Assignment #9
Chapter 11
Shuyang Sun
2021-11-18
Chapter 11 - God Spiked the Integers
This chapter described some of the most common generalized linear models, those used to model counts. It is important to never convert counts to proportions before analysis, because doing so destroys information about sample size. A fundamental difficulty with these models is that parameters are on a different scale, typically log-odds (for binomial) or log-rate (for Poisson), than the outcome variable they describe. Therefore computing implied predictions is even more important than before.
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.
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
11-1. As explained in the chapter, binomial data can be organized in aggregated and disaggregated forms, without any impact on inference. But the likelihood of the data does change when the data are converted between the two formats. Can you explain why?
In the disaggregated form, all outcomes are simply binary values, the distribution of binary values is very extreme between 0 and 1 (concentrated on both ends). In aggregated form, the outcome follows a more even distribution from 0 to n.11-2. Use quap to construct a quadratic approximate posterior distribution for the chimpanzee model that includes a unique intercept for each actor, m11.4 (page 330). Plot and compare the quadratic approximation to the posterior distribution produced instead from MCMC. Can you explain both the differences and the similarities between the approximate and the MCMC distributions? Relax the prior on the actor intercepts to Normal(0,10). Re-estimate the posterior using both ulam and quap. Plot and compare the posterior distributions. Do the differences increase or decrease? Why?
The quap model are very similar to MCMC model in both situation. When we relaxed the prior, the mean of R/P moved to the left side (from -0.4 to -0.5). This makes sense because previous we bounded the prior to be norm(0, 1.5), with the new prior it was able to fit the data better.data(chimpanzees)
d = chimpanzees
d$treatment = 1 + d$prosoc_left + 2 * d$condition
dat_list = list(
pulled_left = d$pulled_left,
actor = d$actor,
treatment = as.integer(d$treatment)
)
# For question 4
m11.3 = quap(
alist(
pulled_left ~ dbinom(1, p),
logit(p) <- a + b[treatment],
a ~ dnorm(0, 1.5),
b[treatment] ~ dnorm(0, 0.5)
), data=dat_list
)
m11.4 = ulam(
alist(
pulled_left ~ dbinom(1, p),
logit(p) <- a[actor] + b[treatment],
a[actor] ~ dnorm(0, 1.5),
b[treatment] ~ dnorm(0, 0.5)
), data=dat_list, chain=4, log_lik=TRUE
)## Running /usr/lib/R/bin/R CMD SHLIB foo.c
## gcc -std=gnu99 -std=gnu11 -I"/usr/share/R/include" -DNDEBUG -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/Rcpp/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/unsupported" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/BH/include" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/src/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppParallel/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DBOOST_NO_AUTO_PTR -include '/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -fpic -g -O2 -fdebug-prefix-map=/build/r-base-i2PIHO/r-base-4.1.2=. -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g -c foo.c -o foo.o
## In file included from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Core:88,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Dense:1,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
## from <command-line>:
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name ‘namespace’
## 628 | namespace Eigen {
## | ^~~~~~~~~
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:17: error: expected ‘=’, ‘,’, ‘;’, ‘asm’ or ‘__attribute__’ before ‘{’ token
## 628 | namespace Eigen {
## | ^
## In file included from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Dense:1,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
## from <command-line>:
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Core:96:10: fatal error: complex: No such file or directory
## 96 | #include <complex>
## | ^~~~~~~~~
## compilation terminated.
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## Chain 4:m11.4.1 = quap(
alist(
pulled_left ~ dbinom(1, p),
logit(p) <- a[actor] + b[treatment],
a[actor] ~ dnorm(0, 1.5),
b[treatment] ~ dnorm(0, 0.5)
), data=dat_list
)
m11.4.2 = ulam(
alist(
pulled_left ~ dbinom(1, p),
logit(p) <- a[actor] + b[treatment],
a[actor] ~ dnorm(0, 10),
b[treatment] ~ dnorm(0, 0.5)
), data=dat_list, chain=4, log_lik=TRUE
)## Running /usr/lib/R/bin/R CMD SHLIB foo.c
## gcc -std=gnu99 -std=gnu11 -I"/usr/share/R/include" -DNDEBUG -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/Rcpp/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/unsupported" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/BH/include" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/src/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppParallel/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DBOOST_NO_AUTO_PTR -include '/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -fpic -g -O2 -fdebug-prefix-map=/build/r-base-i2PIHO/r-base-4.1.2=. -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g -c foo.c -o foo.o
## In file included from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Core:88,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Dense:1,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
## from <command-line>:
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name ‘namespace’
## 628 | namespace Eigen {
## | ^~~~~~~~~
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:17: error: expected ‘=’, ‘,’, ‘;’, ‘asm’ or ‘__attribute__’ before ‘{’ token
## 628 | namespace Eigen {
## | ^
## In file included from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Dense:1,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
## from <command-line>:
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Core:96:10: fatal error: complex: No such file or directory
