srrs2 <- read.table("./data/ARM_Data/radon/srrs2.dat", header=TRUE, sep=",")
mn <- srrs2$state == "MN"
radon <- srrs2$activity[mn]
y <- log(ifelse(radon == 0, 0.1, radon))
n <- length(radon)
x <- srrs2$floor[mn] # 0 for basement, 1 for first floor
srrs2.fips <- srrs2$stfips * 1000 + srrs2$cntyfips
county.name <- as.vector(srrs2$county[mn])
uniq.name <- unique(county.name)
J <- length(uniq.name)
county <- rep(NA, J)
for (i in 1:J) {
county[county.name == uniq.name[i]] <- i
}
srrs2.fips <- srrs2$stfips*1000 + srrs2$cntyfips
cty <- read.table ("data/ARM_Data/radon/cty.dat", header=T, sep=",")
usa.fips <- 1000*cty[,"stfips"] + cty[,"ctfips"]
usa.rows <- match (unique(srrs2.fips[mn]), usa.fips)
uranium <- cty[usa.rows,"Uppm"]
u <- log (uranium)
u.full <- u[county]
data_jags <- list(y = y, x = x, county = county, n = n, J = J)
data_jags$u <- u.full5 Chapter 17-Part A
5.1 Varying intercepts and slopes
Carga de datos:
5.1.1 Varying-Intercept, Varying-Slope Model
We begin with a varying-intercept, varying-slope model that includes (x) but without the county-level uranium predictor. The model is structured as:
\[ y_i \sim N(\alpha_{j[i]} + \beta_{j[i]} x_i , \sigma_y^2), \quad \text{for} \ i = 1, \dots, n \]
Where:
\[ \begin{pmatrix} \alpha_j \\ \beta_j \end{pmatrix} \sim N\left( \begin{pmatrix} \mu_\alpha \\ \mu_\beta \end{pmatrix}, \begin{pmatrix} \sigma_\alpha^2 & \rho\sigma_\alpha\sigma_\beta \\ \rho\sigma_\alpha\sigma_\beta & \sigma_\beta^2 \end{pmatrix}\right), \quad \text{for} \ j = 1, \dots, J \]
This model includes variation in both \(\alpha_j\) and \(\beta_j\) with a between-group correlation parameter \(\rho\).
5.1.2 Simple Model with No Correlation Between Intercepts and Slopes
We begin with the varying-intercept, varying-slope radon model (from Section 13.1), simplified by ignoring the correlation between intercepts and slopes. This assumes independence between the intercepts and slopes.
# Load necessary library
library(R2jags)Loading required package: rjagsLoading required package: codaLinked to JAGS 4.3.2Loaded modules: basemod,bugs
Attaching package: 'R2jags'The following object is masked from 'package:coda':
traceplot# Prepare the data for JAGS
data_jags <- list(y = y, x = x, county = county, n = n, J = J, u = u.full)
# Initial values
inits <- function() {
list(a = rnorm(J), b = rnorm(J), mu.a = rnorm(1), mu.b = rnorm(1), sigma.y = runif(1), sigma.a = runif(1), sigma.b = runif(1))
}
# Parameters to monitor
params <- c("a", "b", "mu.a", "mu.b", "sigma.y", "sigma.a", "sigma.b")
# Run JAGS model
jags_fit <- jags(data = data_jags,
inits = inits,
parameters.to.save = params,
model.file = "codigoJAGS/radon_model_slope1.jags",
n.chains = 3,
n.iter = 5000,
n.burnin = 1000,
n.thin = 10)module glm loadedWarning in jags.model(model.file, data = data, inits = init.values, n.chains =
n.chains, : Unused variable "u" in dataCompiling model graph
Resolving undeclared variables
Allocating nodes
Graph information:
Observed stochastic nodes: 919
Unobserved stochastic nodes: 175
Total graph size: 3232
Initializing model# Check the summary
print(jags_fit)Inference for Bugs model at "codigoJAGS/radon_model_slope1.jags", fit using jags,
3 chains, each with 5000 iterations (first 1000 discarded), n.thin = 10
n.sims = 1200 iterations saved
mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat
a[1] 1.164 0.262 0.632 0.980 1.166 1.345 1.666 1.000
a[2] 0.936 0.103 0.735 0.867 0.935 1.007 1.134 1.001
a[3] 1.479 0.284 0.929 1.292 1.478 1.670 2.022 1.000
a[4] 1.513 0.227 1.072 1.361 1.517 1.661 1.961 1.000
a[5] 1.437 0.259 0.916 1.276 1.438 1.610 1.936 1.001
a[6] 1.474 0.272 0.915 1.293 1.477 1.656 1.993 1.001
a[7] 1.832 0.175 1.494 1.718 1.830 1.945 2.196 1.001
a[8] 1.696 0.268 1.202 1.512 1.694 1.872 2.223 1.002
a[9] 1.140 0.201 0.748 1.004 1.139 1.274 1.529 1.001
a[10] 1.531 0.234 1.077 1.371 1.529 1.684 1.994 1.001
a[11] 1.431 0.239 0.950 1.276 1.440 1.590 1.884 1.005
a[12] 1.588 0.254 1.083 1.419 1.579 1.759 2.082 1.000
a[13] 1.231 0.217 0.821 1.081 1.233 1.379 1.661 1.003
a[14] 1.850 0.185 1.486 1.726 1.852 1.974 2.203 1.007
a[15] 1.396 0.261 0.878 1.215 1.394 1.575 1.881 1.000
a[16] 1.229 0.293 0.658 1.022 1.227 1.419 1.825 1.006
a[17] 1.402 0.266 0.873 1.224 1.401 1.574 1.940 1.001
a[18] 1.190 0.184 0.821 1.072 1.193 1.306 1.561 1.001
a[19] 1.355 0.094 1.175 1.293 1.355 1.417 1.539 1.002
a[20] 1.606 0.269 1.104 1.432 1.603 1.778 2.140 1.000
a[21] 1.623 0.201 1.240 1.486 1.620 1.759 2.026 1.001
a[22] 1.011 0.230 0.562 0.853 1.013 1.168 1.456 1.003
a[23] 1.436 0.298 0.854 1.243 1.437 1.628 2.027 1.000
a[24] 1.867 0.212 1.456 1.723 1.867 2.011 2.279 1.003
a[25] 1.780 0.184 1.436 1.660 1.778 1.901 2.153 1.001
a[26] 1.371 0.074 1.226 1.320 1.373 1.421 1.517 1.001
a[27] 1.614 0.233 1.164 1.461 1.613 1.767 2.061 1.004
a[28] 1.326 0.253 0.823 1.159 1.325 1.496 1.805 1.005
a[29] 1.299 0.277 0.764 1.111 1.303 1.490 1.829 1.005
a[30] 1.094 0.192 0.697 0.967 1.096 1.222 1.468 1.002
a[31] 1.746 0.243 1.266 1.576 1.746 1.902 2.207 1.001
a[32] 1.361 0.251 0.859 1.182 1.366 1.540 1.830 1.002
a[33] 1.731 0.265 1.225 1.554 1.732 1.904 2.251 1.000
a[34] 1.504 0.276 0.947 1.321 1.508 1.690 2.010 1.000
a[35] 1.050 0.243 0.582 0.876 1.054 1.227 1.515 1.009
a[36] 1.875 0.306 1.320 1.667 1.856 2.073 2.492 1.001
a[37] 0.786 0.215 0.347 0.637 0.788 0.932 1.203 1.002
a[38] 1.626 0.261 1.137 1.455 1.620 1.814 2.135 1.000
a[39] 1.601 0.239 1.126 1.445 1.593 1.772 2.047 1.003
a[40] 1.834 0.261 1.350 1.658 1.826 2.005 2.346 1.000
a[41] 1.757 0.212 1.344 1.612 1.752 1.906 2.156 1.000
a[42] 1.421 0.309 0.826 1.205 1.424 1.625 2.026 1.004
a[43] 1.668 0.233 1.221 1.508 1.664 1.820 2.141 1.002
a[44] 1.220 0.223 0.798 1.065 1.221 1.370 1.635 1.004
a[45] 1.358 0.188 0.990 1.238 1.358 1.485 1.726 1.000
a[46] 1.335 0.238 0.868 1.180 1.340 1.485 1.816 1.000
