001                     -0.619  0.500  0.500  0.500  0.500
Year2002                     -0.619  0.500  0.500  0.500  0.500  0.500
Year2003                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2004                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2005                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2006                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2007                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2008                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2009                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2010                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2011                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2012                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2013                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
Year2014                     -0.619  0.500  0.500  0.500  0.500  0.500  0.500
RegionB:Managementb           0.438 -0.707 -0.354 -0.354 -0.707 -0.354 -0.354
RegionC:Managementb           0.438 -0.354 -0.707 -0.354 -0.707 -0.354 -0.354
RegionD:Managementb           0.438 -0.354 -0.354 -0.707 -0.707 -0.354 -0.354
RegionB:Year2001              0.438 -0.707 -0.354 -0.354 -0.354 -0.707 -0.354
RegionC:Year2001              0.438 -0.354 -0.707 -0.354 -0.354 -0.707 -0.354
RegionD:Year2001              0.438 -0.354 -0.354 -0.707 -0.354 -0.707 -0.354
RegionB:Year2002              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.707
RegionC:Year2002              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.707
RegionD:Year2002              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.707
RegionB:Year2003              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2003              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2003              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2004              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2004              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2004              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2005              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2005              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2005              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2006              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2006              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2006              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2007              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2007              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2007              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2008              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2008              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2008              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2009              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2009              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2009              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2010              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2010              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2010              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2011              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2011              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2011              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2012              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2012              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2012              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2013              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2013              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2013              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2014              0.438 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2014              0.438 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2014              0.438 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
Managementb:Year2001          0.438 -0.354 -0.354 -0.354 -0.707 -0.707 -0.354
Managementb:Year2002          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.707
Managementb:Year2003          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2004          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2005          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2006          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2007          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2008          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2009          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2010          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2011          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2012          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2013          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2014          0.438 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
RegionB:Managementb:Year2001 -0.309  0.500  0.250  0.250  0.500  0.500  0.250
RegionC:Managementb:Year2001 -0.309  0.250  0.500  0.250  0.500  0.500  0.250
RegionD:Managementb:Year2001 -0.309  0.250  0.250  0.500  0.500  0.500  0.250
RegionB:Managementb:Year2002 -0.309  0.500  0.250  0.250  0.500  0.250  0.500
RegionC:Managementb:Year2002 -0.309  0.250  0.500  0.250  0.500  0.250  0.500
RegionD:Managementb:Year2002 -0.309  0.250  0.250  0.500  0.500  0.250  0.500
RegionB:Managementb:Year2003 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2003 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2003 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2004 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2004 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2004 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2005 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2005 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2005 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2006 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2006 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2006 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2007 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2007 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2007 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2008 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2008 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2008 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2009 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2009 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2009 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2010 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2010 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2010 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2011 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2011 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2011 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2012 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2012 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2012 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2013 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2013 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2013 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
RegionB:Managementb:Year2014 -0.309  0.500  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2014 -0.309  0.250  0.500  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2014 -0.309  0.250  0.250  0.500  0.500  0.250  0.250
Yr2003 Yr2004 Yr2005 Yr2006 Yr2007 Yr2008 Yr2009
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004                      0.500
Year2005                      0.500  0.500
Year2006                      0.500  0.500  0.500
Year2007                      0.500  0.500  0.500  0.500
Year2008                      0.500  0.500  0.500  0.500  0.500
Year2009                      0.500  0.500  0.500  0.500  0.500  0.500
Year2010                      0.500  0.500  0.500  0.500  0.500  0.500  0.500
Year2011                      0.500  0.500  0.500  0.500  0.500  0.500  0.500
Year2012                      0.500  0.500  0.500  0.500  0.500  0.500  0.500
Year2013                      0.500  0.500  0.500  0.500  0.500  0.500  0.500
Year2014                      0.500  0.500  0.500  0.500  0.500  0.500  0.500
RegionB:Managementb          -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Managementb          -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionD:Managementb          -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionB:Year2001             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2001             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionD:Year2001             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionB:Year2002             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2002             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionD:Year2002             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionB:Year2003             -0.707 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2003             -0.707 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionD:Year2003             -0.707 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionB:Year2004             -0.354 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2004             -0.354 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionD:Year2004             -0.354 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
RegionB:Year2005             -0.354 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionC:Year2005             -0.354 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionD:Year2005             -0.354 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
RegionB:Year2006             -0.354 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionC:Year2006             -0.354 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionD:Year2006             -0.354 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
RegionB:Year2007             -0.354 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
RegionC:Year2007             -0.354 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
RegionD:Year2007             -0.354 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
RegionB:Year2008             -0.354 -0.354 -0.354 -0.354 -0.354 -0.707 -0.354
RegionC:Year2008             -0.354 -0.354 -0.354 -0.354 -0.354 -0.707 -0.354
RegionD:Year2008             -0.354 -0.354 -0.354 -0.354 -0.354 -0.707 -0.354
RegionB:Year2009             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.707
RegionC:Year2009             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.707
RegionD:Year2009             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.707
RegionB:Year2010             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2010             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionD:Year2010             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionB:Year2011             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2011             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionD:Year2011             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionB:Year2012             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2012             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionD:Year2012             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionB:Year2013             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2013             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionD:Year2013             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionB:Year2014             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Year2014             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionD:Year2014             -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
Managementb:Year2001         -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
Managementb:Year2002         -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
Managementb:Year2003         -0.707 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
Managementb:Year2004         -0.354 -0.707 -0.354 -0.354 -0.354 -0.354 -0.354
Managementb:Year2005         -0.354 -0.354 -0.707 -0.354 -0.354 -0.354 -0.354
Managementb:Year2006         -0.354 -0.354 -0.354 -0.707 -0.354 -0.354 -0.354
Managementb:Year2007         -0.354 -0.354 -0.354 -0.354 -0.707 -0.354 -0.354
Managementb:Year2008         -0.354 -0.354 -0.354 -0.354 -0.354 -0.707 -0.354
Managementb:Year2009         -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.707
Managementb:Year2010         -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
Managementb:Year2011         -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
Managementb:Year2012         -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
Managementb:Year2013         -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
Managementb:Year2014         -0.354 -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
RegionB:Managementb:Year2001  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionC:Managementb:Year2001  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionD:Managementb:Year2001  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionB:Managementb:Year2002  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionC:Managementb:Year2002  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionD:Managementb:Year2002  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionB:Managementb:Year2003  0.500  0.250  0.250  0.250  0.250  0.250  0.250
RegionC:Managementb:Year2003  0.500  0.250  0.250  0.250  0.250  0.250  0.250
RegionD:Managementb:Year2003  0.500  0.250  0.250  0.250  0.250  0.250  0.250
RegionB:Managementb:Year2004  0.250  0.500  0.250  0.250  0.250  0.250  0.250
RegionC:Managementb:Year2004  0.250  0.500  0.250  0.250  0.250  0.250  0.250
RegionD:Managementb:Year2004  0.250  0.500  0.250  0.250  0.250  0.250  0.250
RegionB:Managementb:Year2005  0.250  0.250  0.500  0.250  0.250  0.250  0.250
RegionC:Managementb:Year2005  0.250  0.250  0.500  0.250  0.250  0.250  0.250
RegionD:Managementb:Year2005  0.250  0.250  0.500  0.250  0.250  0.250  0.250
RegionB:Managementb:Year2006  0.250  0.250  0.250  0.500  0.250  0.250  0.250
RegionC:Managementb:Year2006  0.250  0.250  0.250  0.500  0.250  0.250  0.250
RegionD:Managementb:Year2006  0.250  0.250  0.250  0.500  0.250  0.250  0.250
RegionB:Managementb:Year2007  0.250  0.250  0.250  0.250  0.500  0.250  0.250
RegionC:Managementb:Year2007  0.250  0.250  0.250  0.250  0.500  0.250  0.250
RegionD:Managementb:Year2007  0.250  0.250  0.250  0.250  0.500  0.250  0.250
RegionB:Managementb:Year2008  0.250  0.250  0.250  0.250  0.250  0.500  0.250
RegionC:Managementb:Year2008  0.250  0.250  0.250  0.250  0.250  0.500  0.250
RegionD:Managementb:Year2008  0.250  0.250  0.250  0.250  0.250  0.500  0.250
RegionB:Managementb:Year2009  0.250  0.250  0.250  0.250  0.250  0.250  0.500
RegionC:Managementb:Year2009  0.250  0.250  0.250  0.250  0.250  0.250  0.500
RegionD:Managementb:Year2009  0.250  0.250  0.250  0.250  0.250  0.250  0.500
RegionB:Managementb:Year2010  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionC:Managementb:Year2010  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionD:Managementb:Year2010  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionB:Managementb:Year2011  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionC:Managementb:Year2011  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionD:Managementb:Year2011  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionB:Managementb:Year2012  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionC:Managementb:Year2012  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionD:Managementb:Year2012  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionB:Managementb:Year2013  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionC:Managementb:Year2013  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionD:Managementb:Year2013  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionB:Managementb:Year2014  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionC:Managementb:Year2014  0.250  0.250  0.250  0.250  0.250  0.250  0.250
RegionD:Managementb:Year2014  0.250  0.250  0.250  0.250  0.250  0.250  0.250
Yr2010 Yr2011 Yr2012 Yr2013 Yr2014 RgnB:M RgnC:M
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011                      0.500
Year2012                      0.500  0.500
Year2013                      0.500  0.500  0.500
Year2014                      0.500  0.500  0.500  0.500
RegionB:Managementb          -0.354 -0.354 -0.354 -0.354 -0.354
RegionC:Managementb          -0.354 -0.354 -0.354 -0.354 -0.354  0.500
RegionD:Managementb          -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.500
RegionB:Year2001             -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2001             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2001             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2002             -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2002             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2002             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2003             -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2003             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2003             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2004             -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2004             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2004             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2005             -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2005             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2005             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2006             -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2006             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2006             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2007             -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2007             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2007             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2008             -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2008             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2008             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2009             -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2009             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2009             -0.354 -0.354 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2010             -0.707 -0.354 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2010             -0.707 -0.354 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2010             -0.707 -0.354 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2011             -0.354 -0.707 -0.354 -0.354 -0.354  0.500  0.250
RegionC:Year2011             -0.354 -0.707 -0.354 -0.354 -0.354  0.250  0.500
RegionD:Year2011             -0.354 -0.707 -0.354 -0.354 -0.354  0.250  0.250
RegionB:Year2012             -0.354 -0.354 -0.707 -0.354 -0.354  0.500  0.250
RegionC:Year2012             -0.354 -0.354 -0.707 -0.354 -0.354  0.250  0.500
RegionD:Year2012             -0.354 -0.354 -0.707 -0.354 -0.354  0.250  0.250
RegionB:Year2013             -0.354 -0.354 -0.354 -0.707 -0.354  0.500  0.250
RegionC:Year2013             -0.354 -0.354 -0.354 -0.707 -0.354  0.250  0.500
RegionD:Year2013             -0.354 -0.354 -0.354 -0.707 -0.354  0.250  0.250
RegionB:Year2014             -0.354 -0.354 -0.354 -0.354 -0.707  0.500  0.250
RegionC:Year2014             -0.354 -0.354 -0.354 -0.354 -0.707  0.250  0.500
RegionD:Year2014             -0.354 -0.354 -0.354 -0.354 -0.707  0.250  0.250
Managementb:Year2001         -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2002         -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2003         -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2004         -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2005         -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2006         -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2007         -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2008         -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2009         -0.354 -0.354 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2010         -0.707 -0.354 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2011         -0.354 -0.707 -0.354 -0.354 -0.354  0.500  0.500
Managementb:Year2012         -0.354 -0.354 -0.707 -0.354 -0.354  0.500  0.500
Managementb:Year2013         -0.354 -0.354 -0.354 -0.707 -0.354  0.500  0.500
Managementb:Year2014         -0.354 -0.354 -0.354 -0.354 -0.707  0.500  0.500
RegionB:Managementb:Year2001  0.250  0.250  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2001  0.250  0.250  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2001  0.250  0.250  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2002  0.250  0.250  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2002  0.250  0.250  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2002  0.250  0.250  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2003  0.250  0.250  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2003  0.250  0.250  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2003  0.250  0.250  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2004  0.250  0.250  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2004  0.250  0.250  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2004  0.250  0.250  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2005  0.250  0.250  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2005  0.250  0.250  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2005  0.250  0.250  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2006  0.250  0.250  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2006  0.250  0.250  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2006  0.250  0.250  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2007  0.250  0.250  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2007  0.250  0.250  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2007  0.250  0.250  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2008  0.250  0.250  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2008  0.250  0.250  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2008  0.250  0.250  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2009  0.250  0.250  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2009  0.250  0.250  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2009  0.250  0.250  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2010  0.500  0.250  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2010  0.500  0.250  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2010  0.500  0.250  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2011  0.250  0.500  0.250  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2011  0.250  0.500  0.250  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2011  0.250  0.500  0.250  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2012  0.250  0.250  0.500  0.250  0.250 -0.707 -0.354
RegionC:Managementb:Year2012  0.250  0.250  0.500  0.250  0.250 -0.354 -0.707
RegionD:Managementb:Year2012  0.250  0.250  0.500  0.250  0.250 -0.354 -0.354
RegionB:Managementb:Year2013  0.250  0.250  0.250  0.500  0.250 -0.707 -0.354
RegionC:Managementb:Year2013  0.250  0.250  0.250  0.500  0.250 -0.354 -0.707
RegionD:Managementb:Year2013  0.250  0.250  0.250  0.500  0.250 -0.354 -0.354
RegionB:Managementb:Year2014  0.250  0.250  0.250  0.250  0.500 -0.707 -0.354
RegionC:Managementb:Year2014  0.250  0.250  0.250  0.250  0.500 -0.354 -0.707
RegionD:Managementb:Year2014  0.250  0.250  0.250  0.250  0.500 -0.354 -0.354
RgnD:M RB:Y2001 RC:Y2001 RD:Y2001 RB:Y2002
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001              0.250
RegionC:Year2001              0.250  0.500
RegionD:Year2001              0.500  0.500    0.500
RegionB:Year2002              0.250  0.500    0.250    0.250
RegionC:Year2002              0.250  0.250    0.500    0.250    0.500
RegionD:Year2002              0.500  0.250    0.250    0.500    0.500
RegionB:Year2003              0.250  0.500    0.250    0.250    0.500
RegionC:Year2003              0.250  0.250    0.500    0.250    0.250
RegionD:Year2003              0.500  0.250    0.250    0.500    0.250
RegionB:Year2004              0.250  0.500    0.250    0.250    0.500
RegionC:Year2004              0.250  0.250    0.500    0.250    0.250
RegionD:Year2004              0.500  0.250    0.250    0.500    0.250
RegionB:Year2005              0.250  0.500    0.250    0.250    0.500
RegionC:Year2005              0.250  0.250    0.500    0.250    0.250
RegionD:Year2005              0.500  0.250    0.250    0.500    0.250
RegionB:Year2006              0.250  0.500    0.250    0.250    0.500
RegionC:Year2006              0.250  0.250    0.500    0.250    0.250
RegionD:Year2006              0.500  0.250    0.250    0.500    0.250
RegionB:Year2007              0.250  0.500    0.250    0.250    0.500
RegionC:Year2007              0.250  0.250    0.500    0.250    0.250
RegionD:Year2007              0.500  0.250    0.250    0.500    0.250
RegionB:Year2008              0.250  0.500    0.250    0.250    0.500
RegionC:Year2008              0.250  0.250    0.500    0.250    0.250
RegionD:Year2008              0.500  0.250    0.250    0.500    0.250
RegionB:Year2009              0.250  0.500    0.250    0.250    0.500
RegionC:Year2009              0.250  0.250    0.500    0.250    0.250
RegionD:Year2009              0.500  0.250    0.250    0.500    0.250
RegionB:Year2010              0.250  0.500    0.250    0.250    0.500
RegionC:Year2010              0.250  0.250    0.500    0.250    0.250
RegionD:Year2010              0.500  0.250    0.250    0.500    0.250
RegionB:Year2011              0.250  0.500    0.250    0.250    0.500
RegionC:Year2011              0.250  0.250    0.500    0.250    0.250
RegionD:Year2011              0.500  0.250    0.250    0.500    0.250
RegionB:Year2012              0.250  0.500    0.250    0.250    0.500
RegionC:Year2012              0.250  0.250    0.500    0.250    0.250
RegionD:Year2012              0.500  0.250    0.250    0.500    0.250
RegionB:Year2013              0.250  0.500    0.250    0.250    0.500
RegionC:Year2013              0.250  0.250    0.500    0.250    0.250
RegionD:Year2013              0.500  0.250    0.250    0.500    0.250
RegionB:Year2014              0.250  0.500    0.250    0.250    0.500
RegionC:Year2014              0.250  0.250    0.500    0.250    0.250
RegionD:Year2014              0.500  0.250    0.250    0.500    0.250
Managementb:Year2001          0.500  0.500    0.500    0.500    0.250
Managementb:Year2002          0.500  0.250    0.250    0.250    0.500
Managementb:Year2003          0.500  0.250    0.250    0.250    0.250
Managementb:Year2004          0.500  0.250    0.250    0.250    0.250
Managementb:Year2005          0.500  0.250    0.250    0.250    0.250
Managementb:Year2006          0.500  0.250    0.250    0.250    0.250
Managementb:Year2007          0.500  0.250    0.250    0.250    0.250
Managementb:Year2008          0.500  0.250    0.250    0.250    0.250
Managementb:Year2009          0.500  0.250    0.250    0.250    0.250
Managementb:Year2010          0.500  0.250    0.250    0.250    0.250
Managementb:Year2011          0.500  0.250    0.250    0.250    0.250
Managementb:Year2012          0.500  0.250    0.250    0.250    0.250
Managementb:Year2013          0.500  0.250    0.250    0.250    0.250
Managementb:Year2014          0.500  0.250    0.250    0.250    0.250
RegionB:Managementb:Year2001 -0.354 -0.707   -0.354   -0.354   -0.354
RegionC:Managementb:Year2001 -0.354 -0.354   -0.707   -0.354   -0.177
RegionD:Managementb:Year2001 -0.707 -0.354   -0.354   -0.707   -0.177
RegionB:Managementb:Year2002 -0.354 -0.354   -0.177   -0.177   -0.707
RegionC:Managementb:Year2002 -0.354 -0.177   -0.354   -0.177   -0.354
RegionD:Managementb:Year2002 -0.707 -0.177   -0.177   -0.354   -0.354
RegionB:Managementb:Year2003 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2003 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2003 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2004 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2004 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2004 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2005 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2005 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2005 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2006 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2006 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2006 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2007 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2007 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2007 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2008 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2008 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2008 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2009 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2009 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2009 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2010 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2010 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2010 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2011 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2011 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2011 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2012 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2012 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2012 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2013 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2013 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2013 -0.707 -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2014 -0.354 -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2014 -0.354 -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2014 -0.707 -0.177   -0.177   -0.354   -0.177
RC:Y2002 RD:Y2002 RB:Y2003 RC:Y2003 RD:Y2003
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002              0.500
RegionB:Year2003              0.250    0.250
RegionC:Year2003              0.500    0.250    0.500
RegionD:Year2003              0.250    0.500    0.500    0.500
RegionB:Year2004              0.250    0.250    0.500    0.250    0.250
RegionC:Year2004              0.500    0.250    0.250    0.500    0.250
RegionD:Year2004              0.250    0.500    0.250    0.250    0.500
RegionB:Year2005              0.250    0.250    0.500    0.250    0.250
RegionC:Year2005              0.500    0.250    0.250    0.500    0.250
RegionD:Year2005              0.250    0.500    0.250    0.250    0.500
RegionB:Year2006              0.250    0.250    0.500    0.250    0.250
RegionC:Year2006              0.500    0.250    0.250    0.500    0.250
RegionD:Year2006              0.250    0.500    0.250    0.250    0.500
RegionB:Year2007              0.250    0.250    0.500    0.250    0.250
RegionC:Year2007              0.500    0.250    0.250    0.500    0.250
RegionD:Year2007              0.250    0.500    0.250    0.250    0.500
RegionB:Year2008              0.250    0.250    0.500    0.250    0.250
RegionC:Year2008              0.500    0.250    0.250    0.500    0.250
RegionD:Year2008              0.250    0.500    0.250    0.250    0.500
RegionB:Year2009              0.250    0.250    0.500    0.250    0.250
RegionC:Year2009              0.500    0.250    0.250    0.500    0.250
RegionD:Year2009              0.250    0.500    0.250    0.250    0.500
RegionB:Year2010              0.250    0.250    0.500    0.250    0.250
RegionC:Year2010              0.500    0.250    0.250    0.500    0.250
RegionD:Year2010              0.250    0.500    0.250    0.250    0.500
RegionB:Year2011              0.250    0.250    0.500    0.250    0.250
RegionC:Year2011              0.500    0.250    0.250    0.500    0.250
RegionD:Year2011              0.250    0.500    0.250    0.250    0.500
RegionB:Year2012              0.250    0.250    0.500    0.250    0.250
RegionC:Year2012              0.500    0.250    0.250    0.500    0.250
RegionD:Year2012              0.250    0.500    0.250    0.250    0.500
RegionB:Year2013              0.250    0.250    0.500    0.250    0.250
RegionC:Year2013              0.500    0.250    0.250    0.500    0.250
