Chapter 13 - Models With Memory

This chapter has been an introduction to the motivation, implementation, and interpretation of basic multilevel models. It focused on varying intercepts, which achieve better estimates of baseline differences among clusters in the data. They achieve better estimates, because they simultaneously model the population of clusters and use inferences about the population to pool information among parameters. From another perspective, varying intercepts are adaptively regularized parameters, relying upon a prior that is itself learned from the data.

Place each answer inside the code chunk (grey box). The code chunks should contain a text response or a code that completes/answers the question or activity requested. Make sure to include plots if the question requests them. Problems are labeled Easy (E), Medium (M), and Hard(H).

Finally, upon completion, name your final output .html file as: YourName_ANLY505-Year-Semester.html and publish the assignment to your R Pubs account and submit the link to Canvas. Each question is worth 5 points.

Questions

13E1. Which of the following priors will produce more shrinkage in the estimates? Show the prior plot. \[\begin{align} \ α_{TANK} ∼ Normal(0, 1) \tag{a} \\ \ α_{TANK} ∼ Normal(0, 2) \tag{b} \\ \end{align}\]

# The first one in a is producing more shrinkage

13E2. Rewrite the following model as a multilevel model. \[\begin{align} y_i ∼ Binomial(1, p_i) \\ logit(p_i) = α_{group[i]} + βx_i \\ α_{group} ∼ Normal(0, 1.5) \\ β ∼ Normal(0, 0.5) \\ \end{align}\]

#yi ~ Binomial(1, pi)
#logit(pi) = αgroup[i] + βxi
#αgroup ∼ Normal(α_bar, σα)
#α_bar ~ Normal(0, 1.5)
#σα ~ Exponential(1)
#β ∼ Normal(0, 0.5)

13E3. Rewrite the following model as a multilevel model. \[\begin{align} y_i ∼ Normal(μ_i, σ) \\ μ_i = α_{group[i]} + βx_i \\ α_{group} ∼ Normal(0, 5) \\ β ∼ Normal(0, 1) \\ σ ∼ Exponential(1) \\ \end{align}\]

#y_i ~ Normal(μ_i, σ)
#μ_i = α_group[i] + βx_i
#α_group[i] ∼ Normal(α_bar, σ_α)
#σ ∼ Exponential(1)
#β ∼ Normal(0, 1)
#α_bar ~ Normal(0, 1)
#σ_α ~ Exponential(1)

13E4. Write a mathematical model formula for a Poisson regression with varying intercepts.

 #y_i ∼ poisson(p_i)
 #log(p_i) = α_{group[i]} + βx_i 
 #α_{group} ∼ Normal(α, α_α)
 #α = Normal(0,1)
 #α_α = Exponential(1)
 #β ∼ Normal(0,1)

13E5. Write a mathematical model formula for a Poisson regression with two different kinds of varying intercepts, a cross-classified model.

 #y_i ∼ poisson(p_i)
 #log(p_i) = α_{group[i]} α1_{group[i]}+ βx_i 
 #α_{group} ∼ Normal(α, α_α)
 #α1_{group} ∼ Normal(α1, α1_α1)
 #α = Normal(0,1)
 #α1 = Normal(0,1)
 #α_α = Exponential(1)
 #α1_α1 = Exponential(1)
 #β ∼ Normal(0,1)

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. Plot the sigma estimates.

library(rethinking)
data(reedfrogs)
data <- reedfrogs
data$pred <- ifelse(data$pred == "yes", 1,0)
data$sizes <- ifelse(data$size == "big", 1, 0)
data$tanks <- 1:nrow(data)