a[47] 1.310 0.288 0.735 1.115 1.316 1.512 1.855 1.002
a[48] 1.250 0.205 0.841 1.118 1.251 1.387 1.640 1.002
a[49] 1.647 0.181 1.290 1.530 1.645 1.767 2.010 1.001
a[50] 1.639 0.320 1.028 1.427 1.623 1.858 2.267 1.005
a[51] 1.783 0.252 1.282 1.614 1.782 1.951 2.280 1.001
a[52] 1.647 0.279 1.091 1.463 1.650 1.829 2.178 1.000
a[53] 1.378 0.280 0.816 1.190 1.384 1.579 1.902 1.001
a[54] 1.342 0.144 1.049 1.249 1.344 1.443 1.622 1.006
a[55] 1.531 0.215 1.099 1.383 1.532 1.683 1.948 1.000
a[56] 1.345 0.279 0.789 1.163 1.341 1.532 1.871 1.001
a[57] 1.089 0.231 0.637 0.932 1.081 1.257 1.526 1.004
a[58] 1.629 0.249 1.143 1.464 1.630 1.802 2.093 1.003
a[59] 1.598 0.263 1.106 1.420 1.594 1.776 2.109 1.000
a[60] 1.401 0.277 0.844 1.223 1.405 1.587 1.941 1.003
a[61] 1.182 0.124 0.944 1.100 1.180 1.266 1.433 1.000
a[62] 1.704 0.248 1.237 1.528 1.701 1.876 2.185 1.003
a[63] 1.528 0.270 0.989 1.348 1.526 1.711 2.051 1.005
a[64] 1.716 0.188 1.348 1.590 1.715 1.840 2.112 1.002
a[65] 1.423 0.281 0.874 1.238 1.417 1.615 1.968 1.002
a[66] 1.570 0.198 1.191 1.439 1.572 1.702 1.972 1.000
a[67] 1.641 0.191 1.275 1.521 1.639 1.762 2.032 1.000
a[68] 1.221 0.210 0.802 1.079 1.227 1.366 1.608 1.001
a[69] 1.368 0.257 0.883 1.195 1.364 1.540 1.869 1.001
a[70] 0.882 0.073 0.735 0.835 0.884 0.929 1.024 1.002
a[71] 1.496 0.140 1.224 1.400 1.497 1.586 1.781 1.003
a[72] 1.546 0.194 1.181 1.413 1.543 1.678 1.924 1.000
a[73] 1.555 0.285 1.004 1.361 1.565 1.749 2.086 1.001
a[74] 1.245 0.249 0.746 1.081 1.246 1.407 1.720 1.000
a[75] 1.545 0.279 0.953 1.356 1.546 1.744 2.071 1.001
a[76] 1.705 0.267 1.179 1.535 1.705 1.888 2.206 1.001
a[77] 1.675 0.219 1.247 1.534 1.671 1.825 2.125 1.002
a[78] 1.384 0.254 0.887 1.216 1.382 1.553 1.883 1.000
a[79] 1.089 0.270 0.547 0.911 1.094 1.272 1.598 1.011
a[80] 1.358 0.105 1.157 1.286 1.360 1.428 1.565 1.000
a[81] 1.874 0.299 1.314 1.658 1.870 2.066 2.472 1.000
a[82] 1.588 0.310 1.009 1.385 1.592 1.793 2.201 1.000
a[83] 1.632 0.189 1.262 1.503 1.631 1.762 2.015 1.002
a[84] 1.592 0.180 1.239 1.472 1.592 1.712 1.947 1.002
a[85] 1.379 0.285 0.812 1.200 1.370 1.557 1.978 1.003
b[1] -0.628 0.298 -1.212 -0.790 -0.659 -0.472 0.072 1.002
b[2] -0.844 0.251 -1.405 -0.993 -0.802 -0.679 -0.415 1.003
b[3] -0.657 0.271 -1.223 -0.817 -0.667 -0.505 -0.077 1.002
b[4] -0.724 0.234 -1.213 -0.860 -0.727 -0.589 -0.263 1.000
b[5] -0.648 0.292 -1.203 -0.816 -0.681 -0.484 -0.048 1.000
b[6] -0.692 0.325 -1.351 -0.864 -0.698 -0.516 -0.037 1.005
b[7] -0.436 0.318 -0.922 -0.677 -0.489 -0.237 0.250 1.002
b[8] -0.641 0.283 -1.187 -0.809 -0.663 -0.489 -0.010 1.002
b[9] -0.511 0.318 -1.013 -0.731 -0.570 -0.329 0.232 1.004
b[10] -0.751 0.256 -1.301 -0.886 -0.743 -0.597 -0.258 1.002
b[11] -0.671 0.317 -1.338 -0.844 -0.690 -0.487 -0.005 1.000
b[12] -0.677 0.331 -1.307 -0.855 -0.697 -0.497 0.024 1.001
b[13] -0.680 0.329 -1.385 -0.844 -0.699 -0.509 0.008 1.000
b[14] -0.745 0.246 -1.267 -0.881 -0.736 -0.602 -0.262 1.000
b[15] -0.687 0.271 -1.202 -0.836 -0.696 -0.535 -0.101 1.005
b[16] -0.683 0.320 -1.376 -0.849 -0.701 -0.501 -0.012 1.001
b[17] -0.796 0.273 -1.450 -0.944 -0.761 -0.634 -0.289 1.003
b[18] -0.550 0.268 -1.035 -0.731 -0.585 -0.384 0.018 1.003
b[19] -0.772 0.228 -1.247 -0.912 -0.761 -0.631 -0.335 1.002
b[20] -0.689 0.331 -1.348 -0.857 -0.701 -0.515 -0.007 1.001
b[21] -0.620 0.304 -1.163 -0.803 -0.657 -0.439 0.032 1.002
b[22] -0.688 0.281 -1.272 -0.844 -0.702 -0.540 -0.053 1.003
b[23] -0.641 0.282 -1.160 -0.808 -0.665 -0.483 -0.060 1.005
b[24] -0.670 0.272 -1.227 -0.827 -0.675 -0.513 -0.096 1.002
b[25] -0.447 0.330 -0.927 -0.686 -0.515 -0.255 0.365 1.001
b[26] -0.775 0.181 -1.148 -0.888 -0.763 -0.665 -0.430 1.010
b[27] -0.629 0.275 -1.158 -0.793 -0.658 -0.469 -0.043 1.000
b[28] -0.657 0.265 -1.190 -0.816 -0.677 -0.507 -0.087 1.002
b[29] -0.685 0.333 -1.393 -0.842 -0.691 -0.509 -0.017 1.002
b[30] -0.673 0.325 -1.314 -0.845 -0.700 -0.500 0.087 1.001
b[31] -0.672 0.331 -1.332 -0.847 -0.695 -0.496 0.088 1.001
b[32] -0.665 0.326 -1.301 -0.841 -0.693 -0.500 0.074 1.000
b[33] -0.687 0.333 -1.362 -0.866 -0.705 -0.503 0.047 1.003
b[34] -0.706 0.278 -1.265 -0.868 -0.706 -0.555 -0.127 1.001
b[35] -0.639 0.251 -1.130 -0.789 -0.666 -0.494 -0.087 1.002
b[36] -0.520 0.330 -1.046 -0.734 -0.583 -0.339 0.237 1.000
b[37] -0.661 0.281 -1.193 -0.817 -0.682 -0.514 -0.020 1.008
b[38] -0.612 0.288 -1.140 -0.782 -0.649 -0.471 0.058 1.003
b[39] -0.610 0.301 -1.148 -0.793 -0.655 -0.446 0.098 1.002
b[40] -0.680 0.302 -1.319 -0.841 -0.697 -0.517 -0.047 1.006
b[41] -0.564 0.310 -1.129 -0.755 -0.602 -0.390 0.144 1.001
b[42] -0.688 0.330 -1.368 -0.849 -0.700 -0.513 -0.036 1.006
b[43] -1.038 0.319 -1.784 -1.230 -0.979 -0.790 -0.574 1.008
b[44] -0.847 0.308 -1.573 -1.006 -0.789 -0.659 -0.320 1.001
b[45] -0.790 0.249 -1.343 -0.926 -0.769 -0.639 -0.317 1.003
b[46] -0.694 0.314 -1.335 -0.856 -0.708 -0.533 -0.033 1.002
b[47] -0.867 0.321 -1.597 -1.046 -0.810 -0.664 -0.311 1.002
b[48] -0.609 0.297 -1.152 -0.785 -0.633 -0.453 0.068 1.002
b[49] -0.861 0.282 -1.487 -1.012 -0.817 -0.681 -0.365 1.002
b[50] -0.676 0.334 -1.370 -0.849 -0.691 -0.492 0.013 1.004
b[51] -0.673 0.324 -1.293 -0.855 -0.696 -0.505 0.052 1.000
b[52] -0.680 0.329 -1.331 -0.864 -0.703 -0.509 0.064 1.002
b[53] -0.711 0.308 -1.364 -0.861 -0.714 -0.532 -0.115 1.004
b[54] -0.869 0.270 -1.455 -1.030 -0.826 -0.698 -0.400 1.002
b[55] -0.603 0.262 -1.093 -0.765 -0.628 -0.443 -0.030 1.002
b[56] -0.831 0.288 -1.521 -0.993 -0.800 -0.656 -0.330 1.003
b[57] -0.712 0.291 -1.376 -0.870 -0.712 -0.546 -0.128 1.003
b[58] -0.603 0.304 -1.183 -0.784 -0.637 -0.425 0.046 1.001
b[59] -0.758 0.268 -1.345 -0.912 -0.747 -0.614 -0.182 1.003
b[60] -0.673 0.339 -1.369 -0.842 -0.690 -0.505 0.019 1.002
b[61] -0.456 0.297 -0.920 -0.676 -0.498 -0.274 0.210 1.002
b[62] -0.395 0.362 -0.902 -0.663 -0.459 -0.200 0.486 1.000
b[63] -0.550 0.295 -1.023 -0.747 -0.595 -0.389 0.111 1.001
