Basic area-level model

The basic area-level model (Fay and Herriot 1979; Rao and Molina 2015) is given by \[ y_i | \theta_i \stackrel{\mathrm{iid}}{\sim} {\cal N} (\theta_i, \psi_i) \,, \\ \theta_i = \beta' x_i + v_i \,, \] where \(i\) runs from 1 to \(m\), the number of areas, \(\beta\) is a vector of regression coefficients for given covariates \(x_i\), and \(v_i \stackrel{\mathrm{iid}}{\sim} {\cal N} (0, \sigma_v^2)\) 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,
  family=f_gaussian(var.prior=pr_fixed(1), var.vec = ~ psi),
  linpred="fitted", data=df
)

The meaning of the arguments used is as follows:

• the first argument is a formula specifying the response variable and the linear predictor to model the mean of the sampling distribution
• the family argument allows to set one of a number of sampling distributions and possibly pass additional family-dependent arguments. In this case the scalar observation level variance parameter is set to a fixed value 1, and unequal variances are set to the vector 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
## llh_ -16.5 5.99   -2.75 0.119   -27 -16.1 -7.28  2535     1
## 
## beta :
##              Mean    SD t-value    MCSE q0.05  q0.5 q0.95 n_eff R_hat
## (Intercept) 0.816 0.122    6.71 0.00224 0.613 0.817  1.02  2933     1
## x           3.295 0.234   14.09 0.00431 2.908 3.294  3.68  2948     1
## 
## v_sigma :
##          Mean     SD t-value    MCSE q0.05  q0.5 q0.95 n_eff R_hat
## v_sigma 0.474 0.0566    8.38 0.00138 0.387 0.471 0.574  1689     1
## 
## linpred_ :
##    Mean    SD t-value    MCSE q0.05  q0.5 q0.95 n_eff R_hat
## 1  4.04 0.220   18.36 0.00402 3.675 4.034  4.40  3000 0.999
## 2  1.44 0.262    5.49 0.00479 1.013 1.445  1.86  3000 1.000
## 3  2.26 0.342    6.60 0.00645 1.692 2.256  2.80  2813 0.999
## 4  1.73 0.250    6.89 0.00469 1.304 1.725  2.14  2852 1.000
## 5  3.73 0.258   14.47 0.00470 3.304 3.731  4.15  3000 0.999
## 6  1.33 0.321    4.14 0.00601 0.807 1.336  1.85  2864 0.999
## 7  1.01 0.297    3.39 0.00547 0.514 1.017  1.48  2940 1.000
## 8  3.25 0.230   14.11 0.00437 2.872 3.252  3.63  2780 1.001
## 9  3.93 0.237   16.58 0.00451 3.543 3.925  4.32  2760 1.000
## 10 0.70 0.310    2.26 0.00565 0.187 0.706  1.20  3000 1.001
## ... 65 elements suppressed ...
## 
## v :
##      Mean    SD t-value    MCSE   q0.05   q0.5   q0.95 n_eff R_hat
## 1   0.178 0.243   0.732 0.00452 -0.2206  0.180  0.5654  2879 1.000
## 2  -0.410 0.266  -1.542 0.00486 -0.8593 -0.407  0.0175  3000 1.000
## 3   0.263 0.344   0.764 0.00644 -0.2983  0.261  0.8147  2860 0.999
## 4  -0.435 0.253  -1.716 0.00474 -0.8543 -0.440 -0.0132  2853 1.000
## 5  -0.118 0.275  -0.431 0.00502 -0.5631 -0.117  0.3359  3000 0.999
## 6   0.504 0.321   1.570 0.00591 -0.0147  0.507  1.0144  2953 0.999
## 7  -0.161 0.297  -0.542 0.00554 -0.6498 -0.156  0.3222  2882 0.999
## 8  -0.821 0.254  -3.228 0.00464 -1.2459 -0.819 -0.4110  3000 1.000
## 9   0.316 0.254   1.243 0.00487 -0.0885  0.308  0.7466  2724 1.001
## 10 -0.549 0.315  -1.745 0.00575 -1.0862 -0.546 -0.0382  3000 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 
## 85.57392 52.64448

compute_WAIC(sim, show.progress=FALSE)

##    WAIC1  p_WAIC1    WAIC2  p_WAIC2 
## 54.35053 21.40862 78.47102 33.46886

Posterior 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,
  family=f_gaussian(var.prior=pr_fixed(1), var.vec = ~ 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")