#model 1 main effect 
m1 <- map2stan(
  alist(
    surv ~ dbinom(density, c),
    logit(c) <- a_tank[tanks],
    a_tank[tanks] ~ dnorm(x, s),
    x ~ dnorm(0, 10), 
    s ~ dexp(1)
  ), 
  data = data, chains = 4 
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
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#model 2 both main effects
m2 <- map2stan(
  alist(
    surv ~ dbinom(density, c),
    logit(c) <- a_tank[tanks] + dp*pred,
    dp ~ dnorm(0,1),
    a_tank[tanks] ~ dnorm(x, s),
    x ~ dnorm(0, 10), 
    s ~ dexp(1)
  ), 
  data = data, chains = 4 
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL 'da51dc1362e9c9ac934bd281ef2b789e' NOW (CHAIN 1).
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#model 3
m3 <- map2stan(
  alist(
    surv ~ dbinom(density, c),
    logit(c) <- a_tank[tanks] + ds*sizes,
    ds ~ dnorm(0,1),
    a_tank[tanks] ~ dnorm(x, s),
    x ~ dnorm(0, 10), 
    s ~ dexp(1)
  ), 
  data = data, chains = 4 
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL 'd4eafb7d4e35948873f92c8b29341162' NOW (CHAIN 1).
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 #model 4
m4 <- map2stan(
  alist(
    surv ~ dbinom(density, c),
    logit(c) <- a_tank[tanks] + dp*pred + ds*sizes,
    dp ~ dnorm(0,1),
    ds ~ dnorm(0,1),
    a_tank[tanks] ~ dnorm(x, s),
    x ~ dnorm(0, 10), 
    s ~ dexp(1)
  ), 
  data = data, chains = 4 
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL 'aceec9a492a49326f0269a6f07738e71' NOW (CHAIN 1).
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## 
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#Model 5
m5 <- map2stan(
  alist(
    surv ~ dbinom(density, c),
    logit(c) <- a_tank[tanks] + dp*pred + ds*sizes + dps*pred*sizes,
    dp ~ dnorm(0,1),
    ds ~ dnorm(0,1),
    dps ~ dnorm(0,1),
    a_tank[tanks] ~ dnorm(x, s),
    x ~ dnorm(0, 10), 
    s ~ dexp(1)
  ), 
  data = data, chains = 4 
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '6940245224e3ff4175531b09eddfa505' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 4e-05 seconds
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## SAMPLING FOR MODEL '6940245224e3ff4175531b09eddfa505' NOW (CHAIN 2).
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## SAMPLING FOR MODEL '6940245224e3ff4175531b09eddfa505' NOW (CHAIN 3).
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precis (m1)
##       mean        sd      5.5%    94.5%    n_eff     Rhat4
## x 1.388994 0.2603375 0.9700529 1.803891 5021.134 0.9998867
## s 1.623424 0.2145903 1.3115436 1.990812 3086.600 1.0010198
precis (m2)
##           mean        sd       5.5%    94.5%    n_eff     Rhat4
## dp 0.004484341 1.0165401 -1.6172955 1.645434 9821.771 0.9994876
## x  1.387692149 0.2591365  0.9831417 1.806184 5147.212 0.9999994
## s  1.624669080 0.2098020  1.3228689 1.987663 3483.633 0.9996337
precis (m3)
##          mean        sd      5.5%    94.5%     n_eff    Rhat4
## ds -0.3734295 0.4484802 -1.066969 0.350980  413.5943 1.004833
## x   1.5696929 0.3406926  1.016606 2.116844  684.3461 1.005056
## s   1.6142790 0.2147254  1.302869 1.975606 1702.9051 1.001786
precis (m4)
##           mean        sd      5.5%     94.5%     n_eff     Rhat4
## dp  0.01037151 0.9887736 -1.591877 1.5960147 5449.1268 0.9992807
## ds -0.40074224 0.4531235 -1.120004 0.3273612  580.8122 1.0150109
## x   1.59022180 0.3435635  1.060724 2.1411227 1013.0376 1.0118297
## s   1.61805243 0.2175912  1.302514 1.9918323 2177.9555 1.0015652
precis (m5)
##             mean        sd      5.5%     94.5%     n_eff     Rhat4
## dp  -0.008477152 0.9867737 -1.574037 1.5758869 3875.2342 0.9999426
## ds  -0.361162440 0.4450390 -1.080434 0.3359259  622.4345 1.0019647
## dps  0.014229853 1.0003104 -1.592278 1.5792057 4835.8110 0.9996054
## x    1.573746985 0.3466385  1.013631 2.1382120  967.5368 1.0013039
## s    1.626784147 0.2251959  1.297616 2.0158988 2022.8627 1.0043329
#With the results of all the models above,what we learn is that adding the predictor variables actually made the results varied. Therefore, predation and size influence the variations in the results of tank variation

13M2. Compare the models you fit just above, using WAIC. Can you reconcile the differences in WAIC with the posterior distributions of the models? Show the WAIC table.

compare(m1, m2, m3, m4, m5)
##        WAIC       SE     dWAIC       dSE    pWAIC    weight
## m2 1009.746 38.19478 0.0000000        NA 37.93381 0.2445757
## m4 1010.075 38.26419 0.3294262 0.7185330 38.10161 0.2074337
## m1 1010.187 38.11288 0.4408416 0.3473721 38.16806 0.1961940
## m3 1010.231 38.18928 0.4851086 0.7301953 38.11409 0.1918993
## m5 1010.596 38.28845 0.8499874 0.7461104 38.34336 0.1598972
#After looking at WAIC, the differences are very minor.