RegionD:Year2013              0.250    0.500    0.250    0.250    0.500
RegionB:Year2014              0.250    0.250    0.500    0.250    0.250
RegionC:Year2014              0.500    0.250    0.250    0.500    0.250
RegionD:Year2014              0.250    0.500    0.250    0.250    0.500
Managementb:Year2001          0.250    0.250    0.250    0.250    0.250
Managementb:Year2002          0.500    0.500    0.250    0.250    0.250
Managementb:Year2003          0.250    0.250    0.500    0.500    0.500
Managementb:Year2004          0.250    0.250    0.250    0.250    0.250
Managementb:Year2005          0.250    0.250    0.250    0.250    0.250
Managementb:Year2006          0.250    0.250    0.250    0.250    0.250
Managementb:Year2007          0.250    0.250    0.250    0.250    0.250
Managementb:Year2008          0.250    0.250    0.250    0.250    0.250
Managementb:Year2009          0.250    0.250    0.250    0.250    0.250
Managementb:Year2010          0.250    0.250    0.250    0.250    0.250
Managementb:Year2011          0.250    0.250    0.250    0.250    0.250
Managementb:Year2012          0.250    0.250    0.250    0.250    0.250
Managementb:Year2013          0.250    0.250    0.250    0.250    0.250
Managementb:Year2014          0.250    0.250    0.250    0.250    0.250
RegionB:Managementb:Year2001 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2001 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2001 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2002 -0.354   -0.354   -0.354   -0.177   -0.177
RegionC:Managementb:Year2002 -0.707   -0.354   -0.177   -0.354   -0.177
RegionD:Managementb:Year2002 -0.354   -0.707   -0.177   -0.177   -0.354
RegionB:Managementb:Year2003 -0.177   -0.177   -0.707   -0.354   -0.354
RegionC:Managementb:Year2003 -0.354   -0.177   -0.354   -0.707   -0.354
RegionD:Managementb:Year2003 -0.177   -0.354   -0.354   -0.354   -0.707
RegionB:Managementb:Year2004 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2004 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2004 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2005 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2005 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2005 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2006 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2006 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2006 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2007 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2007 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2007 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2008 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2008 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2008 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2009 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2009 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2009 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2010 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2010 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2010 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2011 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2011 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2011 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2012 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2012 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2012 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2013 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2013 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2013 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2014 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2014 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2014 -0.177   -0.354   -0.177   -0.177   -0.354
RB:Y2004 RC:Y2004 RD:Y2004 RB:Y2005 RC:Y2005
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004              0.500
RegionD:Year2004              0.500    0.500
RegionB:Year2005              0.500    0.250    0.250
RegionC:Year2005              0.250    0.500    0.250    0.500
RegionD:Year2005              0.250    0.250    0.500    0.500    0.500
RegionB:Year2006              0.500    0.250    0.250    0.500    0.250
RegionC:Year2006              0.250    0.500    0.250    0.250    0.500
RegionD:Year2006              0.250    0.250    0.500    0.250    0.250
RegionB:Year2007              0.500    0.250    0.250    0.500    0.250
RegionC:Year2007              0.250    0.500    0.250    0.250    0.500
RegionD:Year2007              0.250    0.250    0.500    0.250    0.250
RegionB:Year2008              0.500    0.250    0.250    0.500    0.250
RegionC:Year2008              0.250    0.500    0.250    0.250    0.500
RegionD:Year2008              0.250    0.250    0.500    0.250    0.250
RegionB:Year2009              0.500    0.250    0.250    0.500    0.250
RegionC:Year2009              0.250    0.500    0.250    0.250    0.500
RegionD:Year2009              0.250    0.250    0.500    0.250    0.250
RegionB:Year2010              0.500    0.250    0.250    0.500    0.250
RegionC:Year2010              0.250    0.500    0.250    0.250    0.500
RegionD:Year2010              0.250    0.250    0.500    0.250    0.250
RegionB:Year2011              0.500    0.250    0.250    0.500    0.250
RegionC:Year2011              0.250    0.500    0.250    0.250    0.500
RegionD:Year2011              0.250    0.250    0.500    0.250    0.250
RegionB:Year2012              0.500    0.250    0.250    0.500    0.250
RegionC:Year2012              0.250    0.500    0.250    0.250    0.500
RegionD:Year2012              0.250    0.250    0.500    0.250    0.250
RegionB:Year2013              0.500    0.250    0.250    0.500    0.250
RegionC:Year2013              0.250    0.500    0.250    0.250    0.500
RegionD:Year2013              0.250    0.250    0.500    0.250    0.250
RegionB:Year2014              0.500    0.250    0.250    0.500    0.250
RegionC:Year2014              0.250    0.500    0.250    0.250    0.500
RegionD:Year2014              0.250    0.250    0.500    0.250    0.250
Managementb:Year2001          0.250    0.250    0.250    0.250    0.250
Managementb:Year2002          0.250    0.250    0.250    0.250    0.250
Managementb:Year2003          0.250    0.250    0.250    0.250    0.250
Managementb:Year2004          0.500    0.500    0.500    0.250    0.250
Managementb:Year2005          0.250    0.250    0.250    0.500    0.500
Managementb:Year2006          0.250    0.250    0.250    0.250    0.250
Managementb:Year2007          0.250    0.250    0.250    0.250    0.250
Managementb:Year2008          0.250    0.250    0.250    0.250    0.250
Managementb:Year2009          0.250    0.250    0.250    0.250    0.250
Managementb:Year2010          0.250    0.250    0.250    0.250    0.250
Managementb:Year2011          0.250    0.250    0.250    0.250    0.250
Managementb:Year2012          0.250    0.250    0.250    0.250    0.250
Managementb:Year2013          0.250    0.250    0.250    0.250    0.250
Managementb:Year2014          0.250    0.250    0.250    0.250    0.250
RegionB:Managementb:Year2001 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2001 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2001 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2002 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2002 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2002 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2003 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2003 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2003 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2004 -0.707   -0.354   -0.354   -0.354   -0.177
RegionC:Managementb:Year2004 -0.354   -0.707   -0.354   -0.177   -0.354
RegionD:Managementb:Year2004 -0.354   -0.354   -0.707   -0.177   -0.177
RegionB:Managementb:Year2005 -0.354   -0.177   -0.177   -0.707   -0.354
RegionC:Managementb:Year2005 -0.177   -0.354   -0.177   -0.354   -0.707
RegionD:Managementb:Year2005 -0.177   -0.177   -0.354   -0.354   -0.354
RegionB:Managementb:Year2006 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2006 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2006 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2007 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2007 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2007 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2008 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2008 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2008 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2009 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2009 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2009 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2010 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2010 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2010 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2011 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2011 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2011 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2012 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2012 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2012 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2013 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2013 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2013 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2014 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2014 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2014 -0.177   -0.177   -0.354   -0.177   -0.177
RD:Y2005 RB:Y2006 RC:Y2006 RD:Y2006 RB:Y2007
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006              0.250
RegionC:Year2006              0.250    0.500
RegionD:Year2006              0.500    0.500    0.500
RegionB:Year2007              0.250    0.500    0.250    0.250
RegionC:Year2007              0.250    0.250    0.500    0.250    0.500
RegionD:Year2007              0.500    0.250    0.250    0.500    0.500
RegionB:Year2008              0.250    0.500    0.250    0.250    0.500
RegionC:Year2008              0.250    0.250    0.500    0.250    0.250
RegionD:Year2008              0.500    0.250    0.250    0.500    0.250
RegionB:Year2009              0.250    0.500    0.250    0.250    0.500
RegionC:Year2009              0.250    0.250    0.500    0.250    0.250
RegionD:Year2009              0.500    0.250    0.250    0.500    0.250
RegionB:Year2010              0.250    0.500    0.250    0.250    0.500
RegionC:Year2010              0.250    0.250    0.500    0.250    0.250
RegionD:Year2010              0.500    0.250    0.250    0.500    0.250
RegionB:Year2011              0.250    0.500    0.250    0.250    0.500
RegionC:Year2011              0.250    0.250    0.500    0.250    0.250
RegionD:Year2011              0.500    0.250    0.250    0.500    0.250
RegionB:Year2012              0.250    0.500    0.250    0.250    0.500
RegionC:Year2012              0.250    0.250    0.500    0.250    0.250
RegionD:Year2012              0.500    0.250    0.250    0.500    0.250
RegionB:Year2013              0.250    0.500    0.250    0.250    0.500
RegionC:Year2013              0.250    0.250    0.500    0.250    0.250
RegionD:Year2013              0.500    0.250    0.250    0.500    0.250
RegionB:Year2014              0.250    0.500    0.250    0.250    0.500
RegionC:Year2014              0.250    0.250    0.500    0.250    0.250
RegionD:Year2014              0.500    0.250    0.250    0.500    0.250
Managementb:Year2001          0.250    0.250    0.250    0.250    0.250
Managementb:Year2002          0.250    0.250    0.250    0.250    0.250
Managementb:Year2003          0.250    0.250    0.250    0.250    0.250
Managementb:Year2004          0.250    0.250    0.250    0.250    0.250
Managementb:Year2005          0.500    0.250    0.250    0.250    0.250
Managementb:Year2006          0.250    0.500    0.500    0.500    0.250
Managementb:Year2007          0.250    0.250    0.250    0.250    0.500
Managementb:Year2008          0.250    0.250    0.250    0.250    0.250
Managementb:Year2009          0.250    0.250    0.250    0.250    0.250
Managementb:Year2010          0.250    0.250    0.250    0.250    0.250
Managementb:Year2011          0.250    0.250    0.250    0.250    0.250
Managementb:Year2012          0.250    0.250    0.250    0.250    0.250
Managementb:Year2013          0.250    0.250    0.250    0.250    0.250
Managementb:Year2014          0.250    0.250    0.250    0.250    0.250
RegionB:Managementb:Year2001 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2001 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2001 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2002 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2002 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2002 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2003 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2003 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2003 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2004 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2004 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2004 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2005 -0.354   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2005 -0.354   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2005 -0.707   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2006 -0.177   -0.707   -0.354   -0.354   -0.354
RegionC:Managementb:Year2006 -0.177   -0.354   -0.707   -0.354   -0.177
RegionD:Managementb:Year2006 -0.354   -0.354   -0.354   -0.707   -0.177
RegionB:Managementb:Year2007 -0.177   -0.354   -0.177   -0.177   -0.707
RegionC:Managementb:Year2007 -0.177   -0.177   -0.354   -0.177   -0.354
RegionD:Managementb:Year2007 -0.354   -0.177   -0.177   -0.354   -0.354
RegionB:Managementb:Year2008 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2008 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2008 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2009 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2009 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2009 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2010 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2010 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2010 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2011 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2011 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2011 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2012 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2012 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2012 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2013 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2013 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2013 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2014 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2014 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2014 -0.354   -0.177   -0.177   -0.354   -0.177
RC:Y2007 RD:Y2007 RB:Y2008 RC:Y2008 RD:Y2008
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007              0.500
RegionB:Year2008              0.250    0.250
RegionC:Year2008              0.500    0.250    0.500
RegionD:Year2008              0.250    0.500    0.500    0.500
RegionB:Year2009              0.250    0.250    0.500    0.250    0.250
RegionC:Year2009              0.500    0.250    0.250    0.500    0.250
RegionD:Year2009              0.250    0.500    0.250    0.250    0.500
RegionB:Year2010              0.250    0.250    0.500    0.250    0.250
RegionC:Year2010              0.500    0.250    0.250    0.500    0.250
RegionD:Year2010              0.250    0.500    0.250    0.250    0.500
RegionB:Year2011              0.250    0.250    0.500    0.250    0.250
RegionC:Year2011              0.500    0.250    0.250    0.500    0.250
RegionD:Year2011              0.250    0.500    0.250    0.250    0.500
RegionB:Year2012              0.250    0.250    0.500    0.250    0.250
RegionC:Year2012              0.500    0.250    0.250    0.500    0.250
RegionD:Year2012              0.250    0.500    0.250    0.250    0.500
RegionB:Year2013              0.250    0.250    0.500    0.250    0.250
RegionC:Year2013              0.500    0.250    0.250    0.500    0.250
RegionD:Year2013              0.250    0.500    0.250    0.250    0.500
RegionB:Year2014              0.250    0.250    0.500    0.250    0.250
RegionC:Year2014              0.500    0.250    0.250    0.500    0.250
RegionD:Year2014              0.250    0.500    0.250    0.250    0.500
Managementb:Year2001          0.250    0.250    0.250    0.250    0.250
Managementb:Year2002          0.250    0.250    0.250    0.250    0.250
Managementb:Year2003          0.250    0.250    0.250    0.250    0.250
Managementb:Year2004          0.250    0.250    0.250    0.250    0.250
Managementb:Year2005          0.250    0.250    0.250    0.250    0.250
Managementb:Year2006          0.250    0.250    0.250    0.250    0.250
Managementb:Year2007          0.500    0.500    0.250    0.250    0.250
Managementb:Year2008          0.250    0.250    0.500    0.500    0.500
Managementb:Year2009          0.250    0.250    0.250    0.250    0.250
Managementb:Year2010          0.250    0.250    0.250    0.250    0.250
Managementb:Year2011          0.250    0.250    0.250    0.250    0.250
Managementb:Year2012          0.250    0.250    0.250    0.250    0.250
Managementb:Year2013          0.250    0.250    0.250    0.250    0.250
Managementb:Year2014          0.250    0.250    0.250    0.250    0.250
RegionB:Managementb:Year2001 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2001 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2001 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2002 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2002 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2002 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2003 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2003 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2003 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2004 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2004 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2004 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2005 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2005 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2005 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2006 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2006 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2006 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2007 -0.354   -0.354   -0.354   -0.177   -0.177
RegionC:Managementb:Year2007 -0.707   -0.354   -0.177   -0.354   -0.177
RegionD:Managementb:Year2007 -0.354   -0.707   -0.177   -0.177   -0.354
RegionB:Managementb:Year2008 -0.177   -0.177   -0.707   -0.354   -0.354
RegionC:Managementb:Year2008 -0.354   -0.177   -0.354   -0.707   -0.354
RegionD:Managementb:Year2008 -0.177   -0.354   -0.354   -0.354   -0.707
RegionB:Managementb:Year2009 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2009 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2009 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2010 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2010 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2010 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2011 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2011 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2011 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2012 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2012 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2012 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2013 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2013 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2013 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2014 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2014 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2014 -0.177   -0.354   -0.177   -0.177   -0.354
RB:Y2009 RC:Y2009 RD:Y2009 RB:Y2010 RC:Y2010
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009              0.500
RegionD:Year2009              0.500    0.500
RegionB:Year2010              0.500    0.250    0.250
RegionC:Year2010              0.250    0.500    0.250    0.500
RegionD:Year2010              0.250    0.250    0.500    0.500    0.500
RegionB:Year2011              0.500    0.250    0.250    0.500    0.250
RegionC:Year2011              0.250    0.500    0.250    0.250    0.500
RegionD:Year2011              0.250    0.250    0.500    0.250    0.250
RegionB:Year2012              0.500    0.250    0.250    0.500    0.250
RegionC:Year2012              0.250    0.500    0.250    0.250    0.500
RegionD:Year2012              0.250    0.250    0.500    0.250    0.250
RegionB:Year2013              0.500    0.250    0.250    0.500    0.250
RegionC:Year2013              0.250    0.500    0.250    0.250    0.500
RegionD:Year2013              0.250    0.250    0.500    0.250    0.250
RegionB:Year2014              0.500    0.250    0.250    0.500    0.250
RegionC:Year2014              0.250    0.500    0.250    0.250    0.500
RegionD:Year2014              0.250    0.250    0.500    0.250    0.250
Managementb:Year2001          0.250    0.250    0.250    0.250    0.250
Managementb:Year2002          0.250    0.250    0.250    0.250    0.250
Managementb:Year2003          0.250    0.250    0.250    0.250    0.250
Managementb:Year2004          0.250    0.250    0.250    0.250    0.250
Managementb:Year2005          0.250    0.250    0.250    0.250    0.250
Managementb:Year2006          0.250    0.250    0.250    0.250    0.250
Managementb:Year2007          0.250    0.250    0.250    0.250    0.250
Managementb:Year2008          0.250    0.250    0.250    0.250    0.250
Managementb:Year2009          0.500    0.500    0.500    0.250    0.250
Managementb:Year2010          0.250    0.250    0.250    0.500    0.500
Managementb:Year2011          0.250    0.250    0.250    0.250    0.250
Managementb:Year2012          0.250    0.250    0.250    0.250    0.250
Managementb:Year2013          0.250    0.250    0.250    0.250    0.250
Managementb:Year2014          0.250    0.250    0.250    0.250    0.250
RegionB:Managementb:Year2001 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2001 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2001 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2002 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2002 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2002 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2003 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2003 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2003 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2004 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2004 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2004 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2005 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2005 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2005 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2006 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2006 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2006 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2007 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2007 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2007 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2008 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2008 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2008 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2009 -0.707   -0.354   -0.354   -0.354   -0.177
RegionC:Managementb:Year2009 -0.354   -0.707   -0.354   -0.177   -0.354
RegionD:Managementb:Year2009 -0.354   -0.354   -0.707   -0.177   -0.177
RegionB:Managementb:Year2010 -0.354   -0.177   -0.177   -0.707   -0.354
RegionC:Managementb:Year2010 -0.177   -0.354   -0.177   -0.354   -0.707
RegionD:Managementb:Year2010 -0.177   -0.177   -0.354   -0.354   -0.354
RegionB:Managementb:Year2011 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2011 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2011 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2012 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2012 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2012 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2013 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2013 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2013 -0.177   -0.177   -0.354   -0.177   -0.177
RegionB:Managementb:Year2014 -0.354   -0.177   -0.177   -0.354   -0.177
RegionC:Managementb:Year2014 -0.177   -0.354   -0.177   -0.177   -0.354
RegionD:Managementb:Year2014 -0.177   -0.177   -0.354   -0.177   -0.177
RD:Y2010 RB:Y2011 RC:Y2011 RD:Y2011 RB:Y2012
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011              0.250
RegionC:Year2011              0.250    0.500
RegionD:Year2011              0.500    0.500    0.500
RegionB:Year2012              0.250    0.500    0.250    0.250
RegionC:Year2012              0.250    0.250    0.500    0.250    0.500
RegionD:Year2012              0.500    0.250    0.250    0.500    0.500
RegionB:Year2013              0.250    0.500    0.250    0.250    0.500
RegionC:Year2013              0.250    0.250    0.500    0.250    0.250
RegionD:Year2013              0.500    0.250    0.250    0.500    0.250
RegionB:Year2014              0.250    0.500    0.250    0.250    0.500
RegionC:Year2014              0.250    0.250    0.500    0.250    0.250
RegionD:Year2014              0.500    0.250    0.250    0.500    0.250
Managementb:Year2001          0.250    0.250    0.250    0.250    0.250
Managementb:Year2002          0.250    0.250    0.250    0.250    0.250
Managementb:Year2003          0.250    0.250    0.250    0.250    0.250
Managementb:Year2004          0.250    0.250    0.250    0.250    0.250
Managementb:Year2005          0.250    0.250    0.250    0.250    0.250
Managementb:Year2006          0.250    0.250    0.250    0.250    0.250
Managementb:Year2007          0.250    0.250    0.250    0.250    0.250
Managementb:Year2008          0.250    0.250    0.250    0.250    0.250
Managementb:Year2009          0.250    0.250    0.250    0.250    0.250
Managementb:Year2010          0.500    0.250    0.250    0.250    0.250
Managementb:Year2011          0.250    0.500    0.500    0.500    0.250
Managementb:Year2012          0.250    0.250    0.250    0.250    0.500
Managementb:Year2013          0.250    0.250    0.250    0.250    0.250
Managementb:Year2014          0.250    0.250    0.250    0.250    0.250
RegionB:Managementb:Year2001 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2001 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2001 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2002 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2002 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2002 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2003 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2003 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2003 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2004 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2004 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2004 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2005 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2005 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2005 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2006 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2006 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2006 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2007 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2007 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2007 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2008 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2008 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2008 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2009 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2009 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2009 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2010 -0.354   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2010 -0.354   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2010 -0.707   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2011 -0.177   -0.707   -0.354   -0.354   -0.354
RegionC:Managementb:Year2011 -0.177   -0.354   -0.707   -0.354   -0.177
RegionD:Managementb:Year2011 -0.354   -0.354   -0.354   -0.707   -0.177
RegionB:Managementb:Year2012 -0.177   -0.354   -0.177   -0.177   -0.707
RegionC:Managementb:Year2012 -0.177   -0.177   -0.354   -0.177   -0.354
RegionD:Managementb:Year2012 -0.354   -0.177   -0.177   -0.354   -0.354
RegionB:Managementb:Year2013 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2013 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2013 -0.354   -0.177   -0.177   -0.354   -0.177
RegionB:Managementb:Year2014 -0.177   -0.354   -0.177   -0.177   -0.354
RegionC:Managementb:Year2014 -0.177   -0.177   -0.354   -0.177   -0.177
RegionD:Managementb:Year2014 -0.354   -0.177   -0.177   -0.354   -0.177
RC:Y2012 RD:Y2012 RB:Y2013 RC:Y2013 RD:Y2013
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012              0.500
RegionB:Year2013              0.250    0.250
RegionC:Year2013              0.500    0.250    0.500
RegionD:Year2013              0.250    0.500    0.500    0.500
RegionB:Year2014              0.250    0.250    0.500    0.250    0.250
RegionC:Year2014              0.500    0.250    0.250    0.500    0.250
RegionD:Year2014              0.250    0.500    0.250    0.250    0.500
Managementb:Year2001          0.250    0.250    0.250    0.250    0.250
Managementb:Year2002          0.250    0.250    0.250    0.250    0.250
Managementb:Year2003          0.250    0.250    0.250    0.250    0.250
Managementb:Year2004          0.250    0.250    0.250    0.250    0.250
Managementb:Year2005          0.250    0.250    0.250    0.250    0.250
Managementb:Year2006          0.250    0.250    0.250    0.250    0.250
Managementb:Year2007          0.250    0.250    0.250    0.250    0.250
Managementb:Year2008          0.250    0.250    0.250    0.250    0.250
Managementb:Year2009          0.250    0.250    0.250    0.250    0.250
Managementb:Year2010          0.250    0.250    0.250    0.250    0.250
Managementb:Year2011          0.250    0.250    0.250    0.250    0.250
Managementb:Year2012          0.500    0.500    0.250    0.250    0.250
Managementb:Year2013          0.250    0.250    0.500    0.500    0.500
Managementb:Year2014          0.250    0.250    0.250    0.250    0.250
RegionB:Managementb:Year2001 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2001 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2001 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2002 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2002 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2002 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2003 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2003 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2003 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2004 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2004 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2004 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2005 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2005 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2005 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2006 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2006 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2006 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2007 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2007 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2007 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2008 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2008 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2008 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2009 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2009 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2009 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2010 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2010 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2010 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2011 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2011 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2011 -0.177   -0.354   -0.177   -0.177   -0.354
RegionB:Managementb:Year2012 -0.354   -0.354   -0.354   -0.177   -0.177
RegionC:Managementb:Year2012 -0.707   -0.354   -0.177   -0.354   -0.177
RegionD:Managementb:Year2012 -0.354   -0.707   -0.177   -0.177   -0.354
RegionB:Managementb:Year2013 -0.177   -0.177   -0.707   -0.354   -0.354
RegionC:Managementb:Year2013 -0.354   -0.177   -0.354   -0.707   -0.354
RegionD:Managementb:Year2013 -0.177   -0.354   -0.354   -0.354   -0.707
RegionB:Managementb:Year2014 -0.177   -0.177   -0.354   -0.177   -0.177