b[64] -0.707 0.287 -1.342 -0.851 -0.714 -0.551 -0.098 1.003
b[65] -0.669 0.325 -1.308 -0.843 -0.685 -0.501 0.061 1.007
b[66] -0.623 0.223 -1.059 -0.763 -0.642 -0.493 -0.135 1.001
b[67] -0.439 0.295 -0.909 -0.660 -0.477 -0.252 0.197 1.002
b[68] -0.672 0.330 -1.380 -0.844 -0.688 -0.491 0.035 1.003
b[69] -0.673 0.334 -1.346 -0.838 -0.694 -0.497 0.054 1.002
b[70] -0.636 0.171 -0.963 -0.754 -0.642 -0.519 -0.303 1.001
b[71] -0.779 0.240 -1.311 -0.920 -0.759 -0.632 -0.337 1.000
b[72] -0.685 0.327 -1.390 -0.854 -0.696 -0.503 0.031 1.005
b[73] -0.673 0.325 -1.299 -0.850 -0.696 -0.498 0.024 1.004
b[74] -0.670 0.349 -1.394 -0.854 -0.685 -0.491 0.084 1.003
b[75] -0.566 0.302 -1.079 -0.763 -0.602 -0.391 0.129 1.000
b[76] -0.672 0.295 -1.304 -0.841 -0.686 -0.506 -0.037 1.000
b[77] -0.806 0.305 -1.497 -0.962 -0.771 -0.628 -0.231 1.005
b[78] -0.750 0.276 -1.394 -0.899 -0.741 -0.594 -0.215 1.003
b[79] -0.756 0.280 -1.379 -0.916 -0.749 -0.598 -0.199 1.000
b[80] -0.823 0.228 -1.328 -0.955 -0.808 -0.674 -0.401 1.001
b[81] -0.476 0.319 -0.980 -0.700 -0.530 -0.289 0.258 1.002
b[82] -0.670 0.324 -1.378 -0.840 -0.694 -0.504 0.068 1.001
b[83] -1.009 0.317 -1.771 -1.202 -0.952 -0.768 -0.529 1.001
b[84] -0.688 0.284 -1.273 -0.842 -0.696 -0.519 -0.111 1.002
b[85] -0.687 0.332 -1.343 -0.863 -0.705 -0.514 0.048 1.004
mu.a 1.461 0.052 1.363 1.426 1.460 1.497 1.563 1.000
mu.b -0.677 0.086 -0.833 -0.740 -0.680 -0.623 -0.506 1.003
sigma.a 0.341 0.048 0.252 0.308 0.340 0.370 0.444 1.000
sigma.b 0.282 0.138 0.038 0.174 0.288 0.383 0.553 1.017
sigma.y 0.749 0.019 0.712 0.736 0.749 0.761 0.786 1.002
deviance 2077.197 17.616 2044.019 2064.986 2076.773 2089.322 2110.707 1.004
n.eff
a[1] 1200
a[2] 1200
a[3] 1200
a[4] 1200
a[5] 1200
a[6] 1200
a[7] 1200
a[8] 1100
a[9] 1200
a[10] 1200
a[11] 430
a[12] 1200
a[13] 1200
a[14] 320
a[15] 1200
a[16] 580
a[17] 1200
a[18] 1200
a[19] 870
a[20] 1200
a[21] 1200
a[22] 810
a[23] 1200
a[24] 710
a[25] 1200
a[26] 1200
a[27] 570
a[28] 660
a[29] 1200
a[30] 870
a[31] 1200
a[32] 1200
a[33] 1200
a[34] 1200
a[35] 270
a[36] 1200
a[37] 1000
a[38] 1200
a[39] 610
a[40] 1200
a[41] 1200
a[42] 460
a[43] 1200
a[44] 1200
a[45] 1200
a[46] 1200
a[47] 1100
a[48] 1200
a[49] 1200
a[50] 390
a[51] 1200
a[52] 1200
a[53] 1200
a[54] 790
a[55] 1200
a[56] 1200
a[57] 590
a[58] 950
a[59] 1200
a[60] 1200
a[61] 1200
a[62] 720
a[63] 450
a[64] 1100
a[65] 1200
a[66] 1200
a[67] 1200
a[68] 1200
a[69] 1200
a[70] 940
a[71] 620
a[72] 1200
a[73] 1200
a[74] 1200
a[75] 1200
a[76] 1200
a[77] 1200
a[78] 1200
a[79] 470
a[80] 1200
a[81] 1200
a[82] 1200
a[83] 1200
a[84] 1200
a[85] 1200
b[1] 820
b[2] 940
b[3] 1100
b[4] 1200
b[5] 1200
b[6] 1000
b[7] 890
b[8] 1100
b[9] 1200
b[10] 1200
b[11] 1200
b[12] 1200
b[13] 1200
b[14] 1200
b[15] 370
b[16] 1200
b[17] 1100
b[18] 630
b[19] 1200
b[20] 1200
b[21] 1000
b[22] 540
b[23] 620
b[24] 1200
b[25] 1200
b[26] 200
b[27] 1200
b[28] 1200
b[29] 1200
b[30] 1200
b[31] 1200
b[32] 1200
b[33] 1200
b[34] 1200
b[35] 850
b[36] 1200
b[37] 370
b[38] 1200
b[39] 1200
b[40] 600
b[41] 1200
b[42] 1200
b[43] 430
b[44] 1200
b[45] 590
b[46] 1200
b[47] 730
b[48] 1200
b[49] 1100
b[50] 470
b[51] 1200
b[52] 1100
b[53] 440
b[54] 930
b[55] 820
b[56] 550
b[57] 590
b[58] 1200
b[59] 680
b[60] 780
b[61] 930
b[62] 1200
b[63] 1200
b[64] 1200
b[65] 290
b[66] 1200
b[67] 900
b[68] 1200
b[69] 860
b[70] 1200
b[71] 1200
b[72] 420
b[73] 460
b[74] 590
b[75] 1200
b[76] 1200
b[77] 1200
b[78] 900
b[79] 1200
b[80] 1200
b[81] 910
b[82] 1200
b[83] 1200
b[84] 1200
b[85] 510
mu.a 1200
mu.b 740
sigma.a 1200
sigma.b 700
sigma.y 1100
deviance 610
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 = 154.9 and DIC = 2232.1
DIC is an estimate of expected predictive error (lower deviance is better).5.1.3 Model with Correlation Between Intercepts and Slopes
# Prepare the data for JAGS
data_jags <- list(y = y, x = x, county = county, n = n, J = J)
# Initial values
inits <- function() {
list(B = array(rnorm(2 * J), dim = c(J, 2)),
mu.a = rnorm(1), mu.b = rnorm(1),
sigma.y = runif(1), sigma.a = runif(1), sigma.b = runif(1), rho = runif(1, -1, 1))
}
# Parameters to monitor
params <- c("a", "b", "mu.a", "mu.b", "sigma.y", "sigma.a", "sigma.b", "rho")
# Run JAGS model
jags_fit <- jags(data = data_jags,
inits = inits,
parameters.to.save = params,
model.file = "codigoJAGS/radon_model_slope2.jags",
n.chains = 3,
n.iter = 5000,
n.burnin = 1000,
n.thin = 10)Compiling model graph
Resolving undeclared variables
Allocating nodes
Graph information:
Observed stochastic nodes: 919
Unobserved stochastic nodes: 91
Total graph size: 3408
Initializing model# Check the summary
print(jags_fit)Inference for Bugs model at "codigoJAGS/radon_model_slope2.jags", fit using jags,
3 chains, each with 5000 iterations (first 1000 discarded), n.thin = 10
n.sims = 1200 iterations saved
mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat
a[1] 1.148 0.275 0.583 0.971 1.156 1.333 1.663 1.007
a[2] 0.931 0.106 0.723 0.854 0.934 0.999 1.129 1.001
a[3] 1.471 0.287 0.917 1.283 1.474 1.652 2.033 1.005
a[4] 1.519 0.243 1.061 1.352 1.530 1.681 2.004 1.002
a[5] 1.438 0.260 0.931 1.258 1.432 1.611 1.991 1.006
a[6] 1.475 0.259 0.959 1.296 1.485 1.644 1.957 1.006
a[7] 1.831 0.177 1.479 1.717 1.830 1.954 2.180 1.005
a[8] 1.697 0.285 1.156 1.504 1.697 1.881 2.260 1.004
a[9] 1.120 0.205 0.734 0.981 1.122 1.257 1.525 1.014
a[10] 1.520 0.258 1.029 1.343 1.521 1.692 2.028 1.006
a[11] 1.431 0.253 0.951 1.263 1.428 1.593 1.929 1.000
a[12] 1.595 0.258 1.111 1.418 1.595 1.773 2.116 1.003
a[13] 1.221 0.232 0.743 1.068 1.230 1.373 1.671 1.001
a[14] 1.869 0.186 1.532 1.741 1.865 1.991 2.249 1.011
a[15] 1.391 0.281 0.853 1.196 1.384 1.576 1.954 1.006
a[16] 1.220 0.315 0.576 1.014 1.216 1.451 1.821 1.004
a[17] 1.417 0.280 0.896 1.228 1.406 1.600 2.001 1.002
a[18] 1.180 0.199 0.773 1.052 1.186 1.314 1.569 1.006
a[19] 1.348 0.092 1.158 1.290 1.352 1.411 1.518 1.001