## 96 | #include <complex>
## | ^~~~~~~~~
## compilation terminated.
## make: *** [/usr/lib/R/etc/Makeconf:168: foo.o] Error 1
##
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## Chain 4:m11.4.3 = quap(
alist(
pulled_left ~ dbinom(1, p),
logit(p) <- a[actor] + b[treatment],
a[actor] ~ dnorm(0, 10),
b[treatment] ~ dnorm(0, 0.5)
), data=dat_list
)
plotPosterior = function(model, title) {
print(title)
labs = c("R/N", "L/N", "R/P", "L/P")
plot(precis(model, depth=2, pars="b"), labels=labs)
}
plotPosterior(m11.4, "MCMC with dnorm(0, 1.5):")## [1] "MCMC with dnorm(0, 1.5):"
plotPosterior(m11.4.1, "quap with dnorm(0, 1.5):")## [1] "quap with dnorm(0, 1.5):"
plotPosterior(m11.4.2, "MCMC with dnorm(0, 10):")## [1] "MCMC with dnorm(0, 10):"
plotPosterior(m11.4.3, "quap with dnorm(0, 10):")## [1] "quap with dnorm(0, 10):"
11-3. Revisit the data(Kline) islands example. This time drop Hawaii from the sample and refit the models. What changes do you observe?
The model precision and WAIC score went up a lot after dropping Hawaii, since it is such an outlier.data(Kline)
d = Kline
d$P = scale(log(d$population))
d$contact_id = ifelse(d$contact == 'high', 2, 1)
dat = list(
T = d$total_tools,
P = d$P,
cid = d$contact_id
)
d_no_hawaii = d[d$culture != 'Hawaii',]
d_no_hawaii$P = scale(log(d_no_hawaii$population))
d_no_hawaii$contact_id = ifelse(d_no_hawaii$contact == 'high', 2, 1)
dat_no_hawaii = list(
T = d_no_hawaii$total_tools,
P = d_no_hawaii$P,
cid = d_no_hawaii$contact_id
)
m11.10 = ulam(
alist(
T ~ dpois(lambda),
log(lambda) <- a[cid] + b[cid] * P,
a[cid] ~ dnorm(3, 0.5),
b[cid] ~ dnorm(0, 0.2)
), data=dat, chains=4, log_lik=TRUE
)## Running /usr/lib/R/bin/R CMD SHLIB foo.c
## gcc -std=gnu99 -std=gnu11 -I"/usr/share/R/include" -DNDEBUG -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/Rcpp/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/unsupported" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/BH/include" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/src/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppParallel/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DBOOST_NO_AUTO_PTR -include '/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -fpic -g -O2 -fdebug-prefix-map=/build/r-base-i2PIHO/r-base-4.1.2=. -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g -c foo.c -o foo.o
## In file included from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Core:88,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Dense:1,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
## from <command-line>:
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name ‘namespace’
## 628 | namespace Eigen {
## | ^~~~~~~~~
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:17: error: expected ‘=’, ‘,’, ‘;’, ‘asm’ or ‘__attribute__’ before ‘{’ token
## 628 | namespace Eigen {
## | ^
## In file included from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Dense:1,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
## from <command-line>:
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Core:96:10: fatal error: complex: No such file or directory
## 96 | #include <complex>
## | ^~~~~~~~~
## compilation terminated.
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## Chain 4:m11.10.1 = ulam(
alist(
T ~ dpois(lambda),
log(lambda) <- a[cid] + b[cid] * P,
a[cid] ~ dnorm(3, 0.5),
b[cid] ~ dnorm(0, 0.2)
), data=dat_no_hawaii, chains=4, log_lik=TRUE
)## Running /usr/lib/R/bin/R CMD SHLIB foo.c
## gcc -std=gnu99 -std=gnu11 -I"/usr/share/R/include" -DNDEBUG -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/Rcpp/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/unsupported" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/BH/include" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/src/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppParallel/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DBOOST_NO_AUTO_PTR -include '/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -fpic -g -O2 -fdebug-prefix-map=/build/r-base-i2PIHO/r-base-4.1.2=. -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g -c foo.c -o foo.o
## In file included from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Core:88,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Dense:1,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
## from <command-line>:
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name ‘namespace’
## 628 | namespace Eigen {
## | ^~~~~~~~~
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:17: error: expected ‘=’, ‘,’, ‘;’, ‘asm’ or ‘__attribute__’ before ‘{’ token
## 628 | namespace Eigen {
## | ^
## In file included from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Dense:1,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
## from <command-line>:
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Core:96:10: fatal error: complex: No such file or directory
## 96 | #include <complex>
## | ^~~~~~~~~
## compilation terminated.
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## SAMPLING FOR MODEL 'c5bf3fcd8c4a2bf16731479ec5cc9e70' NOW (CHAIN 1).
