---
title: Basic area-level model
output: rmarkdown::html_vignette
bibliography: mcmcsae.bib
vignette: >
  %\VignetteIndexEntry{Basic area-level model}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, echo=FALSE}
options(width=100)  # width of output
```


The basic area-level model [@Fay1979; @rao2015small] 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:
```{r}
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
```{r, message=FALSE}
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:

* the first argument is a formula specifying the response variable and the linear predictor part of the model
* `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`.
* the function expects precisions rather than variances, and with `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:
```{r}
sim <- MCMCsim(sampler, store.all=TRUE, verbose=FALSE)
```

A summary of the results is obtained by
```{r}
(summ <- summary(sim))
```

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.
```{r, fig.show='hold'}
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
```{r}
compute_DIC(sim)
compute_WAIC(sim, show.progress=FALSE)
```

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 $x$:
```{r, fig.align="center"}
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.
```{r}
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
`r m` $\times$ `r m` 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.
```{r, fig.show='hold'}
plot(summ$linpred_[, "Mean"], crossprod(weights(sim), df$y),
     xlab="estimate", ylab="weighted average")
abline(0, 1)
plot(psi, diag(weights(sim)), ylab="weight")
```

## References