a[20] 1.600 0.274 1.078 1.410 1.600 1.779 2.153 1.000
a[21] 1.627 0.206 1.224 1.488 1.627 1.758 2.033 1.003
a[22] 0.988 0.245 0.475 0.822 0.991 1.156 1.449 1.006
a[23] 1.429 0.307 0.820 1.235 1.420 1.618 2.027 1.000
a[24] 1.879 0.210 1.479 1.738 1.879 2.014 2.300 1.001
a[25] 1.790 0.184 1.426 1.671 1.791 1.914 2.176 1.000
a[26] 1.367 0.073 1.223 1.317 1.365 1.417 1.508 1.000
a[27] 1.626 0.249 1.168 1.457 1.623 1.792 2.121 1.001
a[28] 1.321 0.275 0.762 1.145 1.324 1.504 1.840 1.007
a[29] 1.298 0.280 0.716 1.126 1.306 1.488 1.820 1.005
a[30] 1.082 0.192 0.709 0.953 1.084 1.208 1.463 1.007
a[31] 1.753 0.254 1.267 1.591 1.750 1.906 2.293 1.001
a[32] 1.361 0.258 0.869 1.190 1.357 1.544 1.851 1.003
a[33] 1.741 0.261 1.254 1.564 1.738 1.919 2.255 1.001
a[34] 1.532 0.289 0.978 1.344 1.523 1.720 2.118 1.005
a[35] 1.032 0.251 0.523 0.855 1.046 1.209 1.478 1.008
a[36] 1.885 0.322 1.273 1.668 1.874 2.091 2.495 1.002
a[37] 0.755 0.220 0.294 0.609 0.761 0.907 1.160 1.007
a[38] 1.612 0.269 1.112 1.423 1.608 1.802 2.136 1.007
a[39] 1.596 0.251 1.124 1.431 1.586 1.765 2.104 1.001
a[40] 1.860 0.272 1.327 1.677 1.851 2.056 2.413 1.003
a[41] 1.766 0.228 1.325 1.617 1.754 1.921 2.215 1.003
a[42] 1.440 0.320 0.857 1.224 1.432 1.634 2.118 1.005
a[43] 1.692 0.268 1.213 1.505 1.667 1.869 2.298 1.046
a[44] 1.237 0.226 0.786 1.089 1.239 1.392 1.675 1.000
a[45] 1.353 0.196 0.971 1.226 1.352 1.481 1.745 1.005
a[46] 1.351 0.241 0.884 1.180 1.353 1.501 1.818 1.000
a[47] 1.338 0.300 0.778 1.141 1.336 1.535 1.941 1.027
a[48] 1.242 0.203 0.834 1.104 1.239 1.378 1.641 1.001
a[49] 1.666 0.186 1.313 1.540 1.665 1.791 2.041 1.004
a[50] 1.652 0.322 1.069 1.433 1.652 1.845 2.343 1.000
a[51] 1.770 0.250 1.297 1.604 1.767 1.934 2.263 1.001
a[52] 1.645 0.281 1.103 1.459 1.642 1.826 2.216 1.000
a[53] 1.390 0.279 0.814 1.215 1.390 1.575 1.942 1.003
a[54] 1.353 0.153 1.049 1.248 1.357 1.457 1.649 1.004
a[55] 1.539 0.227 1.098 1.392 1.542 1.684 1.975 1.005
a[56] 1.353 0.304 0.762 1.152 1.350 1.562 1.934 1.003
a[57] 1.069 0.235 0.595 0.912 1.075 1.223 1.516 1.008
a[58] 1.625 0.261 1.126 1.444 1.627 1.804 2.131 1.000
a[59] 1.588 0.278 1.052 1.402 1.580 1.776 2.122 1.002
a[60] 1.408 0.286 0.820 1.222 1.418 1.587 1.960 1.000
a[61] 1.171 0.127 0.910 1.084 1.171 1.257 1.407 1.025
a[62] 1.669 0.259 1.176 1.498 1.674 1.836 2.175 1.004
a[63] 1.512 0.287 0.959 1.315 1.508 1.704 2.059 1.004
a[64] 1.726 0.192 1.347 1.602 1.725 1.850 2.117 1.001
a[65] 1.408 0.299 0.789 1.210 1.411 1.616 1.980 1.000
a[66] 1.586 0.203 1.196 1.446 1.587 1.721 1.991 1.014
a[67] 1.634 0.200 1.260 1.499 1.632 1.774 2.003 1.025
a[68] 1.229 0.218 0.820 1.081 1.228 1.373 1.653 1.001
a[69] 1.369 0.257 0.827 1.198 1.379 1.538 1.857 1.002
a[70] 0.877 0.071 0.737 0.831 0.876 0.921 1.021 1.012
a[71] 1.501 0.140 1.236 1.410 1.497 1.591 1.788 1.000
a[72] 1.542 0.190 1.170 1.419 1.543 1.668 1.906 1.002
a[73] 1.564 0.299 0.993 1.365 1.563 1.771 2.129 1.007
a[74] 1.246 0.257 0.742 1.072 1.254 1.426 1.710 1.002
a[75] 1.525 0.271 0.999 1.343 1.521 1.707 2.046 1.002
a[76] 1.721 0.268 1.216 1.547 1.717 1.900 2.280 1.000
a[77] 1.702 0.224 1.280 1.545 1.698 1.860 2.120 1.005
a[78] 1.389 0.259 0.866 1.219 1.391 1.563 1.913 1.002
a[79] 1.086 0.266 0.540 0.913 1.089 1.272 1.581 1.001
a[80] 1.354 0.106 1.151 1.280 1.354 1.427 1.566 1.001
a[81] 1.875 0.316 1.291 1.653 1.868 2.089 2.496 1.002
a[82] 1.615 0.347 0.945 1.396 1.620 1.828 2.293 1.002
a[83] 1.652 0.198 1.299 1.516 1.646 1.780 2.065 1.013
a[84] 1.603 0.182 1.248 1.479 1.604 1.722 1.974 1.002
a[85] 1.396 0.293 0.802 1.202 1.410 1.596 1.927 1.006
b[1] -0.594 0.295 -1.070 -0.779 -0.645 -0.482 0.161 1.063
b[2] -0.743 0.243 -1.297 -0.863 -0.713 -0.608 -0.284 1.022
b[3] -0.683 0.271 -1.311 -0.812 -0.685 -0.572 -0.091 1.046
b[4] -0.725 0.242 -1.244 -0.849 -0.708 -0.612 -0.246 1.025
b[5] -0.654 0.279 -1.217 -0.796 -0.660 -0.536 -0.043 1.036
b[6] -0.688 0.278 -1.223 -0.820 -0.692 -0.582 -0.048 1.022
b[7] -0.559 0.294 -1.017 -0.760 -0.647 -0.397 0.164 1.051
b[8] -0.713 0.273 -1.321 -0.847 -0.705 -0.598 -0.108 1.048
b[9] -0.512 0.319 -0.965 -0.726 -0.614 -0.344 0.254 1.075
b[10] -0.743 0.255 -1.365 -0.846 -0.715 -0.621 -0.268 1.042
b[11] -0.678 0.298 -1.301 -0.818 -0.684 -0.581 0.021 1.028
b[12] -0.713 0.286 -1.306 -0.848 -0.702 -0.595 -0.106 1.021
b[13] -0.654 0.295 -1.282 -0.799 -0.659 -0.530 0.004 1.032
b[14] -0.800 0.237 -1.349 -0.917 -0.768 -0.652 -0.370 1.036
b[15] -0.676 0.262 -1.192 -0.816 -0.684 -0.574 -0.076 1.041
b[16] -0.645 0.310 -1.236 -0.803 -0.668 -0.511 0.094 1.037
b[17] -0.792 0.264 -1.425 -0.902 -0.752 -0.644 -0.325 1.028
b[18] -0.557 0.271 -0.987 -0.738 -0.622 -0.414 0.065 1.069
b[19] -0.748 0.207 -1.221 -0.849 -0.731 -0.638 -0.327 1.021
b[20] -0.716 0.302 -1.329 -0.847 -0.707 -0.598 -0.071 1.031
b[21] -0.676 0.282 -1.227 -0.829 -0.683 -0.561 -0.035 1.025
b[22] -0.619 0.279 -1.135 -0.793 -0.645 -0.493 0.034 1.052
b[23] -0.674 0.274 -1.275 -0.814 -0.676 -0.552 -0.067 1.024
b[24] -0.745 0.263 -1.320 -0.859 -0.724 -0.622 -0.214 1.038
b[25] -0.561 0.306 -1.025 -0.756 -0.649 -0.400 0.166 1.058
b[26] -0.755 0.161 -1.130 -0.848 -0.744 -0.649 -0.468 1.027
b[27] -0.682 0.262 -1.219 -0.818 -0.684 -0.571 -0.105 1.026
b[28] -0.660 0.254 -1.201 -0.800 -0.662 -0.553 -0.097 1.036
b[29] -0.664 0.278 -1.248 -0.810 -0.664 -0.543 -0.015 1.039
b[30] -0.623 0.301 -1.150 -0.796 -0.651 -0.489 0.094 1.028
b[31] -0.750 0.301 -1.459 -0.868 -0.729 -0.630 -0.133 1.029
b[32] -0.675 0.296 -1.328 -0.818 -0.672 -0.549 -0.020 1.031
b[33] -0.746 0.321 -1.437 -0.885 -0.726 -0.624 -0.049 1.047
b[34] -0.732 0.268 -1.297 -0.852 -0.715 -0.611 -0.152 1.022
b[35] -0.600 0.256 -1.024 -0.769 -0.642 -0.483 0.007 1.038
b[36] -0.629 0.309 -1.249 -0.793 -0.659 -0.493 0.068 1.041
b[37] -0.555 0.316 -1.103 -0.749 -0.621 -0.387 0.221 1.054
b[38] -0.655 0.272 -1.184 -0.802 -0.673 -0.546 0.005 1.059