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## Chain 4:compare(m11.10, m11.10.1)## WAIC SE dWAIC dSE pWAIC weight
## m11.10.1 75.72979 11.10414 0.000000 NA 5.878786 0.98236515
## m11.10 83.76996 12.45799 8.040172 3.594034 6.244377 0.01763485plot(coeftab(m11.10, m11.10.1))
11-4. Use WAIC or PSIS to compare the chimpanzee model that includes a unique intercept for each actor, m11.4 (page 330), to the simpler models fit in the same section. Interpret the results.
From the WAIC score we can see that m11.4 (unique intercept for each actor) has a better fit comparing to m11.3, which only has a unique intercept for treatment.compare(m11.3, m11.4)## WAIC SE dWAIC dSE pWAIC weight
## m11.4 532.4402 19.010671 0.0000 NA 8.601406 1.000000e+00
## m11.3 682.5241 9.121908 150.0838 18.46097 3.663223 2.568662e-3311-5. The data contained in data(salamanders) are counts of salamanders (Plethodon elongatus) from 47 different 49-m2 plots in northern California. The column SALAMAN is the count in each plot, and the columns PCTCOVER and FORESTAGE are percent of ground cover and age of trees in the plot, respectively. You will model SALAMAN as a Poisson variable. (a) Model the relationship between density and percent cover, using a log-link (same as the example in the book and lecture). Use weakly informative priors of your choosing. Check the quadratic approximation again, by comparing quap to ulam. Then plot the expected counts and their 89% interval against percent cover. In which ways does the model do a good job? A bad job? (b) Can you improve the model by using the other predictor, FORESTAGE? Try any models you think useful. Can you explain why FORESTAGE helps or does not help with prediction?
The quap model and mcmc model performed similarly given the same variables and priors, however FORESTAGE does not help in this case while using quap model.data(salamanders)
d = salamanders
d = d[d$SALAMAN > 0,]
d$P = scale(log(d$PCTCOVER))
d$Q = scale(log(d$FORESTAGE))
dat_list = list(
S = d$SALAMAN,
P = d$P,
Q = d$Q
)
salamander_model_quap = quap(
alist(
S ~ dpois(lambda),
log(lambda) <- a + p * P,
a ~ dnorm(3, 10),
p~ dnorm(0, 10)
), data=dat_list
)
salamander_model_quap_2 = quap(
alist(
S ~ dpois(lambda),
log(lambda) <- a + p * P + q * Q,
a ~ dnorm(3, 10),
p ~ dnorm(0, 10),
q ~ dnorm(0, 10)
), data=dat_list
)
salamander_model_mcmc = ulam(
alist(
S ~ dpois(lambda),
log(lambda) <- a + p * P,
a ~ dnorm(3, 10),
p ~ dnorm(0, 10)
), data=dat_list, chains=4, log_lik=TRUE
)## Running /usr/lib/R/bin/R CMD SHLIB foo.c
## gcc -std=gnu99 -std=gnu11 -I"/usr/share/R/include" -DNDEBUG -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/Rcpp/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/unsupported" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/BH/include" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/src/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppParallel/include/" -I"/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DBOOST_NO_AUTO_PTR -include '/home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -fpic -g -O2 -fdebug-prefix-map=/build/r-base-i2PIHO/r-base-4.1.2=. -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g -c foo.c -o foo.o
## In file included from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Core:88,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Dense:1,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
## from <command-line>:
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name ‘namespace’
## 628 | namespace Eigen {
## | ^~~~~~~~~
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:17: error: expected ‘=’, ‘,’, ‘;’, ‘asm’ or ‘__attribute__’ before ‘{’ token
## 628 | namespace Eigen {
## | ^
## In file included from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Dense:1,
## from /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
## from <command-line>:
## /home/shuyang/R/x86_64-pc-linux-gnu-library/4.1/RcppEigen/include/Eigen/Core:96:10: fatal error: complex: No such file or directory
## 96 | #include <complex>
## | ^~~~~~~~~
## compilation terminated.
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## Chain 4:
## Chain 4: Elapsed Time: 0.011231 seconds (Warm-up)
## Chain 4: 0.010787 seconds (Sampling)
## Chain 4: 0.022018 seconds (Total)
## Chain 4:compare(salamander_model_quap, salamander_model_quap_2, salamander_model_mcmc)## WAIC SE dWAIC dSE pWAIC
## salamander_model_quap 153.2412 15.07503 0.000000 NA 3.913940
## salamander_model_mcmc 153.5327 15.35874 0.291565 0.6115992 4.148078
## salamander_model_quap_2 156.9762 15.98824 3.735018 1.5537696 6.332977
## weight
## salamander_model_quap 0.49533059
## salamander_model_mcmc 0.42813684
## salamander_model_quap_2 0.07653257