13M3. Re-estimate the basic Reed frog varying intercept model, but now using a Cauchy distribution in place of the Gaussian distribution for the varying intercepts. That is, fit this model: \[\begin{align} s_i ∼ Binomial(n_i, p_i) \\ logit(p_i) = α_{tank[i]} \\ α_{tank} ∼ Cauchy(α, σ) \\ α ∼ Normal(0, 1) \\ σ ∼ Exponential(1) \\ \end{align}\]

(You are likely to see many divergent transitions for this model. Can you figure out why? Can you fix them?) Plot and compare the posterior means of the intercepts, αtank, to the posterior means produced in the chapter, using the customary Gaussian prior. Can you explain the pattern of differences? Take note of any change in the mean α as well.

m_cauchy <- map2stan(
  alist(
    surv ~ dbinom(density, c),
    logit(c) <- a_tank[tanks],
    a_tank[tanks] ~ dnorm(x, s),
    x ~ dnorm(0, 10), 
    s ~ dexp(1)
  ), 
  data = data, chains = 4 
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '30e6c2b9eed793d10ef50bfa01abe3c7' NOW (CHAIN 1).
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## SAMPLING FOR MODEL '30e6c2b9eed793d10ef50bfa01abe3c7' NOW (CHAIN 2).
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precis(m_cauchy)
##       mean        sd      5.5%    94.5%    n_eff    Rhat4
## x 1.382281 0.2608376 0.9689187 1.805722 5502.478 1.000206
## s 1.620151 0.2092464 1.3143210 1.970623 3149.217 1.000289
compare(m1,m2,m3,m4,m5,m_cauchy)
##              WAIC       SE     dWAIC       dSE    pWAIC    weight
## m2       1009.746 38.19478 0.0000000        NA 37.93381 0.2014800
## m_cauchy 1010.014 38.18359 0.2680742 0.3734869 38.06993 0.1762059
## m4       1010.075 38.26419 0.3294262 0.7185330 38.10161 0.1708827
## m1       1010.187 38.11288 0.4408416 0.3473721 38.16806 0.1616235
## m3       1010.231 38.18928 0.4851086 0.7301953 38.11409 0.1580855
## m5       1010.596 38.28845 0.8499874 0.7461104 38.34336 0.1317224
# There is no significant differences among the models in terms of WAIC, including the Cauchy model. Thefefore, it is not significantly different from others.

13M4. Now use a Student-t distribution with ν = 2 for the intercepts: \[\begin{align} α_{tank} ∼ Student(2, α, σ) \end{align}\]

Refer back to the Student-t example in Chapter 7 (page 234), if necessary. Plot and compare the resulting posterior to both the original model and the Cauchy model in 13M3. Can you explain the differences and similarities in shrinkage in terms of the properties of these distributions?

