Bayesian analyses made easy: GLMMs in R package brms
Here I illustrate how to fit GLMMs with the R package brms, and compare to Jags and lme4.
Motivation
I regularly give
a course on Bayesian statistics with R for non-specialists. To illustrate the course, we analyse data with
generalized linear, often mixed, models or GLMMs.
So far, I’ve been using Jags to fit these models. This requires some programming skills, like e.g. coding a loop, to be able to write down the model likelihood. Although students learn a lot from going through that process, it can be daunting.
This year, I thought I’d show them the
R package brms developed by
Paul-Christian Bürkner. In brief, brms allows fitting GLMMs (but not only) in a lme4-like syntax within the Bayesian framework and MCMC methods with Stan.
I’m not a Stan user, but it doesn’t matter. The
vignettes were more than enough to get me started. I also recommend the
list of blog posts about brms.
First things first, we load the packages we will need:
library(brms)
library(posterior) # tools for working with posterior and prior distributions
library(R2jags) # run Jags from within R
library(lme4) # fit GLMM in frequentist framework
Beta-Binomial model
The first example is about a survival experiment. We captured, marked and released 57 individuals (total) out of which 19 made it through the year (alive).
dat <- data.frame(alive = 19, total = 57)
dat
An obvious estimate of the probability of a binomial is the proportion of cases, $19/57 = 0.3333333$ here, but let’s use R built-in functions for the sake of illustration.
Maximum-likelihood estimation
To get an estimate of survival, we fit a logistic regression using the glm() function. The data are grouped (or aggregated), so we need to specify the number of alive and dead individuals as a two-column matrix on the left hand side of the model formula:
mle <- glm(cbind(alive, total - alive) ~ 1,
family = binomial("logit"), # family = binomial("identity") would be more straightforward
data = dat)
On the logit scale, survival is estimated at:
coef(mle) # logit scale
## (Intercept)
## -0.6931472
After back-transformation using the reciprocal logit, we obtain:
plogis(coef(mle))
## (Intercept)
## 0.3333333
So far, so good.
Bayesian analysis with Jags
In Jags, you can fit this model with:
# model
betabin <- function(){
alive ~ dbin(survival, total) # binomial likelihood
logit(survival) <- beta # logit link
beta ~ dnorm(0, 1/1.5) # prior
}
# data
datax <- list(total = 57,
alive = 19)
# initial values
inits <- function() list(beta = rnorm(1, 0, sd = 1.5))
# parameter to monitor
params <- c("survival")
# run jags
bayes.jags <- jags(data = datax,
inits = inits,
parameters.to.save = params,
model.file = betabin,
n.chains = 2,
n.iter = 5000,
n.burnin = 1000,
n.thin = 1)
## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 1
## Unobserved stochastic nodes: 1
## Total graph size: 8
##
## Initializing model
# display results
bayes.jags
## Inference for Bugs model at "/var/folders/r7/j0wqj1k95vz8w44sdxzm986c0000gn/T//RtmpP7MZ1A/modela81d72dd23e2.txt", fit using jags,
## 2 chains, each with 5000 iterations (first 1000 discarded)
## n.sims = 8000 iterations saved
## mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat n.eff
## survival 0.342 0.061 0.228 0.300 0.341 0.382 0.466 1.001 2600
## deviance 5.354 1.390 4.388 4.486 4.819 5.662 9.333 1.001 8000
##
## For each parameter, n.eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor (at convergence, Rhat=1).
##
## DIC info (using the rule, pD = var(deviance)/2)
## pD = 1.0 and DIC = 6.3
## DIC is an estimate of expected predictive error (lower deviance is better).
Bayesian analysis with brms
In brms, you write:
bayes.brms <- brm(alive | trials(total) ~ 1,
family = binomial("logit"), # binomial("identity") would be more straightforward
data = dat,
chains = 2, # nb of chains
iter = 5000, # nb of iterations, including burnin
warmup = 1000, # burnin
thin = 1) # thinning
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/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/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.1/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 '782790eabbd10296f779d85fa03cf1e5' NOW (CHAIN 1).
## Chain 1:
## Chain 1: Gradient evaluation took 2e-05 seconds
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## Chain 1:
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## Chain 1:
##
## SAMPLING FOR MODEL '782790eabbd10296f779d85fa03cf1e5' NOW (CHAIN 2).
## Chain 2:
## Chain 2: Gradient evaluation took 1.3e-05 seconds
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## Chain 2: Adjust your expectations accordingly!