RegionC:Managementb:Year2014 -0.354   -0.177   -0.177   -0.354   -0.177
RegionD:Managementb:Year2014 -0.177   -0.354   -0.177   -0.177   -0.354
RB:Y2014 RC:Y2014 RD:Y2014 M:Y2001 M:Y2002 M:Y2003
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014              0.500
RegionD:Year2014              0.500    0.500
Managementb:Year2001          0.250    0.250    0.250
Managementb:Year2002          0.250    0.250    0.250    0.500
Managementb:Year2003          0.250    0.250    0.250    0.500   0.500
Managementb:Year2004          0.250    0.250    0.250    0.500   0.500   0.500
Managementb:Year2005          0.250    0.250    0.250    0.500   0.500   0.500
Managementb:Year2006          0.250    0.250    0.250    0.500   0.500   0.500
Managementb:Year2007          0.250    0.250    0.250    0.500   0.500   0.500
Managementb:Year2008          0.250    0.250    0.250    0.500   0.500   0.500
Managementb:Year2009          0.250    0.250    0.250    0.500   0.500   0.500
Managementb:Year2010          0.250    0.250    0.250    0.500   0.500   0.500
Managementb:Year2011          0.250    0.250    0.250    0.500   0.500   0.500
Managementb:Year2012          0.250    0.250    0.250    0.500   0.500   0.500
Managementb:Year2013          0.250    0.250    0.250    0.500   0.500   0.500
Managementb:Year2014          0.500    0.500    0.500    0.500   0.500   0.500
RegionB:Managementb:Year2001 -0.354   -0.177   -0.177   -0.707  -0.354  -0.354
RegionC:Managementb:Year2001 -0.177   -0.354   -0.177   -0.707  -0.354  -0.354
RegionD:Managementb:Year2001 -0.177   -0.177   -0.354   -0.707  -0.354  -0.354
RegionB:Managementb:Year2002 -0.354   -0.177   -0.177   -0.354  -0.707  -0.354
RegionC:Managementb:Year2002 -0.177   -0.354   -0.177   -0.354  -0.707  -0.354
RegionD:Managementb:Year2002 -0.177   -0.177   -0.354   -0.354  -0.707  -0.354
RegionB:Managementb:Year2003 -0.354   -0.177   -0.177   -0.354  -0.354  -0.707
RegionC:Managementb:Year2003 -0.177   -0.354   -0.177   -0.354  -0.354  -0.707
RegionD:Managementb:Year2003 -0.177   -0.177   -0.354   -0.354  -0.354  -0.707
RegionB:Managementb:Year2004 -0.354   -0.177   -0.177   -0.354  -0.354  -0.354
RegionC:Managementb:Year2004 -0.177   -0.354   -0.177   -0.354  -0.354  -0.354
RegionD:Managementb:Year2004 -0.177   -0.177   -0.354   -0.354  -0.354  -0.354
RegionB:Managementb:Year2005 -0.354   -0.177   -0.177   -0.354  -0.354  -0.354
RegionC:Managementb:Year2005 -0.177   -0.354   -0.177   -0.354  -0.354  -0.354
RegionD:Managementb:Year2005 -0.177   -0.177   -0.354   -0.354  -0.354  -0.354
RegionB:Managementb:Year2006 -0.354   -0.177   -0.177   -0.354  -0.354  -0.354
RegionC:Managementb:Year2006 -0.177   -0.354   -0.177   -0.354  -0.354  -0.354
RegionD:Managementb:Year2006 -0.177   -0.177   -0.354   -0.354  -0.354  -0.354
RegionB:Managementb:Year2007 -0.354   -0.177   -0.177   -0.354  -0.354  -0.354
RegionC:Managementb:Year2007 -0.177   -0.354   -0.177   -0.354  -0.354  -0.354
RegionD:Managementb:Year2007 -0.177   -0.177   -0.354   -0.354  -0.354  -0.354
RegionB:Managementb:Year2008 -0.354   -0.177   -0.177   -0.354  -0.354  -0.354
RegionC:Managementb:Year2008 -0.177   -0.354   -0.177   -0.354  -0.354  -0.354
RegionD:Managementb:Year2008 -0.177   -0.177   -0.354   -0.354  -0.354  -0.354
RegionB:Managementb:Year2009 -0.354   -0.177   -0.177   -0.354  -0.354  -0.354
RegionC:Managementb:Year2009 -0.177   -0.354   -0.177   -0.354  -0.354  -0.354
RegionD:Managementb:Year2009 -0.177   -0.177   -0.354   -0.354  -0.354  -0.354
RegionB:Managementb:Year2010 -0.354   -0.177   -0.177   -0.354  -0.354  -0.354
RegionC:Managementb:Year2010 -0.177   -0.354   -0.177   -0.354  -0.354  -0.354
RegionD:Managementb:Year2010 -0.177   -0.177   -0.354   -0.354  -0.354  -0.354
RegionB:Managementb:Year2011 -0.354   -0.177   -0.177   -0.354  -0.354  -0.354
RegionC:Managementb:Year2011 -0.177   -0.354   -0.177   -0.354  -0.354  -0.354
RegionD:Managementb:Year2011 -0.177   -0.177   -0.354   -0.354  -0.354  -0.354
RegionB:Managementb:Year2012 -0.354   -0.177   -0.177   -0.354  -0.354  -0.354
RegionC:Managementb:Year2012 -0.177   -0.354   -0.177   -0.354  -0.354  -0.354
RegionD:Managementb:Year2012 -0.177   -0.177   -0.354   -0.354  -0.354  -0.354
RegionB:Managementb:Year2013 -0.354   -0.177   -0.177   -0.354  -0.354  -0.354
RegionC:Managementb:Year2013 -0.177   -0.354   -0.177   -0.354  -0.354  -0.354
RegionD:Managementb:Year2013 -0.177   -0.177   -0.354   -0.354  -0.354  -0.354
RegionB:Managementb:Year2014 -0.707   -0.354   -0.354   -0.354  -0.354  -0.354
RegionC:Managementb:Year2014 -0.354   -0.707   -0.354   -0.354  -0.354  -0.354
RegionD:Managementb:Year2014 -0.354   -0.354   -0.707   -0.354  -0.354  -0.354
M:Y2004 M:Y2005 M:Y2006 M:Y2007 M:Y2008 M:Y2009
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005          0.500
Managementb:Year2006          0.500   0.500
Managementb:Year2007          0.500   0.500   0.500
Managementb:Year2008          0.500   0.500   0.500   0.500
Managementb:Year2009          0.500   0.500   0.500   0.500   0.500
Managementb:Year2010          0.500   0.500   0.500   0.500   0.500   0.500
Managementb:Year2011          0.500   0.500   0.500   0.500   0.500   0.500
Managementb:Year2012          0.500   0.500   0.500   0.500   0.500   0.500
Managementb:Year2013          0.500   0.500   0.500   0.500   0.500   0.500
Managementb:Year2014          0.500   0.500   0.500   0.500   0.500   0.500
RegionB:Managementb:Year2001 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2001 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionD:Managementb:Year2001 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionB:Managementb:Year2002 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2002 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionD:Managementb:Year2002 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionB:Managementb:Year2003 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2003 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionD:Managementb:Year2003 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionB:Managementb:Year2004 -0.707  -0.354  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2004 -0.707  -0.354  -0.354  -0.354  -0.354  -0.354
RegionD:Managementb:Year2004 -0.707  -0.354  -0.354  -0.354  -0.354  -0.354
RegionB:Managementb:Year2005 -0.354  -0.707  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2005 -0.354  -0.707  -0.354  -0.354  -0.354  -0.354
RegionD:Managementb:Year2005 -0.354  -0.707  -0.354  -0.354  -0.354  -0.354
RegionB:Managementb:Year2006 -0.354  -0.354  -0.707  -0.354  -0.354  -0.354
RegionC:Managementb:Year2006 -0.354  -0.354  -0.707  -0.354  -0.354  -0.354
RegionD:Managementb:Year2006 -0.354  -0.354  -0.707  -0.354  -0.354  -0.354
RegionB:Managementb:Year2007 -0.354  -0.354  -0.354  -0.707  -0.354  -0.354
RegionC:Managementb:Year2007 -0.354  -0.354  -0.354  -0.707  -0.354  -0.354
RegionD:Managementb:Year2007 -0.354  -0.354  -0.354  -0.707  -0.354  -0.354
RegionB:Managementb:Year2008 -0.354  -0.354  -0.354  -0.354  -0.707  -0.354
RegionC:Managementb:Year2008 -0.354  -0.354  -0.354  -0.354  -0.707  -0.354
RegionD:Managementb:Year2008 -0.354  -0.354  -0.354  -0.354  -0.707  -0.354
RegionB:Managementb:Year2009 -0.354  -0.354  -0.354  -0.354  -0.354  -0.707
RegionC:Managementb:Year2009 -0.354  -0.354  -0.354  -0.354  -0.354  -0.707
RegionD:Managementb:Year2009 -0.354  -0.354  -0.354  -0.354  -0.354  -0.707
RegionB:Managementb:Year2010 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2010 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionD:Managementb:Year2010 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionB:Managementb:Year2011 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2011 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionD:Managementb:Year2011 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionB:Managementb:Year2012 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2012 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionD:Managementb:Year2012 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionB:Managementb:Year2013 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2013 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionD:Managementb:Year2013 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionB:Managementb:Year2014 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2014 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
RegionD:Managementb:Year2014 -0.354  -0.354  -0.354  -0.354  -0.354  -0.354
M:Y2010 M:Y2011 M:Y2012 M:Y2013 M:Y2014 RB:M:Y2001
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011          0.500
Managementb:Year2012          0.500   0.500
Managementb:Year2013          0.500   0.500   0.500
Managementb:Year2014          0.500   0.500   0.500   0.500
RegionB:Managementb:Year2001 -0.354  -0.354  -0.354  -0.354  -0.354
RegionC:Managementb:Year2001 -0.354  -0.354  -0.354  -0.354  -0.354   0.500
RegionD:Managementb:Year2001 -0.354  -0.354  -0.354  -0.354  -0.354   0.500
RegionB:Managementb:Year2002 -0.354  -0.354  -0.354  -0.354  -0.354   0.500
RegionC:Managementb:Year2002 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionD:Managementb:Year2002 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionB:Managementb:Year2003 -0.354  -0.354  -0.354  -0.354  -0.354   0.500
RegionC:Managementb:Year2003 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionD:Managementb:Year2003 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionB:Managementb:Year2004 -0.354  -0.354  -0.354  -0.354  -0.354   0.500
RegionC:Managementb:Year2004 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionD:Managementb:Year2004 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionB:Managementb:Year2005 -0.354  -0.354  -0.354  -0.354  -0.354   0.500
RegionC:Managementb:Year2005 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionD:Managementb:Year2005 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionB:Managementb:Year2006 -0.354  -0.354  -0.354  -0.354  -0.354   0.500
RegionC:Managementb:Year2006 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionD:Managementb:Year2006 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionB:Managementb:Year2007 -0.354  -0.354  -0.354  -0.354  -0.354   0.500
RegionC:Managementb:Year2007 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionD:Managementb:Year2007 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionB:Managementb:Year2008 -0.354  -0.354  -0.354  -0.354  -0.354   0.500
RegionC:Managementb:Year2008 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionD:Managementb:Year2008 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionB:Managementb:Year2009 -0.354  -0.354  -0.354  -0.354  -0.354   0.500
RegionC:Managementb:Year2009 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionD:Managementb:Year2009 -0.354  -0.354  -0.354  -0.354  -0.354   0.250
RegionB:Managementb:Year2010 -0.707  -0.354  -0.354  -0.354  -0.354   0.500
RegionC:Managementb:Year2010 -0.707  -0.354  -0.354  -0.354  -0.354   0.250
RegionD:Managementb:Year2010 -0.707  -0.354  -0.354  -0.354  -0.354   0.250
RegionB:Managementb:Year2011 -0.354  -0.707  -0.354  -0.354  -0.354   0.500
RegionC:Managementb:Year2011 -0.354  -0.707  -0.354  -0.354  -0.354   0.250
RegionD:Managementb:Year2011 -0.354  -0.707  -0.354  -0.354  -0.354   0.250
RegionB:Managementb:Year2012 -0.354  -0.354  -0.707  -0.354  -0.354   0.500
RegionC:Managementb:Year2012 -0.354  -0.354  -0.707  -0.354  -0.354   0.250
RegionD:Managementb:Year2012 -0.354  -0.354  -0.707  -0.354  -0.354   0.250
RegionB:Managementb:Year2013 -0.354  -0.354  -0.354  -0.707  -0.354   0.500
RegionC:Managementb:Year2013 -0.354  -0.354  -0.354  -0.707  -0.354   0.250
RegionD:Managementb:Year2013 -0.354  -0.354  -0.354  -0.707  -0.354   0.250
RegionB:Managementb:Year2014 -0.354  -0.354  -0.354  -0.354  -0.707   0.500
RegionC:Managementb:Year2014 -0.354  -0.354  -0.354  -0.354  -0.707   0.250
RegionD:Managementb:Year2014 -0.354  -0.354  -0.354  -0.354  -0.707   0.250
RC:M:Y2001 RD:M:Y2001 RB:M:Y2002 RC:M:Y2002
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011
Managementb:Year2012
Managementb:Year2013
Managementb:Year2014
RegionB:Managementb:Year2001
RegionC:Managementb:Year2001
RegionD:Managementb:Year2001  0.500
RegionB:Managementb:Year2002  0.250      0.250
RegionC:Managementb:Year2002  0.500      0.250      0.500
RegionD:Managementb:Year2002  0.250      0.500      0.500      0.500
RegionB:Managementb:Year2003  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2003  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2003  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2004  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2004  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2004  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2005  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2005  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2005  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2006  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2006  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2006  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2007  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2007  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2007  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2008  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2008  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2008  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2009  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2009  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2009  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2010  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2010  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2010  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2011  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2011  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2011  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2012  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2012  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2012  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2013  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2013  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2013  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2014  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2014  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2014  0.250      0.500      0.250      0.250
RD:M:Y2002 RB:M:Y2003 RC:M:Y2003 RD:M:Y2003
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011
Managementb:Year2012
Managementb:Year2013
Managementb:Year2014
RegionB:Managementb:Year2001
RegionC:Managementb:Year2001
RegionD:Managementb:Year2001
RegionB:Managementb:Year2002
RegionC:Managementb:Year2002
RegionD:Managementb:Year2002
RegionB:Managementb:Year2003  0.250
RegionC:Managementb:Year2003  0.250      0.500
RegionD:Managementb:Year2003  0.500      0.500      0.500
RegionB:Managementb:Year2004  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2004  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2004  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2005  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2005  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2005  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2006  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2006  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2006  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2007  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2007  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2007  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2008  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2008  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2008  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2009  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2009  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2009  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2010  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2010  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2010  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2011  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2011  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2011  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2012  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2012  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2012  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2013  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2013  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2013  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2014  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2014  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2014  0.500      0.250      0.250      0.500
RB:M:Y2004 RC:M:Y2004 RD:M:Y2004 RB:M:Y2005
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011
Managementb:Year2012
Managementb:Year2013
Managementb:Year2014
RegionB:Managementb:Year2001
RegionC:Managementb:Year2001
RegionD:Managementb:Year2001
RegionB:Managementb:Year2002
RegionC:Managementb:Year2002
RegionD:Managementb:Year2002
RegionB:Managementb:Year2003
RegionC:Managementb:Year2003
RegionD:Managementb:Year2003
RegionB:Managementb:Year2004
RegionC:Managementb:Year2004  0.500
RegionD:Managementb:Year2004  0.500      0.500
RegionB:Managementb:Year2005  0.500      0.250      0.250
RegionC:Managementb:Year2005  0.250      0.500      0.250      0.500
RegionD:Managementb:Year2005  0.250      0.250      0.500      0.500
RegionB:Managementb:Year2006  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2006  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2006  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2007  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2007  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2007  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2008  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2008  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2008  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2009  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2009  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2009  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2010  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2010  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2010  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2011  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2011  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2011  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2012  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2012  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2012  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2013  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2013  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2013  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2014  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2014  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2014  0.250      0.250      0.500      0.250
RC:M:Y2005 RD:M:Y2005 RB:M:Y2006 RC:M:Y2006
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011
Managementb:Year2012
Managementb:Year2013
Managementb:Year2014
RegionB:Managementb:Year2001
RegionC:Managementb:Year2001
RegionD:Managementb:Year2001
RegionB:Managementb:Year2002
RegionC:Managementb:Year2002
RegionD:Managementb:Year2002
RegionB:Managementb:Year2003
RegionC:Managementb:Year2003
RegionD:Managementb:Year2003
RegionB:Managementb:Year2004
RegionC:Managementb:Year2004
RegionD:Managementb:Year2004
RegionB:Managementb:Year2005
RegionC:Managementb:Year2005
RegionD:Managementb:Year2005  0.500
RegionB:Managementb:Year2006  0.250      0.250
RegionC:Managementb:Year2006  0.500      0.250      0.500
RegionD:Managementb:Year2006  0.250      0.500      0.500      0.500
RegionB:Managementb:Year2007  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2007  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2007  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2008  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2008  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2008  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2009  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2009  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2009  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2010  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2010  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2010  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2011  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2011  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2011  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2012  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2012  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2012  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2013  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2013  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2013  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2014  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2014  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2014  0.250      0.500      0.250      0.250
RD:M:Y2006 RB:M:Y2007 RC:M:Y2007 RD:M:Y2007
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011
Managementb:Year2012
Managementb:Year2013
Managementb:Year2014
RegionB:Managementb:Year2001
RegionC:Managementb:Year2001
RegionD:Managementb:Year2001
RegionB:Managementb:Year2002
RegionC:Managementb:Year2002
RegionD:Managementb:Year2002
RegionB:Managementb:Year2003
RegionC:Managementb:Year2003
RegionD:Managementb:Year2003
RegionB:Managementb:Year2004
RegionC:Managementb:Year2004
RegionD:Managementb:Year2004
RegionB:Managementb:Year2005
RegionC:Managementb:Year2005
RegionD:Managementb:Year2005
RegionB:Managementb:Year2006
RegionC:Managementb:Year2006
RegionD:Managementb:Year2006
RegionB:Managementb:Year2007  0.250
RegionC:Managementb:Year2007  0.250      0.500
RegionD:Managementb:Year2007  0.500      0.500      0.500
RegionB:Managementb:Year2008  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2008  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2008  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2009  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2009  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2009  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2010  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2010  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2010  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2011  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2011  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2011  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2012  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2012  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2012  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2013  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2013  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2013  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2014  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2014  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2014  0.500      0.250      0.250      0.500
RB:M:Y2008 RC:M:Y2008 RD:M:Y2008 RB:M:Y2009
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011
Managementb:Year2012
Managementb:Year2013
Managementb:Year2014
RegionB:Managementb:Year2001
RegionC:Managementb:Year2001
RegionD:Managementb:Year2001
RegionB:Managementb:Year2002
RegionC:Managementb:Year2002
RegionD:Managementb:Year2002
RegionB:Managementb:Year2003
RegionC:Managementb:Year2003
RegionD:Managementb:Year2003
RegionB:Managementb:Year2004
RegionC:Managementb:Year2004
RegionD:Managementb:Year2004
RegionB:Managementb:Year2005
RegionC:Managementb:Year2005
RegionD:Managementb:Year2005
RegionB:Managementb:Year2006
RegionC:Managementb:Year2006
RegionD:Managementb:Year2006
RegionB:Managementb:Year2007
RegionC:Managementb:Year2007
RegionD:Managementb:Year2007
RegionB:Managementb:Year2008
RegionC:Managementb:Year2008  0.500
RegionD:Managementb:Year2008  0.500      0.500
RegionB:Managementb:Year2009  0.500      0.250      0.250
RegionC:Managementb:Year2009  0.250      0.500      0.250      0.500
RegionD:Managementb:Year2009  0.250      0.250      0.500      0.500
RegionB:Managementb:Year2010  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2010  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2010  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2011  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2011  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2011  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2012  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2012  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2012  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2013  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2013  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2013  0.250      0.250      0.500      0.250
RegionB:Managementb:Year2014  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2014  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2014  0.250      0.250      0.500      0.250
RC:M:Y2009 RD:M:Y2009 RB:M:Y2010 RC:M:Y2010
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011
Managementb:Year2012
Managementb:Year2013
Managementb:Year2014
RegionB:Managementb:Year2001
RegionC:Managementb:Year2001
RegionD:Managementb:Year2001
RegionB:Managementb:Year2002
RegionC:Managementb:Year2002
RegionD:Managementb:Year2002
RegionB:Managementb:Year2003
RegionC:Managementb:Year2003
RegionD:Managementb:Year2003
RegionB:Managementb:Year2004
RegionC:Managementb:Year2004
RegionD:Managementb:Year2004
RegionB:Managementb:Year2005
RegionC:Managementb:Year2005
RegionD:Managementb:Year2005
RegionB:Managementb:Year2006
RegionC:Managementb:Year2006
RegionD:Managementb:Year2006
RegionB:Managementb:Year2007
RegionC:Managementb:Year2007
RegionD:Managementb:Year2007
RegionB:Managementb:Year2008
RegionC:Managementb:Year2008
RegionD:Managementb:Year2008
RegionB:Managementb:Year2009
RegionC:Managementb:Year2009
RegionD:Managementb:Year2009  0.500
RegionB:Managementb:Year2010  0.250      0.250
RegionC:Managementb:Year2010  0.500      0.250      0.500
RegionD:Managementb:Year2010  0.250      0.500      0.500      0.500
RegionB:Managementb:Year2011  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2011  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2011  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2012  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2012  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2012  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2013  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2013  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2013  0.250      0.500      0.250      0.250
RegionB:Managementb:Year2014  0.250      0.250      0.500      0.250
RegionC:Managementb:Year2014  0.500      0.250      0.250      0.500
RegionD:Managementb:Year2014  0.250      0.500      0.250      0.250
RD:M:Y2010 RB:M:Y2011 RC:M:Y2011 RD:M:Y2011
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011
Managementb:Year2012
Managementb:Year2013
Managementb:Year2014
RegionB:Managementb:Year2001
RegionC:Managementb:Year2001
RegionD:Managementb:Year2001
RegionB:Managementb:Year2002
RegionC:Managementb:Year2002
RegionD:Managementb:Year2002
RegionB:Managementb:Year2003
RegionC:Managementb:Year2003
RegionD:Managementb:Year2003
RegionB:Managementb:Year2004
RegionC:Managementb:Year2004
RegionD:Managementb:Year2004
RegionB:Managementb:Year2005
RegionC:Managementb:Year2005
RegionD:Managementb:Year2005
RegionB:Managementb:Year2006
RegionC:Managementb:Year2006
RegionD:Managementb:Year2006
RegionB:Managementb:Year2007
RegionC:Managementb:Year2007
RegionD:Managementb:Year2007
RegionB:Managementb:Year2008
RegionC:Managementb:Year2008
RegionD:Managementb:Year2008
RegionB:Managementb:Year2009
RegionC:Managementb:Year2009
RegionD:Managementb:Year2009
RegionB:Managementb:Year2010
RegionC:Managementb:Year2010
RegionD:Managementb:Year2010
RegionB:Managementb:Year2011  0.250
RegionC:Managementb:Year2011  0.250      0.500
RegionD:Managementb:Year2011  0.500      0.500      0.500
RegionB:Managementb:Year2012  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2012  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2012  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2013  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2013  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2013  0.500      0.250      0.250      0.500
RegionB:Managementb:Year2014  0.250      0.500      0.250      0.250
RegionC:Managementb:Year2014  0.250      0.250      0.500      0.250
RegionD:Managementb:Year2014  0.500      0.250      0.250      0.500
RB:M:Y2012 RC:M:Y2012 RD:M:Y2012 RB:M:Y2013
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011
Managementb:Year2012
Managementb:Year2013
Managementb:Year2014
RegionB:Managementb:Year2001
RegionC:Managementb:Year2001
RegionD:Managementb:Year2001
RegionB:Managementb:Year2002
RegionC:Managementb:Year2002
RegionD:Managementb:Year2002
RegionB:Managementb:Year2003
RegionC:Managementb:Year2003
RegionD:Managementb:Year2003
RegionB:Managementb:Year2004
RegionC:Managementb:Year2004
RegionD:Managementb:Year2004
RegionB:Managementb:Year2005
RegionC:Managementb:Year2005
RegionD:Managementb:Year2005
RegionB:Managementb:Year2006
RegionC:Managementb:Year2006
RegionD:Managementb:Year2006
RegionB:Managementb:Year2007
RegionC:Managementb:Year2007
RegionD:Managementb:Year2007
RegionB:Managementb:Year2008
RegionC:Managementb:Year2008
RegionD:Managementb:Year2008
RegionB:Managementb:Year2009
RegionC:Managementb:Year2009
RegionD:Managementb:Year2009
RegionB:Managementb:Year2010
RegionC:Managementb:Year2010
RegionD:Managementb:Year2010
RegionB:Managementb:Year2011
RegionC:Managementb:Year2011
RegionD:Managementb:Year2011
RegionB:Managementb:Year2012
RegionC:Managementb:Year2012  0.500
RegionD:Managementb:Year2012  0.500      0.500
RegionB:Managementb:Year2013  0.500      0.250      0.250
RegionC:Managementb:Year2013  0.250      0.500      0.250      0.500
RegionD:Managementb:Year2013  0.250      0.250      0.500      0.500
RegionB:Managementb:Year2014  0.500      0.250      0.250      0.500
RegionC:Managementb:Year2014  0.250      0.500      0.250      0.250
RegionD:Managementb:Year2014  0.250      0.250      0.500      0.250
RC:M:Y2013 RD:M:Y2013 RB:M:Y2014 RC:M:Y2014
RegionB
RegionC
RegionD
Managementb
Year2001
Year2002
Year2003
Year2004
Year2005
Year2006
Year2007
Year2008
Year2009
Year2010
Year2011
Year2012
Year2013
Year2014
RegionB:Managementb
RegionC:Managementb
RegionD:Managementb
RegionB:Year2001
RegionC:Year2001
RegionD:Year2001
RegionB:Year2002
RegionC:Year2002
RegionD:Year2002
RegionB:Year2003
RegionC:Year2003
RegionD:Year2003
RegionB:Year2004
RegionC:Year2004
RegionD:Year2004
RegionB:Year2005
RegionC:Year2005
RegionD:Year2005
RegionB:Year2006
RegionC:Year2006
RegionD:Year2006
RegionB:Year2007
RegionC:Year2007
RegionD:Year2007
RegionB:Year2008
RegionC:Year2008
RegionD:Year2008
RegionB:Year2009
RegionC:Year2009
RegionD:Year2009
RegionB:Year2010
RegionC:Year2010
RegionD:Year2010
RegionB:Year2011
RegionC:Year2011
RegionD:Year2011
RegionB:Year2012
RegionC:Year2012
RegionD:Year2012
RegionB:Year2013
RegionC:Year2013
RegionD:Year2013
RegionB:Year2014
RegionC:Year2014
RegionD:Year2014
Managementb:Year2001
Managementb:Year2002
Managementb:Year2003
Managementb:Year2004
Managementb:Year2005
Managementb:Year2006
Managementb:Year2007
Managementb:Year2008
Managementb:Year2009
Managementb:Year2010
Managementb:Year2011
Managementb:Year2012
Managementb:Year2013
Managementb:Year2014
RegionB:Managementb:Year2001
RegionC:Managementb:Year2001
RegionD:Managementb:Year2001
RegionB:Managementb:Year2002
RegionC:Managementb:Year2002
RegionD:Managementb:Year2002
RegionB:Managementb:Year2003
RegionC:Managementb:Year2003
RegionD:Managementb:Year2003
RegionB:Managementb:Year2004
RegionC:Managementb:Year2004
RegionD:Managementb:Year2004
RegionB:Managementb:Year2005
RegionC:Managementb:Year2005
RegionD:Managementb:Year2005
RegionB:Managementb:Year2006
RegionC:Managementb:Year2006
RegionD:Managementb:Year2006
RegionB:Managementb:Year2007
RegionC:Managementb:Year2007
RegionD:Managementb:Year2007
RegionB:Managementb:Year2008
RegionC:Managementb:Year2008
RegionD:Managementb:Year2008
RegionB:Managementb:Year2009
RegionC:Managementb:Year2009
RegionD:Managementb:Year2009
RegionB:Managementb:Year2010
RegionC:Managementb:Year2010
RegionD:Managementb:Year2010
RegionB:Managementb:Year2011
RegionC:Managementb:Year2011
RegionD:Managementb:Year2011
RegionB:Managementb:Year2012
RegionC:Managementb:Year2012
RegionD:Managementb:Year2012
RegionB:Managementb:Year2013
RegionC:Managementb:Year2013
RegionD:Managementb:Year2013  0.500
RegionB:Managementb:Year2014  0.250      0.250
RegionC:Managementb:Year2014  0.500      0.250      0.500
RegionD:Managementb:Year2014  0.250      0.500      0.500      0.500