b[39] -0.659 0.260 -1.179 -0.803 -0.670 -0.545 -0.057 1.039
b[40] -0.753 0.291 -1.414 -0.871 -0.727 -0.618 -0.161 1.030
b[41] -0.648 0.271 -1.137 -0.812 -0.667 -0.530 -0.035 1.036
b[42] -0.690 0.282 -1.214 -0.826 -0.686 -0.571 -0.046 1.035
b[43] -1.017 0.361 -1.858 -1.239 -0.889 -0.722 -0.589 1.111
b[44] -0.793 0.291 -1.513 -0.917 -0.748 -0.633 -0.307 1.032
b[45] -0.758 0.221 -1.285 -0.861 -0.738 -0.634 -0.376 1.040
b[46] -0.661 0.299 -1.267 -0.817 -0.674 -0.539 0.040 1.029
b[47] -0.816 0.307 -1.580 -0.934 -0.772 -0.644 -0.267 1.031
b[48] -0.593 0.277 -1.082 -0.764 -0.643 -0.464 0.061 1.043
b[49] -0.864 0.286 -1.565 -1.008 -0.803 -0.662 -0.454 1.036
b[50] -0.744 0.316 -1.445 -0.866 -0.719 -0.606 -0.138 1.025
b[51] -0.757 0.293 -1.443 -0.874 -0.737 -0.628 -0.149 1.027
b[52] -0.738 0.299 -1.355 -0.862 -0.717 -0.616 -0.089 1.021
b[53] -0.695 0.277 -1.281 -0.830 -0.692 -0.575 -0.081 1.034
b[54] -0.832 0.251 -1.418 -0.960 -0.787 -0.653 -0.419 1.060
b[55] -0.648 0.244 -1.094 -0.791 -0.661 -0.540 -0.087 1.029
b[56] -0.794 0.265 -1.446 -0.911 -0.758 -0.641 -0.329 1.033
b[57] -0.655 0.282 -1.203 -0.800 -0.660 -0.537 -0.018 1.037
b[58] -0.654 0.264 -1.128 -0.804 -0.672 -0.541 -0.028 1.048
b[59] -0.761 0.269 -1.364 -0.876 -0.738 -0.632 -0.240 1.024
b[60] -0.692 0.290 -1.314 -0.835 -0.686 -0.569 -0.082 1.024
b[61] -0.478 0.291 -0.863 -0.672 -0.557 -0.313 0.234 1.135
b[62] -0.499 0.342 -0.981 -0.723 -0.618 -0.322 0.351 1.065
b[63] -0.589 0.296 -1.064 -0.773 -0.648 -0.438 0.117 1.057
b[64] -0.757 0.268 -1.332 -0.892 -0.736 -0.633 -0.190 1.038
b[65] -0.684 0.306 -1.260 -0.823 -0.683 -0.569 0.037 1.014
b[66] -0.657 0.222 -1.063 -0.796 -0.667 -0.544 -0.162 1.037
b[67] -0.508 0.292 -0.894 -0.712 -0.595 -0.330 0.183 1.113
b[68] -0.646 0.310 -1.259 -0.809 -0.658 -0.527 0.081 1.033
b[69] -0.668 0.291 -1.256 -0.817 -0.673 -0.555 0.004 1.019
b[70] -0.618 0.161 -0.892 -0.733 -0.634 -0.515 -0.283 1.074
b[71] -0.776 0.219 -1.274 -0.882 -0.753 -0.648 -0.338 1.029
b[72] -0.711 0.292 -1.354 -0.845 -0.706 -0.596 -0.040 1.040
b[73] -0.706 0.318 -1.386 -0.844 -0.702 -0.589 -0.048 1.014
b[74] -0.648 0.291 -1.254 -0.802 -0.664 -0.529 0.006 1.055
b[75] -0.605 0.292 -1.118 -0.782 -0.651 -0.480 0.110 1.037
b[76] -0.735 0.284 -1.395 -0.852 -0.708 -0.615 -0.142 1.035
b[77] -0.832 0.295 -1.543 -0.974 -0.775 -0.652 -0.352 1.057
b[78] -0.746 0.255 -1.328 -0.859 -0.723 -0.619 -0.223 1.017
b[79] -0.678 0.268 -1.218 -0.817 -0.669 -0.554 -0.060 1.025
b[80] -0.778 0.204 -1.249 -0.873 -0.757 -0.649 -0.404 1.019
b[81] -0.588 0.306 -1.096 -0.774 -0.650 -0.447 0.188 1.061
b[82] -0.734 0.304 -1.431 -0.858 -0.713 -0.610 -0.100 1.041
b[83] -0.982 0.328 -1.738 -1.196 -0.880 -0.719 -0.578 1.085
b[84] -0.709 0.266 -1.271 -0.843 -0.702 -0.582 -0.145 1.036
b[85] -0.681 0.286 -1.321 -0.816 -0.682 -0.555 -0.065 1.042
mu.a 1.463 0.055 1.357 1.424 1.462 1.500 1.570 1.004
mu.b -0.692 0.086 -0.850 -0.753 -0.689 -0.639 -0.521 1.033
rho -0.296 0.380 -0.947 -0.555 -0.320 -0.083 0.629 1.028
sigma.a 0.351 0.052 0.261 0.312 0.351 0.386 0.460 1.018
sigma.b 0.236 0.173 0.005 0.042 0.244 0.371 0.564 1.661
sigma.y 0.751 0.019 0.716 0.739 0.750 0.763 0.790 1.004
deviance 2080.543 18.128 2043.587 2068.567 2081.551 2092.926 2114.104 1.108
n.eff
a[1] 430
a[2] 1200
a[3] 730
a[4] 1200
a[5] 670
a[6] 400
a[7] 410
a[8] 1200
a[9] 180
a[10] 1200
a[11] 1200
a[12] 670
a[13] 1200
a[14] 180
a[15] 460
a[16] 1000
a[17] 790
a[18] 320
a[19] 1200
a[20] 1200
a[21] 640
a[22] 520
a[23] 1200
a[24] 1200
a[25] 1200
a[26] 1200
a[27] 1200
a[28] 450
a[29] 410
a[30] 730
a[31] 1200
a[32] 810
a[33] 1200
a[34] 710
a[35] 250
a[36] 1200
a[37] 360
a[38] 270
a[39] 1200
a[40] 690
a[41] 590
a[42] 460
a[43] 47
a[44] 1200
a[45] 380
a[46] 1200
a[47] 1200
a[48] 1200
a[49] 500
a[50] 1200
a[51] 1200
a[52] 1200
a[53] 610
a[54] 420
a[55] 410
a[56] 680
a[57] 400
a[58] 1200
a[59] 740
a[60] 1200
a[61] 82
a[62] 430
a[63] 530
a[64] 1200
a[65] 1200
a[66] 150
a[67] 83
a[68] 1200
a[69] 1000
a[70] 170
a[71] 1200
a[72] 1200
a[73] 1200
a[74] 1200
a[75] 1100
a[76] 1200
a[77] 760
a[78] 1200
a[79] 1200
a[80] 1200
a[81] 1000
a[82] 900
a[83] 170
a[84] 1100
a[85] 730
b[1] 65
b[2] 990
b[3] 320
b[4] 520
b[5] 250
b[6] 650
b[7] 51
b[8] 350
b[9] 33
b[10] 900
b[11] 580
b[12] 1100
b[13] 440
b[14] 140
b[15] 250
b[16] 150
b[17] 250
b[18] 36
b[19] 600
b[20] 920
b[21] 270
b[22] 71
b[23] 480
b[24] 1200
b[25] 46
b[26] 250
b[27] 680
b[28] 180
b[29] 120
b[30] 180
b[31] 630
b[32] 510
b[33] 740
b[34] 1200
b[35] 65
b[36] 170
b[37] 50
b[38] 78
b[39] 140
b[40] 450
b[41] 150
b[42] 280
b[43] 23
b[44] 230
b[45] 290
b[46] 260
b[47] 190
b[48] 76
b[49] 84
b[50] 1200
b[51] 1200
b[52] 1200
b[53] 420
b[54] 62
b[55] 140
b[56] 150
b[57] 270
b[58] 110
b[59] 290
b[60] 970
b[61] 19
b[62] 39
b[63] 73
b[64] 970
b[65] 780
b[66] 160
b[67] 23
b[68] 200
b[69] 620
b[70] 32
b[71] 220
b[72] 710
b[73] 780
b[74] 130
b[75] 94
b[76] 450
b[77] 91
b[78] 480
b[79] 400
b[80] 430
b[81] 58
b[82] 1200
b[83] 29
b[84] 1200
b[85] 210
mu.a 450
mu.b 66
rho 770
sigma.a 110
sigma.b 7
sigma.y 430
deviance 23
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 = 149.6 and DIC = 2230.1
DIC is an estimate of expected predictive error (lower deviance is better).jags_result <- jags_fit$BUGSoutput$summary5.1.4 Model with County-Level Uranium Predictor
We can add group-level predictors to the varying-intercept, varying-slope Bugs models of the previous section by replacing mu.a and mu.b by group-level regressions. Simplest varying-intercept, varying-slope model. For example, we can add a group- level predictor u to the very first model of this chapter by replacing the expressions for a.hat[j] and b.hat[j] with:
- a.hat[j] <- g.a.0 + g.a.1*u[j]
- b.hat[j] <- g.b.0 + g.b.1*u[j] and then removing the prior distributions for mu.a and mu.b and replacing with dnorm (0, .0001) prior distributions for each of g.a.0, g.a.1, g.b.0, and g.b.1.