m6<- map2stan( 
  alist(
    surv ~ dbinom(density , c),
    logit(c) <- a_tank[tanks] ,
    a_tank[tanks] ~ dstudent(2, a , sigma ),
    a ~ dnorm(0, 1) ,
    sigma ~ dcauchy(0,1)
  ), 
  data=data , chains = 4 
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL 'dc8c2921b326ab7dcfc3c09d91d6d88a' NOW (CHAIN 1).
## Chain 1: Rejecting initial value:
## Chain 1:   Error evaluating the log probability at the initial value.
## Chain 1: Exception: student_t_lpdf: Scale parameter is -0.56312, but must be > 0!  (in 'modeldbe898f24c3_dc8c2921b326ab7dcfc3c09d91d6d88a' at line 18)
## 
## Chain 1: Rejecting initial value:
## Chain 1:   Error evaluating the log probability at the initial value.
## Chain 1: Exception: student_t_lpdf: Scale parameter is -1.0749, but must be > 0!  (in 'modeldbe898f24c3_dc8c2921b326ab7dcfc3c09d91d6d88a' at line 18)
## 
## Chain 1: 
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## 
## SAMPLING FOR MODEL 'dc8c2921b326ab7dcfc3c09d91d6d88a' NOW (CHAIN 2).
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## 
## SAMPLING FOR MODEL 'dc8c2921b326ab7dcfc3c09d91d6d88a' NOW (CHAIN 3).
## Chain 3: Rejecting initial value:
## Chain 3:   Error evaluating the log probability at the initial value.
## Chain 3: Exception: student_t_lpdf: Scale parameter is -0.823744, but must be > 0!  (in 'modeldbe898f24c3_dc8c2921b326ab7dcfc3c09d91d6d88a' at line 18)
## 
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## 
## SAMPLING FOR MODEL 'dc8c2921b326ab7dcfc3c09d91d6d88a' NOW (CHAIN 4).
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## Chain 4:
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
precis (m6)
##           mean        sd      5.5%    94.5%    n_eff    Rhat4
## a     1.339921 0.2827537 0.8810578 1.791860 4606.575 1.000098
## sigma 1.272308 0.2234747 0.9480739 1.652167 2778.954 1.000121
compare(m_cauchy,m6)
##              WAIC       SE    dWAIC      dSE    pWAIC    weight
## m_cauchy 1010.014 38.18359 0.000000       NA 38.06993 0.7000848
## m6       1011.709 37.99008 1.695403 1.646614 39.15621 0.2999152
#There is no significant difference in this comparison. And so, student-t model has just a slightly higher WAIC than that of the cauchy model.

13M5. Modify the cross-classified chimpanzees model m13.4 so that the adaptive prior for blocks contains a parameter \(\bar{γ}\) for its mean: \[\begin{align} γ_j ∼ Normal(\bar{γ}, σ_γ) \\ \bar{γ} ∼ Normal(0, 1.5) \\ \end{align}\]

Compare the precis output of this model to m13.4. What has including \(\bar{γ}\) done?

data(chimpanzees)
data=chimpanzees
m7 <- map2stan( 
  alist(
    pulled_left ~ dnorm(mu , 1),
    mu~dnorm(10,1)
  ), 
  data=data , chains = 4
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL 'f984f5c9d566d6e8ac01a0e847a5ffb3' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 1.5e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.15 seconds.
## Chain 1: Adjust your expectations accordingly!
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## Chain 1: 
## 
## SAMPLING FOR MODEL 'f984f5c9d566d6e8ac01a0e847a5ffb3' NOW (CHAIN 2).
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## Chain 2: 
## 
## SAMPLING FOR MODEL 'f984f5c9d566d6e8ac01a0e847a5ffb3' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'f984f5c9d566d6e8ac01a0e847a5ffb3' NOW (CHAIN 4).
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## Chain 4: 
## Error in .local(fit, data, n, ...) : 
##   There appear to be no linear models here
precis(m7)
##         mean         sd     5.5%     94.5%    n_eff    Rhat4
## mu 0.5979434 0.04389997 0.528962 0.6682104 1116.243 1.006429
m8 <- map2stan( 
  alist(
    pulled_left ~ dstudent(2, mu , 1),
    mu~dnorm(10,1)
  ), 
  data=data , chains = 4 
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL 'be249fcfe22dfcab6b0287d2543344ef' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 4.2e-05 seconds
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## Chain 1: 
## 
## SAMPLING FOR MODEL 'be249fcfe22dfcab6b0287d2543344ef' NOW (CHAIN 2).
## Chain 2: 
## Chain 2: Gradient evaluation took 1.7e-05 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.17 seconds.
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## Chain 2: 
## 
## SAMPLING FOR MODEL 'be249fcfe22dfcab6b0287d2543344ef' NOW (CHAIN 3).
## Chain 3: 
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## Chain 3: 
## 
## SAMPLING FOR MODEL 'be249fcfe22dfcab6b0287d2543344ef' NOW (CHAIN 4).
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## Chain 4: 
## Error in .local(fit, data, n, ...) : 
##   There appear to be no linear models here
precis (m8)
##         mean         sd      5.5%     94.5%    n_eff    Rhat4
## mu 0.6196659 0.04278489 0.5527183 0.6893214 1396.107 1.001355
m9 <- map2stan( 
  alist(
    pulled_left ~ dnorm(mu , 1),
    mu~dstudent(2,10,1)
  ), 
  data=data , chains = 4
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '6f2b3345213f16fb6b0158f15ab49519' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 1.2e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
## Chain 1: Adjust your expectations accordingly!
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## Chain 1: 
## 
## SAMPLING FOR MODEL '6f2b3345213f16fb6b0158f15ab49519' NOW (CHAIN 2).
## Chain 2: 
## Chain 2: Gradient evaluation took 5e-06 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
## Chain 2: Adjust your expectations accordingly!
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## Chain 2: 
## 
## SAMPLING FOR MODEL '6f2b3345213f16fb6b0158f15ab49519' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 5e-06 seconds
## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
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## Chain 3: 
## 
## SAMPLING FOR MODEL '6f2b3345213f16fb6b0158f15ab49519' NOW (CHAIN 4).
## Chain 4: 
## Chain 4: Gradient evaluation took 6e-06 seconds
## Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
## Chain 4: Adjust your expectations accordingly!
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## Chain 4: 
## Error in .local(fit, data, n, ...) : 
##   There appear to be no linear models here
precis (m9)
##        mean         sd      5.5%     94.5%    n_eff    Rhat4
## mu 0.580835 0.04507854 0.5083397 0.6537881 1413.911 1.002351
m10 <- map2stan( 
  alist(
    pulled_left ~ dstudent(2,mu , 1),
    mu~dstudent(2,10,1)
  ), 
  data=data , chains = 4
)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
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## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
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## Error in .local(fit, data, n, ...) : 
##   There appear to be no linear models here
precis (m10)
##         mean         sd      5.5%     94.5%    n_eff    Rhat4
## mu 0.6032243 0.04328094 0.5347674 0.6714317 1134.798 1.005061