Standardized Within-Group Residuals:
Min          Q1         Med          Q3         Max
-2.89703388 -0.66461438 -0.02232575  0.61298998  3.74346179

Number of Observations: 3600
Number of Groups:
Block               Site %in% Block
5                            10
Transect %in% Site %in% Block
30

Warning, the following code takes about 40 minutes to run!!
##Please not, this is going to take some time....
modelString=[1012 chars quoted with '"']

data.mlm <- transform(data.mlm, fBlock=interaction(Region,Block),
fSite=interaction(Region,Block, Management,Site),
fTransect=interaction(Region,Block, Management, Site, Transect)
)
X <- model.matrix(~Region*Management*Year, data.mlm)
Z1 <- model.matrix(~-1+fBlock, data.mlm)
Z2 <- model.matrix(~-1+fSite, data.mlm)
Z3 <- model.matrix(~-1+fTransect, data.mlm)
data.list <- with(data.mlm,
list(y=y,
X=X, nX=ncol(X),
Z1=Z1, nZ1=ncol(Z1),
Z2=Z2, nZ2=ncol(Z2),
Z3=Z3, nZ3=ncol(Z3),
n=nrow(data.mlm)
)
)

library(R2jags)
t1 <- proc.time()
data.jags <- jags(data=data.list,
inits=NULL,
parameters.to.save=c('beta','sigma.res','sigma.Z1','sigma.Z2','sigma.Z3'),
model.file=textConnection(modelString),
n.chains=3,
n.iter=1000,
n.burnin=200,
n.thin=10
)

