13E3. Rewrite the following model as a multilevel model.

yi~Normal(mui, sigma)

mui=alphagroup[i] + beta*xi

alphagroup[i]~Normal(0,5)

beta~Normal(0,1)

sigma~Exponential(1)

Solution:

yi ∼Normal(µi, σ)

µi = αgroup[i] + βxi

αgroup ∼Normal(α, σα)

β∼Normal(0, 1)

σ ∼Exponential(0, 1)

α ∼Normal(0, 5)

σα ∼Exponential(0, 1)

The key point is that in the original model, αgroup has only a prior distribution. In the multilevel version, αgroup has a distribution with higher level or hyperparameters. We can learn about these hyperparameters from the data in the second version as opposed to having to set them apriori in the original model.

13M1. Revisit the Reed frog survival data, data(reedfrogs), and add the predation and size treatment variables to the varying intercepts model. Consider models with either main effect alone, both main effects, as well as a model including both and their interaction. Instead of focusing on inferences about these two predictor variables, focus on the inferred variation across tanks. Explain why it changes as it does across models.

First, let’s set up the data list:

library(rethinking)
## Loading required package: cmdstanr
## This is cmdstanr version 0.8.0
## - CmdStanR documentation and vignettes: mc-stan.org/cmdstanr
## - CmdStan path: /Users/brianbeckage/.cmdstan/cmdstan-2.36.0
## - CmdStan version: 2.36.0
## Loading required package: posterior
## This is posterior version 1.6.0
## 
## Attaching package: 'posterior'
## The following objects are masked from 'package:stats':
## 
##     mad, sd, var
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## 
##     %in%, match
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## rethinking (Version 2.42)
## 
## Attaching package: 'rethinking'
## The following object is masked from 'package:stats':
## 
##     rstudent
data(reedfrogs)
d <- reedfrogs
dat <- list(
S = d$surv,
n = d$density,
tank = 1:nrow(d),
pred = ifelse( d$pred=="no" , 0L , 1L ),
size_ = ifelse( d$size=="small" , 1L , 2L )
)

Now to define a series of models. The first is just the varying intercepts model from the text:

m1.1 <- ulam(
alist(
S ~ binomial( n , p ),
logit(p) <- a[tank],
a[tank] ~ normal( a_bar , sigma ),
a_bar ~ normal( 0 , 1.5 ),
sigma ~ exponential( 1 )
), data=dat , chains=4 , cores=4 , log_lik=TRUE )
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The other models just incorporate the predictors, as ordinary regression terms.

# pred
m1.2 <- ulam(
alist(
S ~ binomial( n , p ),
logit(p) <- a[tank] + bp*pred,
a[tank] ~ normal( a_bar , sigma ),
bp ~ normal( -0.5 , 1 ),
a_bar ~ normal( 0 , 1.5 ),
sigma ~ exponential( 1 )
), data=dat , chains=4 , cores=4 , log_lik=TRUE )
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# size
m1.3 <- ulam(
alist(
S ~ binomial( n , p ),
logit(p) <- a[tank] + s[size_],
a[tank] ~ normal( a_bar , sigma ),
s[size_] ~ normal( 0 , 0.5 ),
a_bar ~ normal( 0 , 1.5 ),
sigma ~ exponential( 1 )
), data=dat , chains=4 , cores=4 , log_lik=TRUE )
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# pred + size
m1.4 <- ulam(
alist(
S ~ binomial( n , p ),
logit(p) <- a[tank] + bp*pred + s[size_],
a[tank] ~ normal( a_bar , sigma ),
bp ~ normal( -0.5 , 1 ),
s[size_] ~ normal( 0 , 0.5 ),
a_bar ~ normal( 0 , 1.5 ),
sigma ~ exponential( 1 )
), data=dat , chains=4 , cores=4 , log_lik=TRUE )
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# pred + size + interaction
m1.5 <- ulam(
alist(
S ~ binomial( n , p ),
logit(p) <- a_bar + z[tank]*sigma + bp[size_]*pred + s[size_],
z[tank] ~ normal( 0 , 1 ),
bp[size_] ~ normal( -0.5 , 1 ),
s[size_] ~ normal( 0 , 0.5 ),
a_bar ~ normal( 0 , 1.5 ),
sigma ~ exponential( 1 )
), data=dat , chains=4 , cores=4 , log_lik=TRUE )
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I coded the interaction model using a non-centered parameterization. The interaction itself is done by creating a bp parameter for each size value. In this way, the effect of pred depends upon size. Let’s look at all the sigma posterior distributions:

plot( coeftab( m1.1 , m1.2 , m1.3 , m1.4 , m1.5 ), pars="sigma" )
## Warning: The ESS has been capped to avoid unstable estimates.
## Warning: The ESS has been capped to avoid unstable estimates.

The two models that omit predation, m1.1 and m1.3, have larger values of sigma. This is because predation explains some of the variation among tanks. So when you add it to the model, the variation in the tank intercepts gets smaller.

The general point is that the model with only intercepts measures the variation among tanks, but does nothing to explain it. As we add treatment variables, the variation should shrink, even though the total variation in the data of course stays the same.