# Prepare the data for JAGS
data_jags <- list(y = y, x = x, county = county, n = n, J = J, u = u.full)
# Initial values
inits <- function() {
list(B = array(rnorm(2 * J), dim = c(J, 2)),
g.a.0 = rnorm(1), g.a.1 = rnorm(1),
g.b.0 = rnorm(1), g.b.1 = rnorm(1),
sigma.y = runif(1), sigma.a = runif(1), sigma.b = runif(1), rho = runif(1, -1, 1))
}
# Parameters to monitor
params <- c("a", "b", "g.a.0", "g.a.1", "g.b.0", "g.b.1", "sigma.y", "sigma.a", "sigma.b", "rho")
# Run JAGS model
jags_fit <- jags(data = data_jags,
inits = inits,
parameters.to.save = params,
model.file = "codigoJAGS/radon_model_slope3.jags",
n.chains = 3,
n.iter = 5000,
n.burnin = 1000,
n.thin = 10)Compiling model graph
Resolving undeclared variables
Allocating nodes
Graph information:
Observed stochastic nodes: 919
Unobserved stochastic nodes: 93
Total graph size: 4357
Initializing model# Check the summary
print(jags_fit)Inference for Bugs model at "codigoJAGS/radon_model_slope3.jags", fit using jags,
3 chains, each with 5000 iterations (first 1000 discarded), n.thin = 10
n.sims = 1200 iterations saved
mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat
a[1] 1.142 0.275 0.583 0.955 1.148 1.334 1.666 1.012
a[2] 0.930 0.099 0.725 0.870 0.930 0.995 1.127 1.001
a[3] 1.466 0.309 0.835 1.257 1.465 1.677 2.070 1.001
a[4] 1.511 0.253 1.030 1.340 1.515 1.670 2.022 1.001
a[5] 1.427 0.267 0.900 1.245 1.435 1.602 1.967 1.005
a[6] 1.461 0.282 0.892 1.284 1.471 1.641 2.018 1.013
a[7] 1.825 0.180 1.457 1.706 1.824 1.942 2.162 1.000
a[8] 1.677 0.275 1.135 1.494 1.682 1.855 2.229 1.002
a[9] 1.105 0.206 0.691 0.966 1.107 1.248 1.484 1.008
a[10] 1.529 0.247 1.045 1.361 1.524 1.695 2.004 1.004
a[11] 1.423 0.243 0.937 1.250 1.420 1.590 1.889 1.004
a[12] 1.574 0.264 1.035 1.398 1.570 1.744 2.085 1.002
a[13] 1.204 0.235 0.752 1.049 1.202 1.359 1.663 1.001
a[14] 1.859 0.196 1.491 1.728 1.856 1.986 2.244 1.005
a[15] 1.360 0.273 0.839 1.182 1.352 1.544 1.892 1.001
a[16] 1.206 0.311 0.580 1.005 1.217 1.420 1.788 1.003
a[17] 1.400 0.290 0.854 1.203 1.398 1.580 1.981 1.001
a[18] 1.159 0.194 0.793 1.027 1.165 1.288 1.531 1.004
a[19] 1.347 0.095 1.156 1.287 1.345 1.409 1.532 1.000
a[20] 1.591 0.280 1.017 1.408 1.596 1.779 2.145 1.007
a[21] 1.627 0.207 1.239 1.479 1.631 1.771 2.024 1.003
a[22] 0.977 0.239 0.514 0.813 0.989 1.144 1.431 1.001
a[23] 1.412 0.305 0.826 1.195 1.420 1.628 1.986 1.000
a[24] 1.873 0.217 1.449 1.730 1.868 2.015 2.302 1.004
a[25] 1.780 0.184 1.432 1.650 1.782 1.914 2.117 1.000
a[26] 1.363 0.073 1.227 1.311 1.366 1.412 1.502 1.006
a[27] 1.606 0.245 1.136 1.443 1.600 1.772 2.108 1.009
a[28] 1.304 0.268 0.766 1.124 1.314 1.489 1.813 1.009
a[29] 1.288 0.288 0.710 1.096 1.282 1.475 1.856 1.005
a[30] 1.092 0.189 0.723 0.968 1.091 1.221 1.452 1.004
a[31] 1.745 0.253 1.244 1.578 1.746 1.920 2.236 1.002
a[32] 1.339 0.262 0.799 1.165 1.342 1.514 1.855 1.001
a[33] 1.744 0.262 1.271 1.564 1.731 1.915 2.271 1.000
a[34] 1.503 0.295 0.902 1.299 1.496 1.706 2.075 1.002
a[35] 1.008 0.266 0.451 0.837 1.020 1.192 1.466 1.007
a[36] 1.858 0.317 1.222 1.642 1.852 2.070 2.488 1.005
a[37] 0.746 0.224 0.313 0.592 0.755 0.890 1.162 1.001
a[38] 1.599 0.286 1.046 1.410 1.598 1.793 2.188 1.006
a[39] 1.581 0.248 1.077 1.422 1.577 1.748 2.074 1.001
a[40] 1.849 0.280 1.312 1.661 1.844 2.033 2.417 1.003
a[41] 1.750 0.215 1.346 1.608 1.754 1.889 2.186 1.003
a[42] 1.438 0.333 0.778 1.224 1.435 1.663 2.102 1.002
a[43] 1.696 0.264 1.226 1.502 1.684 1.870 2.264 1.044
a[44] 1.218 0.234 0.748 1.069 1.218 1.369 1.681 1.000
a[45] 1.353 0.194 0.949 1.231 1.347 1.481 1.742 1.005
a[46] 1.319 0.241 0.832 1.166 1.326 1.482 1.806 1.002
a[47] 1.309 0.312 0.694 1.103 1.306 1.515 1.952 1.003
a[48] 1.235 0.216 0.806 1.090 1.241 1.391 1.653 1.001
a[49] 1.669 0.193 1.282 1.542 1.663 1.791 2.063 1.005
a[50] 1.656 0.328 0.984 1.450 1.658 1.867 2.290 1.005
a[51] 1.785 0.264 1.287 1.613 1.772 1.962 2.304 1.006
a[52] 1.648 0.272 1.126 1.475 1.645 1.823 2.206 1.001
a[53] 1.377 0.287 0.837 1.181 1.379 1.566 1.948 1.002
a[54] 1.354 0.150 1.061 1.250 1.355 1.454 1.653 1.005
a[55] 1.519 0.223 1.054 1.372 1.516 1.663 1.963 1.002
a[56] 1.362 0.310 0.765 1.155 1.358 1.570 1.963 1.006
a[57] 1.082 0.254 0.552 0.917 1.088 1.242 1.571 1.014
a[58] 1.615 0.273 1.076 1.425 1.625 1.807 2.128 1.004
a[59] 1.620 0.283 1.065 1.425 1.612 1.812 2.185 1.001
a[60] 1.402 0.287 0.841 1.210 1.403 1.585 1.988 1.001
a[61] 1.162 0.129 0.913 1.072 1.159 1.250 1.422 1.006
a[62] 1.668 0.261 1.142 1.490 1.666 1.840 2.173 1.005
a[63] 1.496 0.290 0.883 1.299 1.494 1.698 2.046 1.002
a[64] 1.744 0.191 1.381 1.605 1.738 1.873 2.128 1.004
a[65] 1.400 0.292 0.788 1.221 1.394 1.591 1.964 1.005
a[66] 1.571 0.209 1.164 1.432 1.569 1.724 1.981 1.001
a[67] 1.649 0.199 1.248 1.516 1.654 1.782 2.046 1.004
a[68] 1.234 0.215 0.838 1.089 1.228 1.387 1.660 1.002
a[69] 1.355 0.257 0.804 1.202 1.358 1.523 1.831 1.003
a[70] 0.881 0.069 0.744 0.835 0.879 0.927 1.021 1.007
a[71] 1.511 0.145 1.226 1.409 1.513 1.613 1.788 1.003
a[72] 1.552 0.205 1.147 1.414 1.547 1.683 1.967 1.003
a[73] 1.590 0.309 1.035 1.382 1.583 1.777 2.203 1.003
a[74] 1.253 0.265 0.715 1.072 1.262 1.442 1.767 1.012
a[75] 1.553 0.294 0.960 1.356 1.554 1.743 2.130 1.001
a[76] 1.742 0.269 1.223 1.565 1.740 1.922 2.268 1.000
a[77] 1.716 0.234 1.273 1.557 1.706 1.873 2.166 1.005
a[78] 1.421 0.269 0.903 1.225 1.427 1.604 1.950 1.001
a[79] 1.108 0.273 0.559 0.921 1.115 1.295 1.633 1.031
a[80] 1.364 0.107 1.162 1.296 1.366 1.434 1.574 1.001
a[81] 1.910 0.312 1.327 1.704 1.899 2.122 2.549 1.001
a[82] 1.630 0.342 1.009 1.398 1.635 1.841 2.362 1.006
a[83] 1.668 0.204 1.260 1.534 1.671 1.806 2.062 1.009
a[84] 1.613 0.191 1.218 1.493 1.612 1.742 1.990 1.002
a[85] 1.386 0.304 0.757 1.196 1.386 1.585 1.979 1.003
b[1] -0.560 0.291 -1.103 -0.731 -0.606 -0.407 0.078 1.023
b[2] -0.723 0.252 -1.287 -0.870 -0.691 -0.573 -0.264 1.029
b[3] -0.650 0.270 -1.238 -0.779 -0.666 -0.526 -0.029 1.010
b[4] -0.707 0.250 -1.280 -0.824 -0.691 -0.563 -0.251 1.015