Compiling model graph
Resolving undeclared variables
Allocating nodes
Graph Size: 1682155

Initializing model

t2<-proc.time()
t2-t1

    user   system  elapsed
5380.352   10.272 5398.207

print(data.jags)

Inference for Bugs model at "5", fit using jags,
3 chains, each with 1000 iterations (first 200 discarded), n.thin = 10
n.sims = 240 iterations saved
mu.vect sd.vect      2.5%       25%       50%       75%     97.5%
beta[1]      25.021   3.041    18.750    23.110    24.851    26.896    31.074
beta[2]      -3.221   4.294   -12.052    -6.251    -3.224    -0.280     4.926
beta[3]      -6.385   4.183   -14.774    -9.241    -6.406    -3.802     1.496
beta[4]     -16.955   4.718   -25.630   -20.004   -17.206   -14.494    -6.886
beta[5]      -0.348   2.934    -6.717    -2.261    -0.075     1.722     4.691
beta[6]       1.115   1.276    -1.443     0.364     1.141     1.921     3.423
beta[7]      -1.286   1.392    -4.132    -2.081    -1.284    -0.475     1.384
beta[8]      -7.204   1.235    -9.521    -8.028    -7.204    -6.359    -4.550
beta[9]      -8.961   1.315   -11.088    -9.882    -9.058    -8.267    -5.962
beta[10]    -10.927   1.243   -13.459   -11.698   -10.935   -10.081    -8.634
beta[11]     -6.264   1.355    -8.674    -7.143    -6.302    -5.406    -3.619
beta[12]     -5.000   1.367    -7.604    -5.924    -5.089    -4.001    -2.476
beta[13]     -5.157   1.286    -7.237    -6.145    -5.243    -4.230    -2.667
beta[14]     -2.861   1.345    -5.396    -3.735    -2.971    -1.857    -0.415
beta[15]      0.116   1.289    -2.597    -0.712     0.120     0.861     2.642
beta[16]     -3.561   1.428    -6.481    -4.577    -3.543    -2.590    -0.973
beta[17]     -6.185   1.244    -8.658    -6.966    -6.068    -5.501    -3.751
beta[18]     -2.947   1.391    -5.594    -3.978    -2.870    -2.027    -0.435
beta[19]     -6.368   1.336    -8.851    -7.354    -6.324    -5.396    -3.763
beta[20]     -0.614   4.054    -8.220    -3.337    -0.610     1.839     7.796
beta[21]     -1.507   3.909    -9.016    -3.634    -1.313     1.311     6.046
beta[22]     -1.531   4.026    -8.934    -4.212    -1.922     0.826     6.974
beta[23]     -4.158   1.905    -7.741    -5.581    -4.098    -2.821    -0.330
beta[24]      2.647   1.715    -0.764     1.526     2.632     3.677     6.308
beta[25]     -1.288   1.912    -5.289    -2.650    -1.253     0.067     2.506
beta[26]      1.235   2.044    -2.399    -0.240     1.163     2.635     5.066
beta[27]      1.864   1.859    -1.950     0.573     1.975     3.033     5.378
beta[28]      3.316   1.858    -0.272     1.994     3.255     4.621     6.656
beta[29]      2.378   1.810    -1.772     1.175     2.591     3.538     5.392
beta[30]      7.498   1.788     3.539     6.399     7.467     8.758    10.753
beta[31]      8.219   1.899     4.258     6.947     8.362     9.546    11.724
beta[32]      1.068   1.948    -2.435    -0.324     1.138     2.509     5.100
beta[33]      7.688   1.759     4.074     6.650     7.660     8.932    10.782
beta[34]      7.987   1.921     4.239     6.998     8.158     9.078    11.777
beta[35]      0.855   1.827    -2.616    -0.319     0.993     1.992     4.756
beta[36]      7.509   1.740     4.234     6.375     7.408     8.736    10.707
beta[37]     10.471   1.930     6.725     9.344    10.466    11.522    14.425
beta[38]     -2.113   2.055    -5.851    -3.397    -2.162    -0.970     1.942
beta[39]      6.234   1.738     2.656     5.064     6.289     7.433     9.402
beta[40]      6.512   1.942     2.496     5.365     6.515     7.684    10.283
beta[41]      2.366   1.910    -1.078     0.921     2.413     3.682     6.161
beta[42]      4.216   1.815     0.483     3.007     4.338     5.400     7.397
beta[43]      2.913   2.079    -1.183     1.437     2.931     4.441     6.864
beta[44]      0.729   1.899    -3.345    -0.571     0.824     1.993     4.434
beta[45]      4.885   1.867     0.898     3.856     4.875     6.013     8.256
beta[46]      3.113   1.848    -0.584     1.982     3.170     4.367     6.392
beta[47]     -1.131   1.943    -5.158    -2.245    -1.215     0.206     2.538
beta[48]     -1.460   1.709    -5.143    -2.560    -1.345    -0.315     1.561
beta[49]     -0.409   1.888    -4.631    -1.675    -0.358     0.875     2.836
beta[50]     -0.925   1.810    -4.377    -2.134    -0.818     0.268     2.262
beta[51]     -2.920   1.809    -6.455    -4.077    -2.925    -1.768     0.287
beta[52]     -1.847   1.938    -5.566    -3.236    -1.690    -0.595     1.803
beta[53]     -0.938   2.018    -4.679    -2.609    -0.764     0.547     2.799
beta[54]     -0.479   1.826    -3.660    -1.903    -0.408     0.867     2.912
beta[55]      2.793   2.001    -0.961     1.390     2.900     4.215     6.365
beta[56]      0.363   1.869    -2.994    -0.967     0.370     1.604     3.876
beta[57]      0.609   1.754    -2.733    -0.420     0.685     1.691     3.885
beta[58]     -0.280   1.889    -3.790    -1.391    -0.175     0.955     3.690
beta[59]      0.779   1.941    -3.171    -0.595     0.646     2.197     4.541
beta[60]      1.134   1.793    -2.460    -0.018     1.231     2.214     4.688
beta[61]      1.259   1.922    -2.605    -0.007     1.400     2.416     5.075
beta[62]     -1.661   1.935    -5.555    -2.870    -1.646    -0.329     1.895
beta[63]     -2.730   1.800    -5.874    -4.079    -2.790    -1.584     1.007
beta[64]      2.244   1.964    -1.615     0.950     2.227     3.718     6.034
beta[65]      0.341   1.804    -3.249    -0.758     0.316     1.627     3.586
beta[66]      0.832   1.850    -2.475    -0.371     0.772     2.003     4.735
beta[67]      2.082   1.965    -1.927     0.903     2.088     3.403     5.832
beta[68]      3.583   1.833     0.321     2.233     3.482     4.891     6.909
beta[69]      8.715   1.834     5.240     7.379     8.681    10.100    12.084
beta[70]     12.478   1.971     8.940    11.047    12.365    13.847    16.659
beta[71]     12.306   1.941     8.803    10.795    12.253    13.711    15.791
beta[72]     14.057   1.879    10.608    12.686    14.018    15.441    17.559
beta[73]     12.997   1.864     8.988    11.797    12.943    14.171    16.605
beta[74]      9.955   1.742     6.912     8.821     9.963    11.244    13.315
beta[75]      9.527   1.978     6.054     8.089     9.493    10.868    13.549
beta[76]      7.538   1.766     4.298     6.355     7.432     8.821    11.009
beta[77]     10.687   1.950     7.260     9.368    10.570    12.003    14.351
beta[78]     14.540   1.913    11.325    13.179    14.471    15.755    18.211
beta[79]      1.128   2.563    -4.457    -0.426     1.366     2.806     5.782
beta[80]      0.824   2.513    -4.045    -0.914     0.833     2.542     5.613
beta[81]      1.234   2.629    -4.061    -0.771     1.513     2.918     6.385
beta[82]     -1.164   2.717    -7.159    -2.859    -1.166     0.692     4.021
beta[83]      0.441   2.569    -4.474    -1.395     0.619     1.986     5.819
beta[84]      2.182   2.421    -3.134     0.773     2.119     3.830     6.529
beta[85]     -4.795   2.764   -10.022    -6.492    -4.855    -2.983     0.849
beta[86]     -1.819   2.702    -7.127    -3.659    -1.995     0.022     2.646
beta[87]     -0.947   2.734    -6.209    -2.799    -0.903     0.823     4.271
beta[88]     -1.875   2.642    -7.338    -3.615    -1.968    -0.146     3.257
beta[89]      0.843   2.455    -3.691    -0.763     0.725     2.621     5.277
beta[90]      1.163   2.543    -3.511    -0.639     1.155     2.595     6.364
beta[91]     -1.613   2.761    -6.705    -3.602    -1.525     0.537     3.168
beta[92]     -1.298   2.494    -5.431    -3.048    -1.573     0.591     3.163
beta[93]      0.440   2.696    -4.977    -1.401     0.367     2.267     5.580
beta[94]     -2.132   2.787    -7.697    -3.887    -1.915    -0.308     2.964
beta[95]     -3.316   2.558    -8.613    -4.928    -3.202    -1.566     0.958
beta[96]     -0.251   2.736    -5.579    -2.008    -0.299     1.408     5.081
beta[97]     -1.238   2.670    -7.459    -2.862    -1.016     0.643     3.576
beta[98]     -3.812   2.653    -9.388    -5.623    -3.572    -1.878     0.762
beta[99]     -1.429   2.756    -7.074    -3.285    -1.302     0.582     3.966
beta[100]    -2.911   2.561    -7.347    -4.777    -2.916    -1.011     1.838
beta[101]    -6.831   2.561   -11.435    -8.737    -6.846    -5.005    -1.909
beta[102]    -3.957   2.623    -8.972    -5.496    -3.960    -2.204     0.990
beta[103]    -1.057   2.741    -6.738    -2.655    -1.154     0.677     4.363
beta[104]    -2.961   2.572    -7.463    -4.754    -2.918    -1.319     1.542
beta[105]    -4.262   2.703    -9.252    -6.019    -4.458    -2.626     1.140
beta[106]     1.189   2.483    -3.832    -0.627     1.229     2.964     5.818
beta[107]    -2.146   2.483    -6.446    -3.895    -2.390    -0.552     2.903
beta[108]    -2.544   2.682    -7.778    -4.490    -2.400    -0.515     2.306
beta[109]     0.235   2.741    -4.676    -1.865     0.105     2.330     5.409
beta[110]    -5.162   2.650   -10.104    -6.819    -5.304    -3.359     0.450
beta[111]    -4.314   2.719    -9.549    -6.294    -4.171    -2.361     0.939
beta[112]     0.685   2.518    -4.228    -0.844     0.808     2.337     5.346
beta[113]    -4.806   2.559    -9.451    -6.640    -4.726    -3.295     1.056
beta[114]     1.083   2.690    -4.549    -0.531     0.916     2.673     6.219
beta[115]    -1.482   2.704    -6.845    -3.241    -1.490     0.485     3.826
beta[116]    -5.749   2.603   -10.534    -7.480    -5.775    -4.017    -0.787
beta[117]    -0.940   2.608    -5.591    -2.685    -0.897     0.686     3.927
beta[118]     1.011   2.637    -3.937    -0.996     1.182     2.834     5.536
beta[119]    -2.527   2.618    -7.589    -4.303    -2.596    -0.681     2.110
beta[120]    -1.901   2.879    -7.472    -3.573    -1.929     0.252     3.446
sigma.Z1      5.149   1.447     2.669     4.265     5.042     5.929     8.242
sigma.Z2      2.195   1.097     0.323     1.387     2.181     3.043     4.255
sigma.Z3      8.845   0.483     7.983     8.536     8.836     9.165     9.788
sigma.res     5.073   0.063     4.959     5.029     5.071     5.114     5.197
deviance  21902.316  26.355 21852.203 21885.074 21901.231 21917.417 21953.612
Rhat n.eff
beta[1]   1.016   240
beta[2]   1.005   230
beta[3]   1.013   240
beta[4]   1.004   240
beta[5]   1.000   240
beta[6]   0.999   240
beta[7]   1.002   240
beta[8]   1.001   240
beta[9]   1.002   240
beta[10]  1.004   240
beta[11]  1.004   240
beta[12]  0.995   240
beta[13]  0.998   240
beta[14]  0.999   240
beta[15]  1.001   240
beta[16]  0.995   240
beta[17]  1.009   240
beta[18]  1.001   240
beta[19]  1.005   240
beta[20]  0.997   240
beta[21]  1.000   240
beta[22]  1.002   240
beta[23]  1.015   150
beta[24]  1.016   190
beta[25]  1.000   240
beta[26]  0.996   240
beta[27]  1.000   240
beta[28]  1.007   210
beta[29]  1.000   240
beta[30]  0.999   240
beta[31]  1.002   240
beta[32]  0.996   240
beta[33]  1.032   190
beta[34]  1.035   210
beta[35]  0.996   240
beta[36]  1.012   240
beta[37]  1.005   240
beta[38]  1.004   240
beta[39]  1.009   240
beta[40]  1.020   240
beta[41]  0.998   240
beta[42]  0.998   240
beta[43]  1.008   170
beta[44]  0.999   240
beta[45]  1.010   240
beta[46]  1.004   240
beta[47]  1.000   240
beta[48]  1.008   240
beta[49]  1.001   240
beta[50]  1.005   220
beta[51]  1.006   240
beta[52]  1.000   240
beta[53]  0.996   240
beta[54]  0.997   240
beta[55]  0.997   240
beta[56]  1.009   230
beta[57]  0.999   240
beta[58]  1.005   240
beta[59]  0.998   240
beta[60]  1.002   240
beta[61]  1.006   240
beta[62]  1.021   180
beta[63]  1.004   240
beta[64]  1.005   240
beta[65]  0.997   240
beta[66]  1.005   240
beta[67]  0.999   240
beta[68]  0.996   240
beta[69]  1.001   240
beta[70]  1.003   240
beta[71]  0.998   240
beta[72]  0.998   240
beta[73]  1.005   240
beta[74]  1.013   240
beta[75]  0.999   240
beta[76]  1.010   190
beta[77]  1.001   240
beta[78]  1.005   230
beta[79]  1.010   200
beta[80]  1.004   240
beta[81]  1.003   240
beta[82]  0.995   240
beta[83]  1.003   240
beta[84]  1.007   180
beta[85]  0.997   240
beta[86]  1.001   240
beta[87]  1.002   240
beta[88]  1.000   240
beta[89]  1.011   140
beta[90]  1.018   110
beta[91]  0.997   240
beta[92]  1.001   240
beta[93]  1.006   240
beta[94]  1.002   240
beta[95]  1.001   240
beta[96]  0.999   240
beta[97]  1.006   240
beta[98]  1.002   240
beta[99]  1.007   210
beta[100] 0.997   240
beta[101] 1.009   150
beta[102] 1.002   240
beta[103] 1.017   160
beta[104] 1.004   240
beta[105] 0.996   240
beta[106] 0.996   240
beta[107] 1.007   240
beta[108] 1.002   240
beta[109] 1.001   240
beta[110] 0.995   240
beta[111] 0.997   240
beta[112] 0.998   240
beta[113] 1.011   120
beta[114] 1.008   180
beta[115] 0.997   240
beta[116] 0.996   240
beta[117] 1.004   240
beta[118] 1.009   190
beta[119] 1.003   240
beta[120] 1.005   240
sigma.Z1  1.139    38
sigma.Z2  1.301    13
sigma.Z3  1.015   120
sigma.res 1.010   220
deviance  0.995   240

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 350.0 and DIC = 22252.3
DIC is an estimate of expected predictive error (lower deviance is better).

library(rstan)
modelString="
data {
int<lower=1> n;
int<lower=1> nX;
int<lower=1> nZ1;
int<lower=1> nZ2;
int<lower=1> nZ3;
vector [n] y;
matrix [n,nX] X;
int Z1[n];
int Z2[n];
int Z3[n];
}
parameters {
vector[nX] beta;
real<lower=0> sigma;
vector [nZ1] gamma1;
vector [nZ2] gamma2;
vector [nZ3] gamma3;
real<lower=0> sigmaZ1;
real<lower=0> sigmaZ2;
real<lower=0> sigmaZ3;
}
transformed parameters {
vector[n] eta;

eta <- X*beta;
for (i in 1:n) {
eta[i] <- eta[i] + gamma1[Z1[i]] + gamma2[Z2[i]] + gamma3[Z3[i]];
}
}
model {
#Likelihood
y~normal(eta,sigma);

#Priors
beta ~ normal(0,1000);
sigma~cauchy(0,5);
gamma1 ~ normal(0,sigmaZ1);
sigmaZ1~cauchy(0,5);
gamma2 ~ normal(0,sigmaZ2);
sigmaZ2~cauchy(0,5);
gamma3 ~ normal(0,sigmaZ3);
sigmaZ3~cauchy(0,5);
}
generated quantities {
vector[n] log_lik;

for (i in 1:n) {
log_lik[i] <- normal_log(y[i], eta, sigma);
}
}
"

data.mlm <- transform(data.mlm, fBlock=interaction(Region,Block),
fSite=interaction(Region,Block, Management,Site),
fTransect=interaction(Region,Block, Management, Site, Transect)
)

Xmat <- model.matrix(~Region*Management*Year,data=data.mlm)
data.mlm.list <- with(data.mlm, list(y = y, X = Xmat, nX=ncol(Xmat),n = nrow(data.mlm),
Z1=as.numeric(fBlock), nZ1=length(levels(fBlock)),
Z2=as.numeric(fSite), nZ2=length(levels(fSite)),
Z3=as.numeric(fTransect), nZ3=length(levels(fTransect))))

library(rstan)
t1 <- proc.time()
data.mlm.rstan <- stan(data=data.mlm.list,
model_code=modelString,
chains=3,
iter=1000,
warmup=500,
thin=2,
save_dso=TRUE
)

SAMPLING FOR MODEL '414cdf629eacfa02c400976a034e2b26' NOW (CHAIN 1).

