Simple simulations to compare stan_lmer() and hand-coded Stan

This project contains a very simple simulation to explore two implementations of Beyesian mixed modeling. The simulation creates data with a linear relationship between one covariate and a response variable with an intercept that varies by site nested within species.

The Stan code is a literal implementation of the simulation code with just a few small changes:

- It uses a
`cauchy(0,5)`prior on the residual variance. - It also uses a
`cauchy(0,5)`prior on the random effect variances.

The `stan_lmer()` model is the the direct analog of the simulation and
the Stan code, except that it uses a `decov()` prior for all of the
random effect.

`results.txt` contains the results from analysis of 100
simulated datasets. Youâ€™ll notice that the bias, root mean squared
error, and coverage (symmetric 80% credible intervals) are very
similar for the intercept (`beta_0`), the regression
coefficient (`beta_1`), and the residual standard deviatiion
(`sigma`). The random effect standard deviations
(`sigma_species` and `sigma_species_site`) are fairly
different from one another. In both cases, the Stan code has a smaller
bias and root mean squared error and it has better coverage
properties.