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) head(data.hier)
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). 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: 0.573899 seconds (Warm-up) # 0.144843 seconds (Sampling) # 0.718742 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: 0.289665 seconds (Warm-up) # 0.115353 seconds (Sampling) # 0.405018 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: 0.404384 seconds (Warm-up) # 0.140678 seconds (Sampling) # 0.545062 seconds (Total)
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 )
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: 0.674769 seconds (Warm-up) # 0.461803 seconds (Sampling) # 1.13657 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: 0.600406 seconds (Warm-up) # 0.549344 seconds (Sampling) # 1.14975 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: 0.703574 seconds (Warm-up) # 0.462358 seconds (Sampling) # 1.16593 seconds (Total)
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),]) head(newdata)
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 head(ndata)
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$marginals.fitted.values[[idx+nr-1]]) change=(rmax-rmin) (change=data.frame(Mean=mean(change), Median=median(change), HPDinterval(as.mcmc(change))))
Mean Median lower upper var1 99.86877 99.84981 75.73204 124.6698
## or as a percentage change (increase) pchange=100*(rmax-rmin)/rmin (pchange=data.frame(Mean=mean(pchange), Median=median(pchange), HPDinterval(as.mcmc(pchange))))
Mean Median lower upper var1 42.88382 42.68041 30.36272 56.36062
Split-plot design
- the number of between block treatments (A) = 3
- the number of blocks = 35
- the number of within block treatments (C) = 3
- the mean of the treatments = 40, 70 and 80 respectively
- the variability (standard deviation) between blocks of the same treatment = 12
- the variability (standard deviation) between treatments withing blocks = 5
library(plyr) set.seed(1) nA <- 3 nC <- 3 nBlock <- 36 sigma <- 5 sigma.block <- 12 n <- nBlock * nC Block <- gl(nBlock, k = 1) C <- gl(nC, k = 1) ## Specify the cell means AC.means <- (rbind(c(40, 70, 80), c(35, 50, 70), c(35, 40, 45))) ## Convert these to effects X <- model.matrix(~A * C, data = expand.grid(A = gl(3, k = 1), C = gl(3, k = 1))) AC <- as.vector(AC.means) AC.effects <- solve(X, AC) A <- gl(nA, nBlock, n) dt <- expand.grid(C = C, Block = Block) dt <- data.frame(dt, A) Xmat <- cbind(model.matrix(~-1 + Block, data = dt), model.matrix(~A * C, data = dt)) block.effects <- rnorm(n = nBlock, mean = 0, sd = sigma.block) all.effects <- c(block.effects, AC.effects) lin.pred <- Xmat %*% all.effects ## the quadrat observations (within sites) are drawn from normal distributions ## with means according to the site means and standard deviations of 5 y <- rnorm(n, lin.pred, sigma) data.splt <- data.frame(y = y, A = A, dt) head(data.splt) #print out the first six rows of the data set
y A C Block A.1 1 30.51110 1 1 1 1 2 62.18599 1 2 1 1 3 77.98268 1 3 1 1 4 46.01960 1 1 2 1 5 71.38110 1 2 2 1 6 80.93691 1 3 2 1
tapply(data.splt$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
- fitted values - called fitted
- fitted values (ignoring Block) - called pred
- cell means predictions - called newdata
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$summary.linear.predictor[(nrow(data.splt)+nrow(fitted)+nrow(pred)+1):nrow(data.pred),]) head(newdata)
A C Block y mean sd 0.025quant 0.5quant 0.975quant predictor.325 1 1 NA NA 44.38840 3.326560 37.81966 44.39321 50.92952 predictor.326 2 1 NA NA 35.60775 3.346761 29.01963 35.60543 42.20765 predictor.327 3 1 NA NA 32.83571 3.346796 26.24966 32.83273 39.43807 predictor.328 1 2 NA NA 75.26213 3.336744 68.67216 75.26738 81.82216 predictor.329 2 2 NA NA 51.47799 3.351807 44.88009 51.47560 58.08802 predictor.330 3 2 NA NA 39.21712 3.351838 32.62154 39.21398 45.82984 mode kld predictor.325 44.40211 1.296486e-09 predictor.326 35.60120 4.667141e-09 predictor.327 32.82730 5.053008e-09 predictor.328 75.27723 1.257169e-09 predictor.329 51.47123 4.692768e-09 predictor.330 39.20821 5.087045e-09
newdata <- reshape:::rename(newdata, c("0.025quant"="lower", "0.975quant"="upper")) fitted <- cbind(fitted, data.inla$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$Res head(ndata)
y A C Block A.1 fit pred Res Pobs 1 30.51110 1 1 1 1 34.31855 44.38840 3.8074422 48.19584 2 62.18599 1 2 1 1 65.19225 75.26213 3.0062668 78.26840 3 77.98268 1 3 1 1 72.85033 82.92021 -5.1323504 77.78786 4 46.01960 1 1 2 1 43.05227 44.38840 -2.9673325 41.42106 5 71.38110 1 2 2 1 73.92597 75.26213 2.5448720 77.80700 6 80.93691 1 3 2 1 81.58405 82.92021 0.6471391 83.56735
library(grid) ggplot(newdata, aes(y=mean, x=C, group=A)) + geom_blank()+ geom_line(aes(linetype=A,x=as.numeric(C)),position=position_dodge(width=0.1)) + geom_linerange(aes(ymin=lower, ymax=upper),position=position_dodge(width=0.1))+ geom_point(aes(fill=A),position=position_dodge(width=0.1), shape=21, size=3)+ scale_fill_manual('A', breaks=c(1,2,3), labels=c('1','2','3'), values=c('black','grey','white'))+ theme_classic()+ theme(legend.key.width=unit(2,'lines'), legend.position=c(0,1), legend.justification=c(0,1))
And as for our specific comparisons..
data.inla$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
- 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) head(data.mlm)
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()
$$ y_i = \alpha + \beta X_{i} + \sum_{j=1}^J{}\gamma_{ij}\mathbf{Z_i} + \varepsilon_i \hspace{4em}\varepsilon_i\sim{}\mathcal{N}(0, \sigma^2) $$
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 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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 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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 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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 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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 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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 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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 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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 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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 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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 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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 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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
##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). 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: 1053.02 seconds (Warm-up) # 645.184 seconds (Sampling) # 1698.2 seconds (Total) SAMPLING FOR MODEL '414cdf629eacfa02c400976a034e2b26' 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: 1051.83 seconds (Warm-up) # 1370.7 seconds (Sampling) # 2422.53 seconds (Total) SAMPLING FOR MODEL '414cdf629eacfa02c400976a034e2b26' 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: 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). 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: 865.425 seconds (Warm-up) # 638.847 seconds (Sampling) # 1504.27 seconds (Total) SAMPLING FOR MODEL 'gaussian(identity) brms-model' 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: 824.412 seconds (Warm-up) # 719.768 seconds (Sampling) # 1544.18 seconds (Total) SAMPLING FOR MODEL 'gaussian(identity) brms-model' 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: 902.438 seconds (Warm-up) # 583.068 seconds (Sampling) # 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