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#  Elapsed Time: 1053.02 seconds (Warm-up)
#                645.184 seconds (Sampling)
#                1698.2 seconds (Total)

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#  Elapsed Time: 1051.83 seconds (Warm-up)
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#                2422.53 seconds (Total)

SAMPLING FOR MODEL '414cdf629eacfa02c400976a034e2b26' NOW (CHAIN 3).

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#  Elapsed Time: 1092.35 seconds (Warm-up)
#                1028.49 seconds (Sampling)
#                2120.84 seconds (Total)

t2<-proc.time()
t2-t1

    user   system  elapsed
6287.744    8.348 6314.412

print(data.mlm.rstan, pars=c('beta','sigmaZ1','sigmaZ2','sigmaZ3','sigma'))

Inference for Stan model: 414cdf629eacfa02c400976a034e2b26.
3 chains, each with iter=1000; warmup=500; thin=2;
post-warmup draws per chain=250, total post-warmup draws=750.

mean se_mean   sd   2.5%    25%    50%    75% 97.5% n_eff Rhat
beta[1]    24.98    0.13 3.33  18.32  22.96  25.00  27.17 31.65   607 1.00
beta[2]    -3.30    0.19 4.67 -12.66  -6.69  -3.29  -0.11  6.06   580 1.00
beta[3]    -6.74    0.18 4.46 -15.47  -9.83  -6.54  -3.51  1.43   611 1.00
beta[4]   -16.80    0.21 4.79 -26.57 -19.96 -16.88 -13.50 -7.79   518 1.00
beta[5]    -0.43    0.12 2.90  -6.13  -2.39  -0.47   1.51  5.20   609 1.00
beta[6]     1.03    0.05 1.22  -1.31   0.18   1.03   1.87  3.37   492 1.01
beta[7]    -1.33    0.07 1.34  -3.85  -2.28  -1.34  -0.45  1.40   343 1.01
beta[8]    -7.29    0.06 1.24  -9.59  -8.12  -7.29  -6.47 -4.81   439 1.01
beta[9]    -9.17    0.06 1.26 -11.64 -10.04  -9.16  -8.35 -6.58   486 1.01
beta[10]  -11.12    0.06 1.34 -13.57 -11.99 -11.14 -10.20 -8.39   480 1.00
beta[11]   -6.40    0.06 1.28  -8.83  -7.29  -6.44  -5.54 -3.84   461 1.01
beta[12]   -5.12    0.06 1.24  -7.44  -5.99  -5.15  -4.28 -2.73   468 1.01
beta[13]   -5.24    0.09 1.32  -7.82  -6.12  -5.29  -4.28 -2.81   241 1.02
beta[14]   -3.06    0.06 1.28  -5.41  -3.93  -3.07  -2.25 -0.54   440 1.00
beta[15]   -0.04    0.05 1.24  -2.41  -0.85  -0.04   0.78  2.46   609 1.01
beta[16]   -3.67    0.06 1.25  -6.17  -4.46  -3.65  -2.92 -1.13   465 1.01
beta[17]   -6.44    0.06 1.30  -8.91  -7.25  -6.40  -5.53 -3.99   428 1.01
beta[18]   -3.08    0.06 1.28  -5.54  -3.88  -3.15  -2.21 -0.47   530 1.00
beta[19]   -6.45    0.06 1.26  -8.94  -7.29  -6.43  -5.61 -4.08   444 1.00
beta[20]   -0.28    0.17 4.21  -8.67  -3.03  -0.15   2.69  7.68   591 1.00
beta[21]   -1.50    0.17 3.93  -9.08  -4.16  -1.65   1.23  6.00   558 1.00
beta[22]   -1.38    0.16 4.00  -9.18  -4.04  -1.38   1.37  6.14   629 1.00
beta[23]   -4.04    0.08 1.81  -7.61  -5.14  -4.06  -2.92 -0.46   507 1.01
beta[24]    2.74    0.07 1.78  -0.67   1.60   2.71   3.97  6.09   638 1.01
beta[25]   -1.16    0.09 1.89  -4.80  -2.41  -1.14   0.02  2.62   485 1.00
beta[26]    1.52    0.09 1.90  -2.09   0.26   1.52   2.85  5.17   428 1.00
beta[27]    1.86    0.09 1.84  -1.89   0.67   1.92   3.03  5.52   422 1.01
beta[28]    3.22    0.12 1.91  -0.68   1.98   3.24   4.43  7.02   243 1.01
beta[29]    2.46    0.09 1.80  -1.21   1.28   2.55   3.64  5.84   390 1.01
beta[30]    7.56    0.08 1.77   4.05   6.47   7.55   8.66 11.06   468 1.01
beta[31]    8.19    0.09 1.81   4.83   6.92   8.12   9.48 11.69   397 1.01
beta[32]    1.38    0.08 1.80  -1.97   0.16   1.24   2.61  4.97   520 1.00
beta[33]    7.90    0.08 1.76   4.64   6.68   7.91   9.14 11.42   453 1.00
beta[34]    8.16    0.10 1.85   4.48   6.91   8.21   9.48 11.66   361 1.00
beta[35]    1.00    0.08 1.95  -2.86  -0.36   0.96   2.32  4.61   601 1.00
beta[36]    7.68    0.08 1.92   3.81   6.41   7.66   9.04 11.41   532 1.00
beta[37]   10.55    0.10 1.89   6.67   9.26  10.63  11.77 14.05   351 1.00
beta[38]   -1.92    0.08 1.82  -5.51  -3.13  -1.93  -0.65  1.56   513 1.00
beta[39]    6.35    0.08 1.78   2.98   5.15   6.25   7.56  9.92   492 1.01
beta[40]    6.55    0.09 1.89   2.81   5.30   6.64   7.73 10.23   439 1.00
beta[41]    2.53    0.07 1.80  -1.17   1.36   2.49   3.74  6.01   577 1.01
beta[42]    4.31    0.09 1.81   0.87   3.02   4.33   5.55  7.85   446 1.01
beta[43]    3.03    0.08 1.85  -0.63   1.78   3.03   4.31  6.64   477 1.01
beta[44]    0.90    0.09 1.86  -2.61  -0.39   0.88   2.15  4.47   410 1.01
beta[45]    5.00    0.09 1.84   1.59   3.80   4.97   6.23  8.63   466 1.01
beta[46]    3.23    0.13 1.84  -0.38   1.99   3.21   4.43  7.00   216 1.02
beta[47]   -0.93    0.09 1.84  -4.45  -2.22  -1.04   0.32  2.76   467 1.01
beta[48]   -1.28    0.08 1.83  -4.74  -2.46  -1.34  -0.11  2.46   497 1.00
beta[49]   -0.24    0.09 1.83  -3.83  -1.56  -0.27   0.97  3.48   382 1.00
beta[50]   -0.72    0.08 1.85  -4.22  -2.00  -0.70   0.50  2.93   519 1.00
beta[51]   -2.86    0.08 1.75  -6.27  -3.99  -2.91  -1.74  0.61   435 1.01
beta[52]   -1.76    0.08 1.85  -5.17  -3.01  -1.79  -0.56  1.84   498 1.00
beta[53]   -0.77    0.08 1.79  -4.29  -1.96  -0.80   0.38  2.97   457 1.00
beta[54]   -0.33    0.08 1.80  -3.72  -1.57  -0.43   0.77  3.43   474 1.01
beta[55]    3.03    0.08 1.86  -0.54   1.80   2.97   4.29  6.72   482 1.01
beta[56]    0.73    0.08 1.83  -2.83  -0.51   0.72   1.94  4.40   522 1.00
beta[57]    0.88    0.08 1.78  -2.46  -0.28   0.80   1.93  4.51   497 1.01
beta[58]    0.04    0.09 1.80  -3.36  -1.15   0.03   1.22  3.63   369 1.01
beta[59]    0.97    0.08 1.79  -2.61  -0.28   1.00   2.20  4.32   501 1.00
beta[60]    1.26    0.09 1.88  -2.67   0.06   1.25   2.47  5.01   485 1.01
beta[61]    1.40    0.08 1.83  -2.58   0.23   1.49   2.62  4.60   479 1.00
beta[62]   -1.41    0.09 1.83  -4.92  -2.71  -1.40  -0.12  2.04   457 1.00
beta[63]   -2.63    0.08 1.75  -5.98  -3.82  -2.59  -1.43  0.96   468 1.00
beta[64]    2.34    0.09 1.85  -1.11   1.10   2.35   3.52  5.94   456 1.00
beta[65]    0.34    0.09 1.84  -3.35  -0.89   0.37   1.58  3.65   421 1.01
beta[66]    0.93    0.10 1.90  -2.97  -0.27   0.91   2.23  4.46   396 1.01
beta[67]    2.10    0.09 1.79  -1.32   0.80   2.13   3.35  5.77   434 1.00
beta[68]    3.80    0.09 1.82   0.21   2.57   3.84   5.07  7.13   418 1.01
beta[69]    8.84    0.09 1.89   4.98   7.58   8.84  10.10 12.49   476 1.00
beta[70]   12.49    0.09 1.88   8.88  11.26  12.53  13.77 16.20   415 1.01
beta[71]   12.40    0.08 1.82   8.94  11.17  12.38  13.60 15.97   486 1.01
beta[72]   14.17    0.09 1.89  10.24  13.06  14.22  15.40 17.83   413 1.02
beta[73]   13.07    0.08 1.78   9.40  11.83  13.17  14.29 16.27   453 1.01
beta[74]   10.05    0.09 1.75   6.70   8.91  10.02  11.26 13.37   410 1.00
beta[75]    9.62    0.09 1.81   6.19   8.51   9.56  10.74 13.32   419 1.01
beta[76]    7.71    0.09 1.84   4.08   6.49   7.78   8.92 11.16   425 1.01
beta[77]   10.76    0.09 1.79   7.38   9.53  10.80  12.00 14.19   408 1.01
beta[78]   14.61    0.08 1.86  10.97  13.31  14.59  15.93 18.11   484 1.01
beta[79]    1.05    0.10 2.50  -3.58  -0.74   1.02   2.80  5.98   572 1.01
beta[80]    0.89    0.11 2.60  -3.88  -1.03   0.87   2.66  6.11   572 1.01
beta[81]    1.22    0.12 2.57  -3.92  -0.54   1.25   3.06  6.10   422 1.00
beta[82]   -1.56    0.12 2.70  -6.70  -3.34  -1.50   0.30  3.66   475 1.00
beta[83]    0.36    0.12 2.59  -4.83  -1.41   0.59   2.02  5.38   469 1.01
beta[84]    2.10    0.14 2.70  -3.11   0.29   2.11   3.71  7.44   387 1.01
beta[85]   -4.96    0.13 2.57  -9.94  -6.71  -4.98  -3.18  0.11   391 1.01
beta[86]   -1.78    0.12 2.62  -7.22  -3.44  -1.80   0.04  3.01   486 1.00
beta[87]   -0.84    0.12 2.59  -5.52  -2.63  -0.88   0.96  3.97   461 1.01
beta[88]   -2.17    0.11 2.54  -7.10  -3.83  -2.21  -0.42  2.51   522 1.00
beta[89]    0.73    0.13 2.59  -4.09  -1.07   0.70   2.28  5.83   424 1.01
beta[90]    0.97    0.15 2.57  -3.94  -0.88   1.04   2.68  5.78   298 1.01
beta[91]   -1.94    0.11 2.69  -6.94  -3.76  -2.02  -0.07  3.59   607 1.00
beta[92]   -1.32    0.12 2.68  -6.73  -3.05  -1.30   0.49  4.49   510 1.00
beta[93]    0.46    0.13 2.69  -5.01  -1.37   0.45   2.22  5.65   432 1.00
beta[94]   -2.31    0.11 2.65  -7.54  -4.11  -2.26  -0.52  2.78   546 1.01
beta[95]   -3.35    0.12 2.67  -8.43  -5.26  -3.28  -1.66  2.03   509 1.01
beta[96]   -0.16    0.13 2.64  -5.14  -1.98  -0.21   1.47  5.22   435 1.01
beta[97]   -1.48    0.12 2.63  -6.78  -3.25  -1.52   0.33  3.76   518 1.01
beta[98]   -3.91    0.11 2.64  -9.04  -5.61  -3.88  -2.15  1.31   547 1.01
beta[99]   -1.56    0.11 2.53  -6.52  -3.34  -1.55   0.16  3.66   508 1.01
beta[100]  -3.35    0.11 2.60  -8.29  -5.22  -3.33  -1.59  1.61   513 1.01
beta[101]  -7.03    0.13 2.75 -12.25  -8.97  -7.02  -5.16 -1.84   480 1.01
beta[102]  -4.10    0.13 2.63  -9.04  -5.92  -4.23  -2.22  0.80   433 1.02
beta[103]  -1.34    0.11 2.54  -6.27  -3.13  -1.32   0.49  3.34   512 1.00
beta[104]  -3.03    0.12 2.65  -8.23  -4.80  -3.13  -1.12  2.68   514 1.00
beta[105]  -4.26    0.12 2.46  -9.01  -5.95  -4.37  -2.59  0.77   422 1.01
beta[106]   0.96    0.12 2.52  -3.84  -0.80   0.91   2.72  5.57   425 1.00
beta[107]  -2.04    0.12 2.52  -7.02  -3.79  -2.00  -0.28  2.93   438 1.00
beta[108]  -2.58    0.12 2.54  -7.49  -4.32  -2.55  -0.86  2.23   455 1.01
beta[109]   0.01    0.12 2.48  -4.86  -1.62   0.05   1.66  4.72   441 1.01
beta[110]  -5.39    0.12 2.54 -10.47  -7.15  -5.31  -3.64 -0.56   438 1.01
beta[111]  -4.55    0.12 2.66  -9.87  -6.38  -4.46  -2.78  0.48   463 1.01
beta[112]   0.34    0.11 2.59  -4.88  -1.41   0.25   2.12  5.39   506 1.01
beta[113]  -4.97    0.11 2.59 -10.12  -6.70  -4.90  -3.25  0.05   513 1.01
beta[114]   0.86    0.12 2.50  -4.09  -0.89   0.79   2.61  5.75   433 1.00
beta[115]  -1.70    0.11 2.48  -6.39  -3.44  -1.78  -0.01  2.95   497 1.00
beta[116]  -5.82    0.13 2.65 -10.80  -7.62  -5.91  -4.06 -0.66   393 1.01
beta[117]  -1.02    0.11 2.59  -5.64  -2.73  -1.04   0.71  4.29   513 1.00
beta[118]   0.81    0.11 2.54  -4.17  -0.90   0.88   2.51  5.57   526 1.00
beta[119]  -2.54    0.12 2.51  -7.51  -4.24  -2.47  -0.87  2.41   439 1.01
beta[120]  -1.96    0.13 2.64  -7.15  -3.75  -1.94  -0.23  3.16   423 1.01
sigmaZ1     5.27    0.05 1.28   3.30   4.37   5.10   5.92  8.34   551 1.00
sigmaZ2     1.90    0.19 1.05   0.33   1.02   1.77   2.62  4.02    32 1.09
sigmaZ3     8.84    0.02 0.46   7.97   8.55   8.81   9.16  9.77   342 1.00
sigma       5.07    0.00 0.06   4.95   5.03   5.07   5.11  5.19   663 1.00

Samples were drawn using NUTS(diag_e) at Thu Dec 17 19:54:14 2015.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).

data.mlm <- transform(data.mlm, fBlock=interaction(Region,Block),
fSite=interaction(Region,Block, Management,Site),
fTransect=interaction(Region,Block, Management, Site, Transect)
)

library(brms)
data.mlm.brm <- brm(y~Region*Management*Year+(1|fBlock)+(1|fSite)+(1|fTransect), data=data.mlm, family='gaussian',
prior=c(set_prior('normal(0,1000)', class='b'),
set_prior('cauchy(0,5)', class='sd')),
n.chains=3, n.iter=1000, warmup=500, n.thin=2
)

SAMPLING FOR MODEL 'gaussian(identity) brms-model' NOW (CHAIN 1).