13M6. Sometimes the prior and the data (through the likelihood) are in conflict, because they concentrate around different regions of parameter space. What happens in these cases depends a lot upon the shape of the tails of the distributions. Likewise, the tails of distributions strongly influence can outliers are shrunk or not towards the mean. I want you to consider four different models to fit to one observation at y = 0. The models differ only in the distributions assigned to the likelihood and prior. Here are the four models:

\[\begin{align} Model \;NN: y &∼ Normal(μ, 1) & Model \;TN: y &∼ Student(2, μ, 1) \\ μ &∼ Normal(10, 1) & μ &∼ Normal(10, 1) \\ Model \;NT: y &∼ Normal(μ, 1) & Model \;TT: y &∼ Student(2, μ, 1) \\ μ &∼ Student(2, 10, 1) & μ &∼ Student(2, 10, 1) \\ \end{align}\]

Estimate and plot the posterior distributions against the likelihoods for these models and compare them. Can you explain the results, using the properties of the distributions?

EXTRA CREDIT (10 POINTS)

13H1. In 1980, a typical Bengali woman could have 5 or more children in her lifetime. By the year 2000, a typical Bengali woman had only 2 or 3. You’re going to look at a historical set of data, when contraception was widely available but many families chose not to use it. These data reside in data(bangladesh) and come from the 1988 Bangladesh Fertility Survey. Each row is one of 1934 women. There are six variables, but you can focus on two of them for this practice problem: (1) district: ID number of administrative district each woman resided in (2) use.contraception: An indicator (0/1) of whether the woman was using contraception The first thing to do is ensure that the cluster variable, district, is a contiguous set of integers. Recall that these values will be index values inside the model. If there are gaps, you’ll have parameters for which there is no data to inform them. Worse, the model probably won’t run. Look at the unique values of the district variable:

data(bangladesh)
d <- bangladesh
sort(unique(d$district)) #District 54 is absent. So district isn’t yet a good index variable, because it’s not contiguous. This is easy to fix. Just make a new variable that is contiguous. This is enough to do it:
##  [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## [26] 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
## [51] 51 52 53 55 56 57 58 59 60 61
d$district_id <- as.integer(as.factor(d$district))
sort(unique(d$district_id))  # Now there are 60 values, contiguous integers 1 to 60. Now, focus on predicting use.contraception, clustered by district_id.
##  [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## [26] 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
## [51] 51 52 53 54 55 56 57 58 59 60

Fit both (1) a traditional fixed-effects model that uses an index variable for district and (2) a multilevel model with varying intercepts for district. Plot the predicted proportions of women in each district using contraception, for both the fixed-effects model and the varying-effects model. That is, make a plot in which district ID is on the horizontal axis and expected proportion using contraception is on the vertical. Make one plot for each model, or layer them on the same plot, as you prefer. How do the models disagree? Can you explain the pattern of disagreement? In particular, can you explain the most extreme cases of disagreement, both why they happen where they do and why the models reach different inferences?