b[5] -0.599 0.276 -1.172 -0.757 -0.614 -0.460 0.014 1.019
b[6] -0.638 0.316 -1.275 -0.782 -0.638 -0.494 0.070 1.028
b[7] -0.503 0.290 -0.945 -0.705 -0.561 -0.346 0.214 1.022
b[8] -0.641 0.280 -1.213 -0.781 -0.651 -0.501 -0.063 1.016
b[9] -0.417 0.336 -0.912 -0.648 -0.505 -0.221 0.404 1.044
b[10] -0.717 0.254 -1.296 -0.844 -0.697 -0.571 -0.238 1.028
b[11] -0.630 0.300 -1.262 -0.772 -0.640 -0.495 0.014 1.012
b[12] -0.666 0.305 -1.304 -0.803 -0.668 -0.527 0.052 1.017
b[13] -0.581 0.316 -1.159 -0.751 -0.615 -0.441 0.184 1.014
b[14] -0.764 0.262 -1.368 -0.908 -0.727 -0.585 -0.294 1.022
b[15] -0.622 0.275 -1.156 -0.772 -0.629 -0.491 0.002 1.011
b[16] -0.583 0.320 -1.162 -0.750 -0.616 -0.429 0.135 1.015
b[17] -0.751 0.281 -1.413 -0.884 -0.709 -0.581 -0.253 1.020
b[18] -0.505 0.263 -0.919 -0.680 -0.560 -0.373 0.173 1.015
b[19] -0.716 0.227 -1.247 -0.837 -0.697 -0.571 -0.303 1.012
b[20] -0.668 0.307 -1.360 -0.804 -0.664 -0.516 -0.018 1.021
b[21] -0.627 0.278 -1.182 -0.777 -0.630 -0.497 0.037 1.011
b[22] -0.562 0.288 -1.093 -0.734 -0.595 -0.419 0.075 1.028
b[23] -0.613 0.287 -1.174 -0.759 -0.627 -0.489 0.022 1.018
b[24] -0.693 0.280 -1.308 -0.848 -0.678 -0.535 -0.131 1.021
b[25] -0.499 0.294 -0.946 -0.702 -0.554 -0.346 0.208 1.033
b[26] -0.725 0.171 -1.135 -0.813 -0.708 -0.606 -0.449 1.030
b[27] -0.630 0.258 -1.155 -0.766 -0.630 -0.505 -0.058 1.030
b[28] -0.617 0.251 -1.098 -0.755 -0.634 -0.486 -0.103 1.026
b[29] -0.608 0.293 -1.214 -0.755 -0.625 -0.482 0.045 1.020
b[30] -0.568 0.313 -1.186 -0.735 -0.598 -0.423 0.149 1.024
b[31] -0.676 0.305 -1.344 -0.824 -0.668 -0.524 -0.081 1.021
b[32] -0.616 0.310 -1.247 -0.763 -0.622 -0.467 0.049 1.020
b[33] -0.694 0.313 -1.364 -0.846 -0.686 -0.531 -0.091 1.019
b[34] -0.693 0.275 -1.298 -0.823 -0.677 -0.544 -0.139 1.021
b[35] -0.547 0.263 -1.003 -0.714 -0.586 -0.419 0.061 1.020
b[36] -0.574 0.313 -1.171 -0.757 -0.601 -0.427 0.129 1.015
b[37] -0.511 0.327 -1.069 -0.717 -0.573 -0.344 0.268 1.028
b[38] -0.605 0.289 -1.178 -0.764 -0.616 -0.472 0.061 1.009
b[39] -0.607 0.291 -1.185 -0.772 -0.629 -0.465 0.062 1.011
b[40] -0.683 0.306 -1.341 -0.828 -0.676 -0.525 -0.033 1.041
b[41] -0.607 0.291 -1.202 -0.775 -0.627 -0.479 0.057 1.023
b[42] -0.643 0.315 -1.252 -0.782 -0.642 -0.498 0.043 1.015
b[43] -0.999 0.365 -1.860 -1.241 -0.928 -0.709 -0.513 1.093
b[44] -0.739 0.286 -1.421 -0.888 -0.703 -0.570 -0.237 1.017
b[45] -0.739 0.246 -1.284 -0.866 -0.712 -0.578 -0.319 1.033
b[46] -0.615 0.301 -1.235 -0.760 -0.627 -0.484 0.067 1.016
b[47] -0.767 0.309 -1.499 -0.903 -0.725 -0.582 -0.261 1.045
b[48] -0.544 0.292 -1.068 -0.714 -0.581 -0.403 0.155 1.009
b[49] -0.820 0.306 -1.491 -0.997 -0.767 -0.620 -0.298 1.033
b[50] -0.688 0.329 -1.419 -0.835 -0.675 -0.527 0.017 1.015
b[51] -0.692 0.333 -1.404 -0.863 -0.683 -0.534 0.024 1.018
b[52] -0.670 0.313 -1.311 -0.818 -0.663 -0.533 0.025 1.022
b[53] -0.657 0.295 -1.306 -0.786 -0.656 -0.514 -0.019 1.031
b[54] -0.794 0.270 -1.417 -0.933 -0.749 -0.610 -0.361 1.037
b[55] -0.578 0.256 -1.054 -0.732 -0.604 -0.447 0.024 1.009
b[56] -0.776 0.311 -1.468 -0.946 -0.725 -0.576 -0.267 1.032
b[57] -0.684 0.283 -1.244 -0.836 -0.679 -0.562 -0.034 1.019
b[58] -0.688 0.272 -1.245 -0.840 -0.691 -0.552 -0.115 1.008
b[59] -0.802 0.293 -1.484 -0.944 -0.757 -0.644 -0.270 1.023
b[60] -0.651 0.286 -1.224 -0.785 -0.659 -0.528 -0.010 1.020
b[61] -0.457 0.290 -0.903 -0.668 -0.536 -0.271 0.190 1.052
b[62] -0.481 0.338 -0.988 -0.704 -0.570 -0.323 0.423 1.032
b[63] -0.561 0.296 -1.117 -0.740 -0.608 -0.420 0.156 1.023
b[64] -0.735 0.268 -1.370 -0.867 -0.714 -0.583 -0.245 1.018
b[65] -0.673 0.298 -1.361 -0.807 -0.673 -0.542 -0.023 1.034
b[66] -0.639 0.218 -1.096 -0.751 -0.647 -0.531 -0.165 1.016
b[67] -0.523 0.288 -0.972 -0.713 -0.595 -0.347 0.147 1.040
b[68] -0.671 0.303 -1.272 -0.824 -0.683 -0.552 0.035 1.007
b[69] -0.712 0.301 -1.336 -0.852 -0.697 -0.581 -0.047 1.019
b[70] -0.623 0.162 -0.936 -0.719 -0.643 -0.524 -0.265 1.033
b[71] -0.836 0.251 -1.364 -0.996 -0.809 -0.658 -0.372 1.015
b[72] -0.800 0.338 -1.580 -0.974 -0.775 -0.622 -0.104 1.023
b[73] -0.809 0.344 -1.579 -1.006 -0.777 -0.621 -0.134 1.020
b[74] -0.723 0.334 -1.447 -0.897 -0.716 -0.569 0.016 1.016
b[75] -0.660 0.321 -1.227 -0.846 -0.679 -0.516 0.147 1.018
b[76] -0.807 0.313 -1.499 -0.984 -0.771 -0.624 -0.183 1.029
b[77] -0.918 0.325 -1.681 -1.092 -0.853 -0.687 -0.418 1.035
b[78] -0.808 0.293 -1.417 -0.983 -0.769 -0.634 -0.259 1.017
b[79] -0.757 0.317 -1.440 -0.930 -0.731 -0.593 -0.129 1.021
b[80] -0.843 0.227 -1.323 -0.990 -0.810 -0.676 -0.483 1.013
b[81] -0.646 0.327 -1.246 -0.830 -0.670 -0.505 0.090 1.018
b[82] -0.805 0.346 -1.561 -0.987 -0.769 -0.621 -0.104 1.022
b[83] -1.055 0.366 -1.867 -1.297 -0.992 -0.756 -0.554 1.077
b[84] -0.803 0.295 -1.440 -0.958 -0.768 -0.633 -0.216 1.008
b[85] -0.754 0.332 -1.464 -0.930 -0.741 -0.597 -0.050 1.008
g.a.0 1.482 0.084 1.318 1.425 1.484 1.534 1.645 1.005
g.a.1 0.036 0.122 -0.203 -0.042 0.037 0.116 0.265 1.003
g.b.0 -0.744 0.130 -0.998 -0.828 -0.736 -0.658 -0.491 1.010
g.b.1 -0.126 0.189 -0.515 -0.252 -0.120 -0.003 0.237 1.008
rho -0.215 0.402 -0.851 -0.504 -0.281 0.016 0.715 1.062
sigma.a 0.356 0.053 0.263 0.320 0.354 0.391 0.463 1.030
sigma.b 0.251 0.163 0.013 0.089 0.263 0.372 0.555 1.465
sigma.y 0.751 0.020 0.714 0.738 0.750 0.764 0.792 1.018
deviance 2080.243 18.666 2042.978 2067.366 2081.228 2094.058 2113.524 1.065
n.eff
a[1] 340
a[2] 1200
a[3] 1200
a[4] 1200
a[5] 1200
a[6] 250
a[7] 1200
a[8] 900
a[9] 320
a[10] 1200
a[11] 710
a[12] 740
a[13] 1200
a[14] 350
a[15] 1200
a[16] 1200
a[17] 1200
a[18] 480
a[19] 1200
a[20] 300
a[21] 1000
a[22] 1200
a[23] 1200
a[24] 440
a[25] 1200
a[26] 330
a[27] 770
a[28] 350
a[29] 700
a[30] 620
a[31] 1000
a[32] 1200
a[33] 1200
a[34] 1200
a[35] 340
a[36] 1200
a[37] 1200
a[38] 1200
a[39] 1200
a[40] 590
a[41] 640
a[42] 880
a[43] 50
a[44] 1200
a[45] 420
a[46] 1000
a[47] 750
a[48] 1200
a[49] 500
a[50] 550
a[51] 320
a[52] 1200
a[53] 1200
a[54] 360
a[55] 1200
a[56] 420
a[57] 730