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#  Elapsed Time: 865.425 seconds (Warm-up)
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#                1485.51 seconds (Total)

summary(data.mlm.brm)

 Family: gaussian (identity)
Formula: y ~ Region * Management * Year + (1 | fBlock) + (1 | fSite) + (1 | fTransect)
Data: data.mlm (Number of observations: 3600)
Samples: 3 chains, each with n.iter = 1000; n.warmup = 500; n.thin = 2;
total post-warmup samples = 750
WAIC: 22265.34

Random Effects:
~fBlock (Number of levels: 20)
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sd(Intercept)     5.35      1.32     3.27     8.26        241 1.03

~fSite (Number of levels: 80)
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sd(Intercept)     1.61      1.18     0.04     4.16         33 1.07

~fTransect (Number of levels: 240)
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sd(Intercept)     8.86      0.46        8     9.79        319    1

Fixed Effects:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample
Intercept                       25.14      3.27    18.55    32.03        439
RegionB                         -3.50      4.45   -12.83     4.67        478
RegionC                         -6.85      4.64   -16.09     1.84        534
RegionD                        -16.91      4.63   -25.83    -8.23        394
Managementb                     -0.58      2.81    -5.98     4.78        323
Year2001                         1.10      1.37    -1.52     3.73        235
Year2002                        -1.27      1.34    -3.78     1.45        218
Year2003                        -7.18      1.32    -9.81    -4.60        148
Year2004                        -9.09      1.30   -11.75    -6.43        187
Year2005                       -11.01      1.30   -13.63    -8.53        298
Year2006                        -6.39      1.28    -8.93    -3.82        217
Year2007                        -5.03      1.31    -7.48    -2.33        211
Year2008                        -5.22      1.39    -7.87    -2.41        206
Year2009                        -3.02      1.39    -5.86    -0.34        202
Year2010                         0.05      1.35    -2.37     2.78        205
Year2011                        -3.61      1.29    -6.20    -1.26        192
Year2012                        -6.33      1.35    -8.93    -3.93        223
Year2013                        -2.95      1.30    -5.36    -0.45        218
Year2014                        -6.42      1.32    -9.07    -3.82        210
RegionB:Managementb             -0.33      4.23    -8.34     8.26        363
RegionC:Managementb             -1.18      3.91    -8.68     6.28        354
RegionD:Managementb             -1.45      3.99    -8.96     6.70        407
RegionB:Year2001                -4.14      1.90    -7.69    -0.25        256
RegionC:Year2001                 2.74      1.84    -1.01     6.31        436
RegionD:Year2001                -1.30      1.91    -4.96     2.21        262
RegionB:Year2002                 1.38      1.92    -2.29     5.31        238
RegionC:Year2002                 1.77      1.81    -1.76     5.05        285
RegionD:Year2002                 3.15      1.92    -0.85     6.97        330
RegionB:Year2003                 2.41      1.83    -1.23     6.09        230
RegionC:Year2003                 7.57      1.81     4.13    11.03        248
RegionD:Year2003                 8.06      1.90     4.19    11.66        161
RegionB:Year2004                 1.17      1.86    -2.46     4.61        265
RegionC:Year2004                 7.82      1.76     4.27    11.26        429
RegionD:Year2004                 8.00      1.90     4.07    11.65        197
RegionB:Year2005                 0.90      1.89    -2.82     4.54        274
RegionC:Year2005                 7.66      1.75     4.24    10.81        440
RegionD:Year2005                10.48      1.87     6.85    14.12        331
RegionB:Year2006                -1.99      1.81    -5.50     1.47        349
RegionC:Year2006                 6.39      1.87     2.84    10.19        334
RegionD:Year2006                 6.61      1.86     3.17    10.29        220
RegionB:Year2007                 2.40      1.84    -1.23     5.77        269
RegionC:Year2007                 4.27      1.78     0.66     7.53        295
RegionD:Year2007                 2.96      1.87    -0.97     6.24        230
RegionB:Year2008                 0.75      1.91    -3.23     4.28        248
RegionC:Year2008                 4.99      1.84     1.39     8.34        365
RegionD:Year2008                 3.12      1.91    -0.53     6.81        205
RegionB:Year2009                -1.05      1.92    -4.66     2.71        202
RegionC:Year2009                -1.26      1.86    -4.94     2.52        322
RegionD:Year2009                -0.26      1.91    -4.10     3.42        293
RegionB:Year2010                -0.78      1.90    -4.57     2.69        269
RegionC:Year2010                -2.93      1.78    -6.28     0.78        221
RegionD:Year2010                -1.92      1.87    -5.89     1.44        224
RegionB:Year2011                -0.84      1.77    -4.14     2.71        221
RegionC:Year2011                -0.35      1.79    -3.99     3.15        246
RegionD:Year2011                 2.93      1.89    -0.85     6.40        217
RegionB:Year2012                 0.55      1.88    -3.05     4.17        285
RegionC:Year2012                 0.91      1.84    -2.67     4.58        310
RegionD:Year2012                -0.19      1.93    -4.05     3.31        267
RegionB:Year2013                 0.78      1.88    -3.19     4.32        296
RegionC:Year2013                 1.22      1.81    -2.08     4.70        298
RegionD:Year2013                 1.27      1.89    -2.32     4.81        302
RegionB:Year2014                -1.52      1.85    -5.16     2.12        232
RegionC:Year2014                -2.69      1.78    -5.99     0.63        413
RegionD:Year2014                 2.32      1.81    -1.32     5.91        226
Managementb:Year2001             0.30      1.90    -3.71     3.90        305
Managementb:Year2002             0.81      1.90    -2.88     4.78        273
Managementb:Year2003             1.99      1.86    -1.68     5.57        203
Managementb:Year2004             3.62      1.90     0.06     7.38        258
Managementb:Year2005             8.77      1.83     5.33    12.57        351
Managementb:Year2006            12.49      1.79     9.03    15.85        204
Managementb:Year2007            12.28      1.84     8.79    15.88        240
Managementb:Year2008            14.22      1.86    10.48    17.77        237
Managementb:Year2009            12.91      1.88     9.39    16.60        260
Managementb:Year2010             9.84      1.88     6.17    13.49        253
Managementb:Year2011             9.53      1.87     5.92    13.35        259
Managementb:Year2012             7.65      1.83     4.23    11.38        235
Managementb:Year2013            10.60      1.93     6.82    14.23        268
Managementb:Year2014            14.55      1.77    10.91    18.02        296
RegionB:Managementb:Year2001     1.19      2.67    -4.03     6.13        286
RegionC:Managementb:Year2001     0.88      2.57    -4.27     5.82        353
RegionD:Managementb:Year2001     1.41      2.56    -3.14     6.56        246
RegionB:Managementb:Year2002    -1.29      2.77    -7.02     3.99        341
RegionC:Managementb:Year2002     0.52      2.57    -4.47     5.44        330
RegionD:Managementb:Year2002     2.32      2.64    -2.98     7.48        383
RegionB:Managementb:Year2003    -4.80      2.66    -9.86     0.46        258
RegionC:Managementb:Year2003    -1.76      2.51    -6.39     3.16        364
RegionD:Managementb:Year2003    -0.63      2.62    -5.42     4.61        185
RegionB:Managementb:Year2004    -1.84      2.73    -7.38     3.41        304
RegionC:Managementb:Year2004     0.89      2.60    -4.02     5.81        384
RegionD:Managementb:Year2004     1.17      2.62    -4.19     6.23        192
RegionB:Managementb:Year2005    -1.70      2.76    -6.87     3.48        293
RegionC:Managementb:Year2005    -1.32      2.48    -6.26     3.58        388
RegionD:Managementb:Year2005     0.56      2.57    -4.15     5.68        366
RegionB:Managementb:Year2006    -2.16      2.56    -7.27     2.80        361
RegionC:Managementb:Year2006    -3.33      2.50    -8.41     1.30        299
RegionD:Managementb:Year2006    -0.23      2.59    -5.57     4.80        213
RegionB:Managementb:Year2007    -1.10      2.64    -6.30     3.94        324
RegionC:Managementb:Year2007    -3.75      2.60    -8.69     1.49        306
RegionD:Managementb:Year2007    -1.40      2.55    -6.10     3.59        163
RegionB:Managementb:Year2008    -3.21      2.71    -8.36     2.53        264
RegionC:Managementb:Year2008    -7.14      2.67   -12.00    -1.98        345
RegionD:Managementb:Year2008    -4.04      2.62    -9.04     1.28        203
RegionB:Managementb:Year2009    -0.99      2.68    -6.37     4.14        253
RegionC:Managementb:Year2009    -3.01      2.47    -7.68     1.74        312
RegionD:Managementb:Year2009    -4.10      2.56    -9.05     0.75        265
RegionB:Managementb:Year2010     1.18      2.64    -4.18     6.28        344
RegionC:Managementb:Year2010    -1.92      2.57    -6.84     3.19        295
RegionD:Managementb:Year2010    -2.27      2.66    -7.42     2.95        230
RegionB:Managementb:Year2011     0.23      2.64    -4.95     5.21        308
RegionC:Managementb:Year2011    -5.29      2.57   -10.14    -0.24        311
RegionD:Managementb:Year2011    -4.39      2.66    -9.57     0.57        221
RegionB:Managementb:Year2012     0.51      2.68    -5.02     5.52        307
RegionC:Managementb:Year2012    -4.98      2.56   -10.01     0.05        346
RegionD:Managementb:Year2012     1.12      2.56    -3.82     6.16        272
RegionB:Managementb:Year2013    -1.48      2.69    -6.44     3.94        300
RegionC:Managementb:Year2013    -5.77      2.65   -10.47    -0.69        333
RegionD:Managementb:Year2013    -0.88      2.65    -5.82     4.29        250
RegionB:Managementb:Year2014     0.96      2.60    -3.99     6.18        335
RegionC:Managementb:Year2014    -2.41      2.46    -6.76     2.38        380
RegionD:Managementb:Year2014    -1.88      2.55    -6.90     3.32        333
Rhat
Intercept                    1.01
RegionB                      1.00
RegionC                      1.01
RegionD                      1.00
Managementb                  1.01
Year2001                     1.01
Year2002                     1.00
Year2003                     1.01
Year2004                     1.00
Year2005                     1.01
Year2006                     1.01
Year2007                     1.01
Year2008                     1.01
Year2009                     1.01
Year2010                     1.01
Year2011                     1.00
Year2012                     1.00
Year2013                     1.01
Year2014                     1.01
RegionB:Managementb          1.00
RegionC:Managementb          1.00
RegionD:Managementb          1.00
RegionB:Year2001             1.00
RegionC:Year2001             1.00
RegionD:Year2001             1.01
RegionB:Year2002             1.00
RegionC:Year2002             1.00
RegionD:Year2002             1.00
RegionB:Year2003             1.00
RegionC:Year2003             1.01
RegionD:Year2003             1.01
RegionB:Year2004             1.00
RegionC:Year2004             1.00
RegionD:Year2004             1.01
RegionB:Year2005             1.00
RegionC:Year2005             1.00
RegionD:Year2005             1.00
RegionB:Year2006             1.00
RegionC:Year2006             1.00
RegionD:Year2006             1.01
RegionB:Year2007             1.00
RegionC:Year2007             1.00
RegionD:Year2007             1.01
RegionB:Year2008             1.00
RegionC:Year2008             1.00
RegionD:Year2008             1.01
RegionB:Year2009             1.00
RegionC:Year2009             1.01
RegionD:Year2009             1.01
RegionB:Year2010             1.00
RegionC:Year2010             1.00
RegionD:Year2010             1.01
RegionB:Year2011             1.00
RegionC:Year2011             1.00
RegionD:Year2011             1.01
RegionB:Year2012             1.00
RegionC:Year2012             1.01
RegionD:Year2012             1.01
RegionB:Year2013             1.00
RegionC:Year2013             1.00
RegionD:Year2013             1.01
RegionB:Year2014             1.00
RegionC:Year2014             1.00
RegionD:Year2014             1.01
Managementb:Year2001         1.00
Managementb:Year2002         1.01
Managementb:Year2003         1.01
Managementb:Year2004         1.01
Managementb:Year2005         1.00
Managementb:Year2006         1.01
Managementb:Year2007         1.01
Managementb:Year2008         1.00
Managementb:Year2009         1.01
Managementb:Year2010         1.01
Managementb:Year2011         1.00
Managementb:Year2012         1.00
Managementb:Year2013         1.01
Managementb:Year2014         1.01
RegionB:Managementb:Year2001 1.00
RegionC:Managementb:Year2001 1.00
RegionD:Managementb:Year2001 1.01
RegionB:Managementb:Year2002 1.00
RegionC:Managementb:Year2002 1.00
RegionD:Managementb:Year2002 1.00
RegionB:Managementb:Year2003 1.00
RegionC:Managementb:Year2003 1.00
RegionD:Managementb:Year2003 1.01
RegionB:Managementb:Year2004 1.00
RegionC:Managementb:Year2004 1.00
RegionD:Managementb:Year2004 1.00
RegionB:Managementb:Year2005 1.00
RegionC:Managementb:Year2005 1.00
RegionD:Managementb:Year2005 1.00
RegionB:Managementb:Year2006 1.00
RegionC:Managementb:Year2006 1.00
RegionD:Managementb:Year2006 1.00
RegionB:Managementb:Year2007 1.00
RegionC:Managementb:Year2007 1.00
RegionD:Managementb:Year2007 1.01
RegionB:Managementb:Year2008 1.00
RegionC:Managementb:Year2008 1.00
RegionD:Managementb:Year2008 1.01
RegionB:Managementb:Year2009 1.00
RegionC:Managementb:Year2009 1.00
RegionD:Managementb:Year2009 1.01
RegionB:Managementb:Year2010 1.00
RegionC:Managementb:Year2010 1.00
RegionD:Managementb:Year2010 1.01
RegionB:Managementb:Year2011 1.00
RegionC:Managementb:Year2011 1.00
RegionD:Managementb:Year2011 1.01
RegionB:Managementb:Year2012 1.00
RegionC:Managementb:Year2012 1.00
RegionD:Managementb:Year2012 1.00
RegionB:Managementb:Year2013 1.00
RegionC:Managementb:Year2013 1.00
RegionD:Managementb:Year2013 1.01
RegionB:Managementb:Year2014 1.00
RegionC:Managementb:Year2014 1.00
RegionD:Managementb:Year2014 1.00

Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sigma(y)     5.07      0.06     4.96     5.19        508    1

Samples were drawn using NUTS(diag_e). For each parameter, Eff.Sample is a
crude measure of effective sample size, and Rhat is the potential scale
reduction factor on split chains (at convergence, Rhat = 1).

stancode(data.mlm.brm)

functions {
}
data {
int<lower=1> N;  # number of observations
vector[N] Y;  # response variable
int<lower=1> K;  # number of fixed effects
matrix[N, K] X;  # FE design matrix
# data for random effects of fBlock
int<lower=1> J_1[N];  # RE levels
int<lower=1> N_1;  # number of levels
int<lower=1> K_1;  # number of REs
real Z_1[N];  # RE design matrix
# data for random effects of fSite
int<lower=1> J_2[N];  # RE levels
int<lower=1> N_2;  # number of levels
int<lower=1> K_2;  # number of REs
real Z_2[N];  # RE design matrix
# data for random effects of fTransect
int<lower=1> J_3[N];  # RE levels
int<lower=1> N_3;  # number of levels
int<lower=1> K_3;  # number of REs
real Z_3[N];  # RE design matrix
}
transformed data {
}
parameters {
real b_Intercept;  # fixed effects Intercept
vector[K] b;  # fixed effects
vector[N_1] pre_1;  # unscaled REs
real<lower=0> sd_1;  # RE standard deviation
vector[N_2] pre_2;  # unscaled REs
real<lower=0> sd_2;  # RE standard deviation
vector[N_3] pre_3;  # unscaled REs
real<lower=0> sd_3;  # RE standard deviation
real<lower=0> sigma;  # residual SD
}
transformed parameters {
vector[N] eta;  # linear predictor
vector[N_1] r_1;  # REs
vector[N_2] r_2;  # REs
vector[N_3] r_3;  # REs
# compute linear predictor
eta <- X * b + b_Intercept;
r_1 <- sd_1 * (pre_1);  # scale REs
r_2 <- sd_2 * (pre_2);  # scale REs
r_3 <- sd_3 * (pre_3);  # scale REs
# if available add REs to linear predictor
for (n in 1:N) {
eta[n] <- eta[n] + Z_1[n] * r_1[J_1[n]] + Z_2[n] * r_2[J_2[n]] + Z_3[n] * r_3[J_3[n]];
}
}
model {
# prior specifications
b_Intercept ~ normal(0,1000);
b ~ normal(0,1000);
sd_1 ~ cauchy(0,5);
pre_1 ~ normal(0, 1);
sd_2 ~ cauchy(0,5);
pre_2 ~ normal(0, 1);
sd_3 ~ cauchy(0,5);
pre_3 ~ normal(0, 1);
sigma ~ cauchy(0, 13);
# likelihood contribution
Y ~ normal(eta, sigma);
}
generated quantities {
}

data.mlm <- transform(data.mlm, fBlock=interaction(Region,Block),
fSite=interaction(Region,Block, Management, Site),
fTransect=interaction(Region,Block, Management, Site, Transect)
)
pred <- fitted <- subset(data.mlm, select=c(Region,fBlock,Management,fSite,fTransect,Year,y))
fitted$y <- pred$y <- pred$fBlock <- pred$fSite <- pred$fTransect <- NA newdata <- expand.grid(Region=levels(data.mlm$Region), fBlock=NA,
Management=levels(data.mlm$Management), fSite=NA, fTransect=NA, Year=levels(data.mlm$Year), y=NA)
data.pred <- rbind(subset(data.mlm, select=c(Region,fBlock,Management,fSite,fTransect,Year,y)),
fitted, pred ,newdata)


Now lets fit the model.