# prep trimmed data list
dlist <- list(
  use_contraception = d$use.contraception,
  district = d$district_id )

# fixed effects model
m12H1f <- map2stan(
  alist(
    use_contraception ~ dbinom( 1 , p ),
    logit(p) <- a_district[district],
    a_district[district] ~ dnorm(0,10)
    ),
  data=dlist )
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
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## SAMPLING FOR MODEL 'b4571e1922f4d013ee76e7653c521068' NOW (CHAIN 1).
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# varying effects model
m12H1v <- map2stan(
  alist(
    use_contraception ~ dbinom( 1 , p ),
    logit(p) <- a + a_district[district],
    a ~ dnorm(0,10),
    a_district[district] ~ dnorm(0,sigma),
    sigma ~ dcauchy(0,1)
    ),
  data=dlist )
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.0/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.0/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
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# get predicted probabilities for each district for each model
# first, set up a new data list to compute predictions for. In this list, each case is just a unique district.
pred.dat <- list(district=1:60)

# call link for each model, passing it the new cases to compute predicted probabilities for:
pred1 <- link(m12H1f,data=pred.dat)
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pred2 <- link(m12H1v,data=pred.dat)
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# have a glimps of the prediction for each district was computed for each of 1000 samples from the posterior distribution
str(pred1)
##  num [1:1000, 1:60] 0.207 0.266 0.303 0.259 0.232 ...
#  get average probabilities in each district:
p1.mean <- apply( pred1 , 2 , mean )
p2.mean <- apply( pred2 , 2 , mean )
round(p1.mean,2)
##  [1] 0.26 0.35 0.95 0.50 0.36 0.29 0.28 0.38 0.31 0.08 0.00 0.34 0.42 0.63 0.36
## [16] 0.55 0.29 0.34 0.39 0.40 0.39 0.20 0.26 0.07 0.45 0.39 0.18 0.25 0.28 0.49
## [31] 0.46 0.21 0.43 0.66 0.50 0.35 0.54 0.29 0.50 0.46 0.50 0.55 0.53 0.22 0.33
## [46] 0.52 0.47 0.52 0.02 0.47 0.46 0.44 0.42 0.17 0.57 0.19 0.46 0.10 0.22 0.21
# plot the estimated proportions using contraception:
#  take the estimated log-odds lists and use the logistic transform on them, to convert to probability scale
plot( 1:60 , p1.mean , col=rangi2 , pch=16 , xlab="District" ,
ylab="probability use contraception" )
points( 1:60 , p2.mean )
abline( h=logistic(coef(m12H1v)[1]) , lty=2 ) # plot line for 'a'

# The blue points are the fixed-effects estimates, and the open black ones are the varying effects. The dashed line is the average proportion of women using contraception, in the entire sample (as estimated by the varying effects model). Notice first that the black points are always closer to the dashedline, as was the case with the tadpole example in lecture. This results from shrinkage, which  resultsfrom pooling information.

# There are cases with rather extreme disagreements, though. The most obvious is district 3, which has a fixed (blue) estimate of 1 but a varying (black) estimate of only 0.44. There are also two districts (11 and 49) for which the fixed estimates are zero, but the varying estimates are 0.18 and 0.30. In these districts, either all sampled women used contraception or none did. As a result, the fixed effects estimates were silly—the parameter estimates are far from zero with very large standard errors. The varying effects model was able to produce more rational estimates, because itpooled information from other districts. Depending upon how many women were sampled in eachdistrict, there was more or less shrinkage (pooling) towards to grand mean. So for example in the case of district 3, there were only 2 women in the sample, and so there is a lot of distance between the blue and black points. In contrast, district 11 had 21 women in the sample, and so while pooling pulls the estimate off of zero to 0.18, it doesn’t pull it nearly as far as district 3.

# Another way to think of this phenomenon is to view the same estimates arranged by number of women in the district sample, on the horizontal axis. Then on the vertical we can plot the distance (absolute value of the difference) between the fixed and varying estimates. Here’s what that looks like:

# compute number of women sampled in each district
n_by_district <- sapply( 1:60 , function(did) length(d$district_id[d$district_id==did]) )

# compute shrinkage
shrinkage <- abs( p1.mean - p2.mean )
plot( n_by_district , shrinkage , col ="slateblue" ,
xlab ="number of women sampled" , ylab ="shrinkage by district" )