a[58] 690
a[59] 1200
a[60] 1200
a[61] 900
a[62] 550
a[63] 750
a[64] 450
a[65] 440
a[66] 1200
a[67] 440
a[68] 1200
a[69] 1200
a[70] 290
a[71] 830
a[72] 1200
a[73] 1000
a[74] 260
a[75] 1200
a[76] 1200
a[77] 410
a[78] 1200
a[79] 1200
a[80] 1200
a[81] 1200
a[82] 360
a[83] 220
a[84] 880
a[85] 1200
b[1] 170
b[2] 170
b[3] 610
b[4] 620
b[5] 1200
b[6] 520
b[7] 170
b[8] 620
b[9] 59
b[10] 130
b[11] 1200
b[12] 1000
b[13] 440
b[14] 140
b[15] 1200
b[16] 530
b[17] 160
b[18] 140
b[19] 270
b[20] 240
b[21] 450
b[22] 160
b[23] 540
b[24] 190
b[25] 92
b[26] 84
b[27] 1200
b[28] 1000
b[29] 1200
b[30] 750
b[31] 1200
b[32] 1200
b[33] 1200
b[34] 1200
b[35] 200
b[36] 1200
b[37] 140
b[38] 1100
b[39] 1200
b[40] 200
b[41] 1200
b[42] 820
b[43] 26
b[44] 210
b[45] 100
b[46] 1200
b[47] 81
b[48] 610
b[49] 72
b[50] 1200
b[51] 370
b[52] 1200
b[53] 290
b[54] 66
b[55] 840
b[56] 97
b[57] 480
b[58] 1200
b[59] 150
b[60] 970
b[61] 43
b[62] 110
b[63] 230
b[64] 320
b[65] 440
b[66] 940
b[67] 61
b[68] 1200
b[69] 770
b[70] 74
b[71] 210
b[72] 1200
b[73] 250
b[74] 1200
b[75] 290
b[76] 190
b[77] 69
b[78] 410
b[79] 350
b[80] 220
b[81] 1100
b[82] 520
b[83] 31
b[84] 330
b[85] 1200
g.a.0 960
g.a.1 1200
g.b.0 280
g.b.1 500
rho 78
sigma.a 69
sigma.b 9
sigma.y 120
deviance 36
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 = 164.4 and DIC = 2244.6
DIC is an estimate of expected predictive error (lower deviance is better).5.2 Modelos no-anidados
This experiment includes n = 40 data points with J = 5 treatment conditions and K = 8 airports. The response variable is modeled using a non-nested multilevel structure:
5.2.1 Model Specification
The model is defined as follows:
\[ y_i \sim N(\mu + \gamma_{j[i]} + \delta_{k[i]}, \sigma_y^2), \quad \text{for } i = 1, \dots, n \]
\[ \gamma_j \sim N(0, \sigma_\gamma^2), \quad \text{for } j = 1, \dots, J \]
\[ \delta_k \sim N(0, \sigma_\delta^2), \quad \text{for } k = 1, \dots, K \]
- \(\mu\): Overall intercept.
- \(\gamma_j\): Treatment effect for each condition, centered at zero to avoid redundancy with \(\mu\).
- \(\delta_k\): Airport effect, also centered at zero.
This model incorporates treatment and airport effects independently to capture their individual contributions.
Carga de datos:
The data are grouped in two different ways (by treatment and airport in the flight simulator example):
library("arm")Loading required package: MASSLoading required package: MatrixLoading required package: lme4
arm (Version 1.14-4, built: 2024-4-1)Working directory is /home/lbarboza/Dropbox/Cursos/Actuales/SP1653/NotasClase/ModelosMixtos/ModelosMixtos
Attaching package: 'arm'The following object is masked from 'package:R2jags':
traceplotThe following object is masked from 'package:coda':
traceplotpilots <- read.table ("data/ARM_Data/pilots/pilots.dat", header=TRUE)
attach (pilots)
group.names <- as.vector(unique(group))
scenario.names <- as.vector(unique(scenario))
n.group <- length(group.names)
n.scenario <- length(scenario.names)
successes <- NULL
failures <- NULL
group.id <- NULL
scenario.id <- NULL
for (j in 1:n.group){
for (k in 1:n.scenario){
ok <- group==group.names[j] & scenario==scenario.names[k]
successes <- c (successes, sum(recovered[ok]==1,na.rm=T))
failures <- c (failures, sum(recovered[ok]==0,na.rm=T))
group.id <- c (group.id, j)
scenario.id <- c (scenario.id, k)
}
}
y <- successes/(successes+failures)# Define treatment and airport group sizes
n.treatment <- length(unique(group.id)) # Number of unique treatment groups
n.airport <- length(unique(scenario.id)) # Number of unique airports
# Prepare the data for JAGS
data_jags <- list(y = successes, treatment = group.id, airport = scenario.id,
n = length(successes), n.treatment = n.treatment, n.airport = n.airport)
# Initial values for the parameters
inits <- function() {
list(mu = rnorm(1), sigma.y = runif(1), sigma.gamma = runif(1), sigma.delta = runif(1))
}
# Parameters to monitor
params <- c("mu", "gamma", "delta", "sigma.y", "sigma.gamma", "sigma.delta")
# Run the JAGS model
jags_fit <- jags(data = data_jags,
inits = inits,
parameters.to.save = params,
model.file = "codigoJAGS/pilots.jags",
n.chains = 3,
n.iter = 5000,
n.burnin = 1000,
n.thin = 10)Compiling model graph
Resolving undeclared variables
Allocating nodes
Graph information:
Observed stochastic nodes: 40
Unobserved stochastic nodes: 17
Total graph size: 188
Initializing model# View the summary of the model fit
print(jags_fit)Inference for Bugs model at "codigoJAGS/pilots.jags", fit using jags,
3 chains, each with 5000 iterations (first 1000 discarded), n.thin = 10
n.sims = 1200 iterations saved
mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat n.eff
delta[1] -0.618 1.243 -3.031 -1.431 -0.593 0.175 1.810 1.001 1200
delta[2] -2.112 1.203 -4.581 -2.849 -2.074 -1.373 0.198 1.000 1200
delta[3] 0.107 1.201 -2.216 -0.629 0.057 0.929 2.455 1.000 1200
delta[4] -1.517 1.248 -4.096 -2.315 -1.487 -0.704 1.041 1.001 1200
delta[5] -0.017 1.213 -2.415 -0.803 -0.023 0.749 2.414 1.000 1200
delta[6] 3.617 1.219 1.291 2.843 3.615 4.362 6.098 1.000 1200
delta[7] -2.070 1.197 -4.666 -2.778 -2.041 -1.291 0.197 1.000 1200
delta[8] 2.690 1.209 0.388 1.921 2.619 3.425 5.148 1.001 1200
gamma[1] -0.158 0.455 -1.246 -0.320 -0.069 0.047 0.620 1.015 570
gamma[2] 0.069 0.436 -0.827 -0.122 0.026 0.255 0.995 1.002 1200
gamma[3] -0.080 0.435 -1.093 -0.252 -0.032 0.102 0.783 1.002 760
gamma[4] -0.059 0.415 -0.945 -0.233 -0.017 0.143 0.769 1.010 330
gamma[5] 0.227 0.458 -0.477 -0.009 0.124 0.395 1.398 1.002 750
mu 3.061 1.107 0.905 2.350 3.054 3.723 5.371 1.000 1200
sigma.delta 2.756 1.013 1.459 2.071 2.538 3.217 5.156 1.001 1200
sigma.gamma 0.491 0.482 0.014 0.163 0.361 0.664 1.787 1.002 930
sigma.y 1.559 0.207 1.215 1.414 1.540 1.675 2.008 1.002 1100
deviance 147.118 5.597 138.785 143.005 146.311 150.442 160.260 1.001 1200
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 = 15.7 and DIC = 162.8
DIC is an estimate of expected predictive error (lower deviance is better).