#fit the model
data.mlm.inla <- inla(y~Region*Management*Year + f(fBlock, model='iid') + f(fSite, model='iid') +
f(fTransect, model='iid'),
data=data.pred,
control.compute=list(dic=TRUE, cpo=TRUE, waic=TRUE))
#examine the regular summary
summary(data.mlm.inla)

Call:
c("inla(formula = y ~ Region * Management * Year + f(fBlock, model = \"iid\") + ",  "    f(fSite, model = \"iid\") + f(fTransect, model = \"iid\"), data = data.pred, ",  "    control.compute = list(dic = TRUE, cpo = TRUE, waic = TRUE))" )

Time used:
Pre-processing    Running inla Post-processing           Total
1.1116         26.0171          1.2412         28.3700

Fixed effects:
mean     sd 0.025quant 0.5quant 0.975quant
(Intercept)                   24.6428 2.0219    20.6702  24.6430    28.6106
RegionB                       -2.7919 2.8649    -8.4191  -2.7922     2.8317
RegionC                       -6.1740 2.8649   -11.8011  -6.1743    -0.5503
RegionD                      -16.3433 2.8649   -21.9701 -16.3438   -10.7193
Managementb                   -0.0063 2.8388    -5.5823  -0.0065     5.5658
Year2001                       1.3526 1.2731    -1.1469   1.3525     3.8503
Year2002                      -1.0240 1.2731    -3.5234  -1.0241     1.4737
Year2003                      -6.9176 1.2731    -9.4171  -6.9177    -4.4199
Year2004                      -8.8187 1.2731   -11.3182  -8.8189    -6.3210
Year2005                     -10.7273 1.2731   -13.2268 -10.7274    -8.2296
Year2006                      -6.1018 1.2731    -8.6013  -6.1019    -3.6041
Year2007                      -4.8061 1.2731    -7.3056  -4.8062    -2.3084
Year2008                      -4.9573 1.2731    -7.4568  -4.9574    -2.4596
Year2009                      -2.7635 1.2731    -5.2630  -2.7636    -0.2658
Year2010                       0.3021 1.2731    -2.1974   0.3020     2.7998
Year2011                      -3.3724 1.2731    -5.8719  -3.3725    -0.8747
Year2012                      -6.1053 1.2731    -8.6047  -6.1054    -3.6076
Year2013                      -2.7125 1.2731    -5.2119  -2.7126    -0.2147
Year2014                      -6.1341 1.2731    -8.6336  -6.1343    -3.6364
RegionB:Managementb           -0.9610 4.0276    -8.8735  -0.9609     6.9433
RegionC:Managementb           -2.0505 4.0276    -9.9630  -2.0504     5.8538
RegionD:Managementb           -1.8585 4.0276    -9.7714  -1.8583     6.0455
RegionB:Year2001              -4.4197 1.8061    -7.9662  -4.4197    -0.8766
RegionC:Year2001               2.4054 1.8061    -1.1410   2.4055     5.9485
RegionD:Year2001              -1.5545 1.8061    -5.1010  -1.5545     1.9886
RegionB:Year2002               1.1082 1.8061    -2.4383   1.1082     4.6513
RegionC:Year2002               1.4631 1.8061    -2.0834   1.4631     5.0062
RegionD:Year2002               2.8907 1.8061    -0.6558   2.8908     6.4338
RegionB:Year2003               2.0695 1.8061    -1.4770   2.0695     5.6126
RegionC:Year2003               7.1736 1.8061     3.6271   7.1736    10.7167
RegionD:Year2003               7.7300 1.8061     4.1835   7.7300    11.2730
RegionB:Year2004               0.9277 1.8061    -2.6188   0.9277     4.4708
RegionC:Year2004               7.4500 1.8061     3.9034   7.4500    10.9930
RegionD:Year2004               7.6769 1.8061     4.1304   7.6769    11.2200
RegionB:Year2005               0.6064 1.8061    -2.9401   0.6064     4.1495
RegionC:Year2005               7.2181 1.8061     3.6715   7.2181    10.7611
RegionD:Year2005              10.1484 1.8061     6.6019  10.1485    13.6915
RegionB:Year2006              -2.2718 1.8061    -5.8182  -2.2717     1.2713
RegionC:Year2006               5.9627 1.8061     2.4162   5.9627     9.5058
RegionD:Year2006               6.2214 1.8061     2.6748   6.2214     9.7644
RegionB:Year2007               2.1887 1.8061    -1.3578   2.1887     5.7318
RegionC:Year2007               3.8832 1.8061     0.3367   3.8833     7.4263
RegionD:Year2007               2.6782 1.8061    -0.8684   2.6782     6.2212
RegionB:Year2008               0.5084 1.8061    -3.0380   0.5085     4.0515
RegionC:Year2008               4.5876 1.8061     1.0410   4.5876     8.1306
RegionD:Year2008               2.8558 1.8061    -0.6907   2.8558     6.3989
RegionB:Year2009              -1.3081 1.8061    -4.8546  -1.3081     2.2350
RegionC:Year2009              -1.6213 1.8061    -5.1678  -1.6213     1.9218
RegionD:Year2009              -0.5952 1.8061    -4.1417  -0.5952     2.9479
RegionB:Year2010              -1.0900 1.8061    -4.6364  -1.0900     2.4531
RegionC:Year2010              -3.2915 1.8061    -6.8380  -3.2915     0.2516
RegionD:Year2010              -2.1868 1.8061    -5.7333  -2.1868     1.3563
RegionB:Year2011              -1.1357 1.8061    -4.6821  -1.1357     2.4074
RegionC:Year2011              -0.7368 1.8061    -4.2833  -0.7368     2.8063
RegionD:Year2011               2.6466 1.8061    -0.8999   2.6467     6.1897
RegionB:Year2012               0.3261 1.8061    -3.2203   0.3261     3.8692
RegionC:Year2012               0.5037 1.8061    -3.0428   0.5037     4.0468
RegionD:Year2012              -0.4155 1.8061    -3.9620  -0.4154     3.1276
RegionB:Year2013               0.5496 1.8061    -2.9969   0.5496     4.0927
RegionC:Year2013               0.8817 1.8061    -2.6648   0.8817     4.4248
RegionD:Year2013               0.9829 1.8061    -2.5636   0.9829     4.5259
RegionB:Year2014              -1.7862 1.8061    -5.3326  -1.7862     1.7569
RegionC:Year2014              -3.0523 1.8061    -6.5989  -3.0523     0.4907
RegionD:Year2014               1.9685 1.8061    -1.5780   1.9686     5.5116
Managementb:Year2001          -0.1168 1.7879    -3.6275  -0.1167     3.3904
Managementb:Year2002           0.4905 1.7879    -3.0202   0.4906     3.9977
Managementb:Year2003           1.5462 1.7879    -1.9645   1.5462     5.0534
Managementb:Year2004           3.2588 1.7879    -0.2520   3.2588     6.7660
Managementb:Year2005           8.2845 1.7879     4.7738   8.2845    11.7917
Managementb:Year2006          12.0897 1.7879     8.5789  12.0897    15.5968
Managementb:Year2007          11.9272 1.7879     8.4165  11.9273    15.4344
Managementb:Year2008          13.7524 1.7879    10.2416  13.7524    17.2595
Managementb:Year2009          12.5840 1.7879     9.0732  12.5840    16.0912
Managementb:Year2010           9.4642 1.7879     5.9535   9.4643    12.9714
Managementb:Year2011           9.1905 1.7879     5.6798   9.1906    12.6977
Managementb:Year2012           7.2610 1.7879     3.7503   7.2610    10.7682
Managementb:Year2013          10.2575 1.7879     6.7468  10.2576    13.7647
Managementb:Year2014          14.1252 1.7879    10.6145  14.1252    17.6324
RegionB:Managementb:Year2001   1.6140 2.5387    -3.3704   1.6139     6.5947
RegionC:Managementb:Year2001   1.3881 2.5387    -3.5962   1.3879     6.3689
RegionD:Managementb:Year2001   1.7717 2.5387    -3.2126   1.7716     6.7525
RegionB:Managementb:Year2002  -0.9127 2.5387    -5.8971  -0.9129     4.0680
RegionC:Managementb:Year2002   1.0245 2.5387    -3.9598   1.0243     6.0053
RegionD:Managementb:Year2002   2.5962 2.5387    -2.3881   2.5960     7.5769
RegionB:Managementb:Year2003  -4.2686 2.5387    -9.2530  -4.2688     0.7121
RegionC:Managementb:Year2003  -1.1608 2.5387    -6.1451  -1.1610     3.8199
RegionD:Managementb:Year2003  -0.1457 2.5387    -5.1301  -0.1459     4.8350
RegionB:Managementb:Year2004  -1.4681 2.5387    -6.4524  -1.4682     3.5127
RegionC:Managementb:Year2004   1.4347 2.5387    -3.5496   1.4345     6.4155
RegionD:Managementb:Year2004   1.6745 2.5387    -3.3099   1.6743     6.6552
RegionB:Managementb:Year2005  -1.2411 2.5387    -6.2255  -1.2413     3.7396
RegionC:Managementb:Year2005  -0.6574 2.5387    -5.6417  -0.6576     4.3234
RegionD:Managementb:Year2005   1.0731 2.5387    -3.9112   1.0729     6.0539
RegionB:Managementb:Year2006  -1.7939 2.5387    -6.7782  -1.7941     3.1868
RegionC:Managementb:Year2006  -2.7158 2.5387    -7.7001  -2.7160     2.2650
RegionD:Managementb:Year2006   0.3185 2.5387    -4.6658   0.3183     5.2993
RegionB:Managementb:Year2007  -0.8219 2.5387    -5.8062  -0.8221     4.1588
RegionC:Managementb:Year2007  -3.2109 2.5387    -8.1952  -3.2111     1.7699
RegionD:Managementb:Year2007  -1.0048 2.5387    -5.9891  -1.0050     3.9760
RegionB:Managementb:Year2008  -2.6783 2.5387    -7.6627  -2.6785     2.3024
RegionC:Managementb:Year2008  -6.4522 2.5387   -11.4365  -6.4524    -1.4714
RegionD:Managementb:Year2008  -3.5675 2.5387    -8.5518  -3.5677     1.4133
RegionB:Managementb:Year2009  -0.7228 2.5387    -5.7071  -0.7230     4.2579
RegionC:Managementb:Year2009  -2.4499 2.5387    -7.4342  -2.4501     2.5309
RegionD:Managementb:Year2009  -3.6791 2.5387    -8.6634  -3.6793     1.3017
RegionB:Managementb:Year2010   1.6038 2.5387    -3.3806   1.6036     6.5845
RegionC:Managementb:Year2010  -1.3120 2.5387    -6.2963  -1.3122     3.6688
RegionD:Managementb:Year2010  -1.8512 2.5387    -6.8356  -1.8514     3.1295
RegionB:Managementb:Year2011   0.6171 2.5387    -4.3672   0.6170     5.5978
RegionC:Managementb:Year2011  -4.6949 2.5387    -9.6792  -4.6951     0.2858
RegionD:Managementb:Year2011  -3.9911 2.5387    -8.9754  -3.9913     0.9897
RegionB:Managementb:Year2012   0.9087 2.5387    -4.0756   0.9085     5.8894
RegionC:Managementb:Year2012  -4.3989 2.5387    -9.3832  -4.3991     0.5818
RegionD:Managementb:Year2012   1.4622 2.5387    -3.5221   1.4620     6.4430
RegionB:Managementb:Year2013  -1.1006 2.5387    -6.0849  -1.1008     3.8801
RegionC:Managementb:Year2013  -5.2287 2.5387   -10.2130  -5.2289    -0.2479
RegionD:Managementb:Year2013  -0.4744 2.5387    -5.4587  -0.4746     4.5063
RegionB:Managementb:Year2014   1.3995 2.5387    -3.5848   1.3994     6.3803
RegionC:Managementb:Year2014  -1.8742 2.5387    -6.8585  -1.8744     3.1066
RegionD:Managementb:Year2014  -1.3992 2.5387    -6.3835  -1.3994     3.5816
mode kld
(Intercept)                   24.6435   0
RegionB                       -2.7926   0
RegionC                       -6.1748   0
RegionD                      -16.3445   0
Managementb                   -0.0068   0
Year2001                       1.3524   0
Year2002                      -1.0242   0
Year2003                      -6.9178   0
Year2004                      -8.8190   0
Year2005                     -10.7276   0
Year2006                      -6.1021   0
Year2007                      -4.8063   0
Year2008                      -4.9576   0
Year2009                      -2.7637   0
Year2010                       0.3019   0
Year2011                      -3.3726   0
Year2012                      -6.1055   0
Year2013                      -2.7127   0
Year2014                      -6.1344   0
RegionB:Managementb           -0.9605   0
RegionC:Managementb           -2.0499   0
RegionD:Managementb           -1.8576   0
RegionB:Year2001              -4.4196   0
RegionC:Year2001               2.4057   0
RegionD:Year2001              -1.5543   0
RegionB:Year2002               1.1084   0
RegionC:Year2002               1.4633   0
RegionD:Year2002               2.8910   0
RegionB:Year2003               2.0697   0
RegionC:Year2003               7.1739   0
RegionD:Year2003               7.7302   0
RegionB:Year2004               0.9279   0
RegionC:Year2004               7.4502   0
RegionD:Year2004               7.6771   0
RegionB:Year2005               0.6066   0
RegionC:Year2005               7.2183   0
RegionD:Year2005              10.1487   0
RegionB:Year2006              -2.2715   0
RegionC:Year2006               5.9630   0
RegionD:Year2006               6.2216   0
RegionB:Year2007               2.1889   0
RegionC:Year2007               3.8835   0
RegionD:Year2007               2.6784   0
RegionB:Year2008               0.5087   0
RegionC:Year2008               4.5879   0
RegionD:Year2008               2.8561   0
RegionB:Year2009              -1.3079   0
RegionC:Year2009              -1.6211   0
RegionD:Year2009              -0.5950   0
RegionB:Year2010              -1.0898   0
RegionC:Year2010              -3.2912   0
RegionD:Year2010              -2.1866   0
RegionB:Year2011              -1.1355   0
RegionC:Year2011              -0.7366   0
RegionD:Year2011               2.6469   0
RegionB:Year2012               0.3263   0
RegionC:Year2012               0.5040   0
RegionD:Year2012              -0.4152   0
RegionB:Year2013               0.5498   0
RegionC:Year2013               0.8820   0
RegionD:Year2013               0.9831   0
RegionB:Year2014              -1.7860   0
RegionC:Year2014              -3.0521   0
RegionD:Year2014               1.9688   0
Managementb:Year2001          -0.1165   0
Managementb:Year2002           0.4908   0
Managementb:Year2003           1.5465   0
Managementb:Year2004           3.2591   0
Managementb:Year2005           8.2848   0
Managementb:Year2006          12.0900   0
Managementb:Year2007          11.9275   0
Managementb:Year2008          13.7527   0
Managementb:Year2009          12.5843   0
Managementb:Year2010           9.4645   0
Managementb:Year2011           9.1908   0
Managementb:Year2012           7.2613   0
Managementb:Year2013          10.2578   0
Managementb:Year2014          14.1255   0
RegionB:Managementb:Year2001   1.6138   0
RegionC:Managementb:Year2001   1.3878   0
RegionD:Managementb:Year2001   1.7714   0
RegionB:Managementb:Year2002  -0.9130   0
RegionC:Managementb:Year2002   1.0242   0
RegionD:Managementb:Year2002   2.5959   0
RegionB:Managementb:Year2003  -4.2689   0
RegionC:Managementb:Year2003  -1.1612   0
RegionD:Managementb:Year2003  -0.1461   0
RegionB:Managementb:Year2004  -1.4684   0
RegionC:Managementb:Year2004   1.4343   0
RegionD:Managementb:Year2004   1.6741   0
RegionB:Managementb:Year2005  -1.2415   0
RegionC:Managementb:Year2005  -0.6577   0
RegionD:Managementb:Year2005   1.0728   0
RegionB:Managementb:Year2006  -1.7942   0
RegionC:Managementb:Year2006  -2.7162   0
RegionD:Managementb:Year2006   0.3181   0
RegionB:Managementb:Year2007  -0.8222   0
RegionC:Managementb:Year2007  -3.2113   0
RegionD:Managementb:Year2007  -1.0051   0
RegionB:Managementb:Year2008  -2.6787   0
RegionC:Managementb:Year2008  -6.4526   0
RegionD:Managementb:Year2008  -3.5679   0
RegionB:Managementb:Year2009  -0.7231   0
RegionC:Managementb:Year2009  -2.4503   0
RegionD:Managementb:Year2009  -3.6795   0
RegionB:Managementb:Year2010   1.6035   0
RegionC:Managementb:Year2010  -1.3123   0
RegionD:Managementb:Year2010  -1.8516   0
RegionB:Managementb:Year2011   0.6168   0
RegionC:Managementb:Year2011  -4.6953   0
RegionD:Managementb:Year2011  -3.9915   0
RegionB:Managementb:Year2012   0.9084   0
RegionC:Managementb:Year2012  -4.3993   0
RegionD:Managementb:Year2012   1.4619   0
RegionB:Managementb:Year2013  -1.1009   0
RegionC:Managementb:Year2013  -5.2290   0
RegionD:Managementb:Year2013  -0.4748   0
RegionB:Managementb:Year2014   1.3992   0
RegionC:Managementb:Year2014  -1.8745   0
RegionD:Managementb:Year2014  -1.3995   0

Random effects:
Name	  Model
fBlock   IID model
fSite   IID model
fTransect   IID model

Model hyperparameters:
mean        sd 0.025quant
Precision for the Gaussian observations 3.900e-02 1.000e-03     0.0371
Precision for fSite                     1.808e+04 1.758e+04  1234.8571
Precision for fTransect                 9.800e-03 9.000e-04     0.0081
Precision for fBlock                    1.936e+04 1.883e+04  1381.6452
0.5quant 0.975quant      mode
Precision for the Gaussian observations 3.900e-02  4.090e-02    0.0389
Precision for fSite                     1.294e+04  6.479e+04 3354.0166
Precision for fTransect                 9.800e-03  1.170e-02    0.0097
Precision for fBlock                    1.386e+04  6.901e+04 3785.0408

Expected number of effective parameters(std dev): 347.67(0.3603)
Number of equivalent replicates : 10.35

Deviance Information Criterion (DIC) ...: 22250.71
Effective number of parameters .........: 348.73

Watanabe-Akaike information criterion (WAIC) ...: 22262.86
Effective number of parameters .................: 332.05

Marginal log-Likelihood:  -11820.99
CPO and PIT are computed

Posterior marginals for linear predictor and fitted values computed

s <- inla.contrib.sd(data.mlm.inla, nsamples=1000)
s\$hyper

                                        mean          sd        2.5%
sd for the Gaussian observations  5.06980496 0.061824722 4.945622500
sd for fSite                      0.01005813 0.004981725 0.004040437
sd for fTransect                 10.11075291 0.471701921 9.229803711
sd for fBlock                     0.00998637 0.005242160 0.003693150
97.5%
sd for the Gaussian observations  5.19643186
sd for fSite                      0.02154145
sd for fTransect                 11.03697015
sd for fBlock                     0.02378770


WOW, that is a fraction of the time of JAGS and STAN

## References

 Exponential family of distributions

The exponential distributions are a class of continuous distribution which can be characterized by two parameters. One of these parameters (the location parameter) is a function of the mean and the other (the dispersion parameter) is a function of the variance of the distribution. Note that recent developments have further extended generalized linear models to accommodate other non-exponential residual distributions.

End of instructions