# Question

What’s the difference between `ranef`

and the b parameters?

According to the https://cran.r-project.org/web/packages/rstanarm/vignettes/glmer.html#glms-with-group-specific-terms “b as the random effects” and

Consequently, b is not a “parameter” to be estimated because parameters are unknown constants that > are fixed in repeated sampling.

I want to make sure I’m interpreting the summary output correctly.

# Reproducible Example

I was trying to recreate a radon model found in Gelman and Hill’s text (pg. 282) but had a question about the posterior summary results.

The model in question has a varying intercept and slope with a single group-level predictor:

also written

If we look at county 85, the b values are not the same as the values produced by ranef. What are the b parameter values?

```
library(rstanarm)
demo(package="rstanarm",topic="ARM_Ch12_13")
tail(ranef(M4)$county,8)
tail(posterior_summary(M4)[,1:3],8)
```

which yields

```
: (Intercept) x
: 78 0.02177865 -0.009344675
: 79 -0.11318990 -0.087323466
: 80 -0.01616251 -0.089506652
: 81 0.10016236 0.156978232
: 82 0.01636097 0.004331113
: 83 -0.05364101 -0.188520895
: 84 0.06537049 0.019689862
: 85 -0.03264432 -0.004255915
: Estimate Est.Error Q2.5
: b[(Intercept) county:84] 0.071846727 0.11608273 -0.140401534
: b[x county:84] 0.027510480 0.23361550 -0.446632534
: b[(Intercept) county:85] -0.038354974 0.14350999 -0.347426522
: b[x county:85] -0.004709328 0.25448657 -0.523348587
: sigma 0.752818158 0.01826484 0.717829578
: Sigma[county:(Intercept),(Intercept)] 0.021750010 0.01406291 0.002438374
: Sigma[county:x,(Intercept)] 0.007244463 0.01625235 -0.028200743
```

I was able to reproduce the calculations for the varying intercept and slope using the `ranef`

function as well which would seem to confirm that the b parameters are not the same as ranef and that ranef behavior aligns with that in the text:

```
# \alpha_{85}
(first_bracket <- fixef(M4)["(Intercept)"] + fixef(M4)["u"]*unique(radon$u)[85] + ranef(M4)$county[85,c("(Intercept)")])
# \beta_{85}
(second_bracket <- fixef(M4)["x"] + fixef(M4)["x:u"] * unique(radon$u)[85] + ranef(M4)$county[85, c("x")])
```

or

```
a.hat.M4 <- coef(M4)$county[,1] + coef(M4)$county[,3]*unique(radon$u)
b.hat.M4 <- coef(M4)$county[,2] + coef(M4)$county[,4]*unique(radon$u)
# Just show for county 85
a.hat.M4[85]
b.hat.M4[85]
```

## Aside

Aside: Is there a nicer way of retrieving the alpha and beta vectors in the presence of a group-level predictor so they can be plotted? I’d really like to do something like the output of mcmc_intervals.

I’d be grateful for any help you could provide. Thank you and thanks to the people working on this project.