The basic area-level model (Fay and Herriot 1979; Rao and Molina 2015) is given by \[ y_i | \theta_i \stackrel{\mathrm{ind}}{\sim} {\cal N} (\theta_i, \psi_i) \,, \\ \theta_i = \beta' x_i + v_i \,, \] where \(i\) runs from 1 to \(m\), the number of areas, \(\beta\) is a vector of regression coefficients for given covariates \(x_i\), and \(v_i \stackrel{\mathrm{ind}}{\sim} {\cal N} (0, \sigma_v^2)\) are independent random area effects. For each area an observation \(y_i\) is available with given variance \(\psi_i\).
First we generate some data according to this model:
m <- 75L # number of areas
df <- data.frame(
area=1:m, # area indicator
x=runif(m) # covariate
)
v <- rnorm(m, sd=0.5) # true area effects
theta <- 1 + 3*df$x + v # quantity of interest
psi <- runif(m, 0.5, 2) / sample(1:25, m, replace=TRUE) # given variances
df$y <- rnorm(m, theta, sqrt(psi))A sampler function for a model with a regression component and a random intercept is created by
library(mcmcsae)
model <- y ~ reg(~ 1 + x, name="beta") + gen(factor = ~iid(area), name="v")
sampler <- create_sampler(model, sigma.fixed=TRUE, Q0=1/psi, linpred="fitted", data=df)The meaning of the arguments used here is as follows:
sigma.fixed=TRUE signifies that the observation level variance parameter is fixed at 1. In this case it means that the variances are known and given by psi.Q0=1/psi the precisions are set to the vector 1/psi.linpred="fitted" indicates that we wish to obtain samples from the posterior distribution for the vector \(\theta\) of small area means.data is the data.frame in which variables used in the model specification are looked up.An MCMC simulation using this sampler function is then carried out as follows:
sim <- MCMCsim(sampler, store.all=TRUE, verbose=FALSE)A summary of the results is obtained by
(summ <- summary(sim))## llh_ :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## -27.958 5.617 -4.977 0.113 -37.599 -27.505 -19.285 2474.167 0.999
##
## linpred_ :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 1 3.21 0.332 9.68 0.00609 2.67 3.21 3.75 2967 0.999
## 2 2.17 0.200 10.84 0.00365 1.84 2.17 2.50 3000 1.001
## 3 3.12 0.248 12.60 0.00471 2.73 3.13 3.52 2770 0.999
## 4 3.16 0.235 13.45 0.00443 2.76 3.17 3.53 2812 1.000
## 5 3.48 0.163 21.37 0.00304 3.21 3.48 3.75 2859 0.999
## 6 3.78 0.289 13.07 0.00529 3.32 3.78 4.25 3000 1.001
## 7 2.49 0.250 9.96 0.00456 2.07 2.48 2.90 3000 1.000
## 8 2.43 0.194 12.50 0.00354 2.11 2.43 2.74 3000 1.000
## 9 1.47 0.252 5.86 0.00467 1.07 1.47 1.90 2898 1.001
## 10 2.21 0.300 7.35 0.00610 1.72 2.20 2.69 2419 1.001
## ... 65 elements suppressed ...
##
## beta :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## (Intercept) 0.979 0.145 6.78 0.00765 0.732 0.983 1.21 357 1
## x 3.112 0.249 12.48 0.01346 2.717 3.107 3.53 343 1
##
## v_sigma :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 4.82e-01 5.88e-02 8.21e+00 1.33e-03 3.96e-01 4.78e-01 5.85e-01 1.95e+03 1.00e+00
##
## v :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 1 0.34996 0.332 1.0555 0.00608 -0.1915 0.342391 0.8942 2969 1.000
## 2 -0.67638 0.206 -3.2781 0.00465 -1.0160 -0.675347 -0.3420 1968 1.000
## 3 0.03780 0.252 0.1498 0.00560 -0.3717 0.034257 0.4504 2029 1.000
## 4 -0.40845 0.249 -1.6381 0.00633 -0.8251 -0.405081 -0.0109 1551 1.004
## 5 0.38716 0.180 2.1455 0.00491 0.0927 0.383639 0.6839 1352 1.002
## 6 0.12312 0.298 0.4134 0.00633 -0.3636 0.117878 0.6188 2214 0.999
## 7 0.24721 0.255 0.9679 0.00510 -0.1711 0.245228 0.6648 2509 1.000
## 8 -0.02883 0.201 -0.1431 0.00485 -0.3492 -0.024925 0.2946 1728 1.000
## 9 0.00438 0.261 0.0168 0.00637 -0.4197 0.000961 0.4469 1679 1.000
## 10 -0.54875 0.300 -1.8279 0.00625 -1.0299 -0.551127 -0.0567 2307 1.001
## ... 65 elements suppressed ...In this example we can compare the model parameter estimates to the ‘true’ parameter values that have been used to generate the data. In the next plots we compare the estimated and ‘true’ random effects, as well as the model estimates and ‘true’ estimands. In the latter plot, the original ‘direct’ estimates are added as red triangles.
plot(v, summ$v[, "Mean"], xlab="true v", ylab="posterior mean"); abline(0, 1)
plot(theta, summ$linpred_[, "Mean"], xlab="true theta", ylab="estimated"); abline(0, 1)
points(theta, df$y, col=2, pch=2)
We can compute model selection measures DIC and WAIC by
compute_DIC(sim)## DIC p_DIC
## 104.62721 48.71086compute_WAIC(sim, show.progress=FALSE)## WAIC1 p_WAIC1 WAIC2 p_WAIC2
## 75.99048 20.06448 96.98012 30.55930Posterior means of residuals can be extracted from the simulation output using method residuals. Here is a plot of (posterior means of) residuals against covariate \(x\):
plot(df$x, residuals(sim, mean.only=TRUE), xlab="x", ylab="residual"); abline(h=0)
A linear predictor in a linear model can be expressed as a weighted sum of the response variable. If we set compute.weights=TRUE then such weights are computed for all linear predictors specified in argument linpred. In this case it means that a set of weights is computed for each area.
sampler <- create_sampler(model, sigma.fixed=TRUE, Q0=1/psi,
linpred="fitted", data=df, compute.weights=TRUE)
sim <- MCMCsim(sampler, store.all=TRUE, verbose=FALSE)Now the weights method returns a matrix of weights, in this case a 75 \(\times\) 75 matrix \(w_{ij}\) holding the weight of direct estimate \(i\) in linear predictor \(j\). To verify that the weights applied to the direct estimates yield the model-based estimates we plot them against each other. Also shown is a plot of the weight of the direct estimate for each area in the predictor for that same area, against the variance of the direct estimate.
plot(summ$linpred_[, "Mean"], crossprod(weights(sim), df$y),
xlab="estimate", ylab="weighted average")
abline(0, 1)
plot(psi, diag(weights(sim)), ylab="weight")