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## Chain 2: Elapsed Time: 0.009763 seconds (Warm-up)
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## Chain 2:
You can display the results:
bayes.brms
## Family: binomial
## Links: mu = logit
## Formula: alive | trials(total) ~ 1
## Data: dat (Number of observations: 1)
## Draws: 2 chains, each with iter = 5000; warmup = 1000; thin = 1;
## total post-warmup draws = 8000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.69 0.28 -1.27 -0.14 1.00 3213 3572
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
And visualize the posterior density and trace of survival (on the logit scale):
plot(bayes.brms)
To get survival on the $[0,1]$ scale, we extract the MCMC values, then apply the reciprocal logit function to each of these values, and summarize its posterior distribution:
draws_fit <- as_draws_matrix(bayes.brms)
draws_fit
## # A draws_matrix: 4000 iterations, 2 chains, and 2 variables
## variable
## draw b_Intercept lp__
## 1 -0.70 -4.2
## 2 -0.67 -4.2
## 3 -0.86 -4.4
## 4 -0.80 -4.3
## 5 -1.32 -6.6
## 6 -0.57 -4.2
## 7 -0.51 -4.4
## 8 -0.56 -4.3
## 9 -0.69 -4.2
## 10 -0.41 -4.6
## # ... with 7990 more draws
summarize_draws(plogis(draws_fit[,1]))
What is the prior used by default?
prior_summary(bayes.brms)
What if I want to use the same prior as in Jags instead?
nlprior <- prior(normal(0, 1.5), class = "Intercept") # new prior
bayes.brms <- brm(alive | trials(total) ~ 1,
family = binomial("logit"),
data = dat,
prior = nlprior, # set prior by hand
chains = 2, # nb of chains
iter = 5000, # nb of iterations, including burnin
warmup = 1000, # burnin
thin = 1)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/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/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.1/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 '59dcc499a2b4c8a1b2c8f4e80e70fcce' NOW (CHAIN 1).
## Chain 1:
## Chain 1: Gradient evaluation took 1.8e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.18 seconds.
## Chain 1: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL '59dcc499a2b4c8a1b2c8f4e80e70fcce' 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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Double-check the prior that was used:
prior_summary(bayes.brms)
Logistic regression with covariates
In this example, we ask whether annual variation in white stork breeding success can be explained by rainfall in the wintering area. Breeding success is measured by the ratio of the number of chicks over the number of pairs.
nbchicks <- c(151,105,73,107,113,87,77,108,118,122,112,120,122,89,69,71,53,41,53,31,35,14,18)
nbpairs <- c(173,164,103,113,122,112,98,121,132,136,133,137,145,117,90,80,67,54,58,39,42,23,23)
rain <- c(67,52,88,61,32,36,72,43,92,32,86,28,57,55,66,26,28,96,48,90,86,78,87)
dat <- data.frame(nbchicks = nbchicks,
nbpairs = nbpairs,
rain = (rain - mean(rain))/sd(rain)) # standardized rainfall
The data are grouped.
Maximum-likelihood estimation
You can get maximum likelihood estimates with:
mle <- glm(cbind(nbchicks, nbpairs - nbchicks) ~ rain,
family = binomial("logit"),
data = dat)
summary(mle)
##
## Call:
## glm(formula = cbind(nbchicks, nbpairs - nbchicks) ~ rain, family = binomial("logit"),
## data = dat)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -5.9630 -1.3354 0.3929 1.6168 3.8800
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 1.55436 0.05565 27.931 <2e-16 ***
## rain -0.14857 0.05993 -2.479 0.0132 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 109.25 on 22 degrees of freedom
## Residual deviance: 103.11 on 21 degrees of freedom
## AIC: 205.84
##
## Number of Fisher Scoring iterations: 4
There is a negative effect of rainfall on breeding success:
visreg::visreg(mle, scale = "response")
Bayesian analysis with Jags
In Jags, we can fit the same model:
# data
dat <- list(N = 23, # nb of years
nbchicks = nbchicks,
nbpairs = nbpairs,
rain = (rain - mean(rain))/sd(rain)) # standardized rainfall
# define model
logistic <- function() {
# likelihood
for(i in 1:N){ # loop over years
nbchicks[i] ~ dbin(p[i], nbpairs[i]) # binomial likelihood
logit(p[i]) <- a + b.rain * rain[i] # prob success is linear fn of rainfall on logit scale
}
# priors
a ~ dnorm(0,0.01) # intercept
b.rain ~ dnorm(0,0.01) # slope for rainfall
}
# function that generates initial values
inits <- function() list(a = rnorm(n = 1, mean = 0, sd = 5),
b.rain = rnorm(n = 1, mean = 0, sd = 5))
# specify parameters that need to be estimated
parameters <- c("a", "b.rain")
# run Jags
bayes.jags <- jags(data = dat,
inits = inits,
parameters.to.save = parameters,
model.file = logistic,
n.chains = 2,
n.iter = 5000, # includes burn-in!
n.burnin = 1000)
## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 23
## Unobserved stochastic nodes: 2
## Total graph size: 134
##
## Initializing model
# display results
bayes.jags
## Inference for Bugs model at "/var/folders/r7/j0wqj1k95vz8w44sdxzm986c0000gn/T//RtmpP7MZ1A/modela81d1b01bd3f.txt", fit using jags,
## 2 chains, each with 5000 iterations (first 1000 discarded), n.thin = 4
## n.sims = 2000 iterations saved
## mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat n.eff
## a 1.546 0.163 1.439 1.514 1.552 1.588 1.657 1.182 2000
## b.rain -0.152 0.121 -0.275 -0.190 -0.148 -0.106 -0.033 1.129 2000
## deviance 213.612 330.910 201.899 202.434 203.197 204.689 209.169 1.121 2000
##
## For each parameter, n.eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor (at convergence, Rhat=1).
##
## DIC info (using the rule, pD = var(deviance)/2)
## pD = 54753.9 and DIC = 54967.5
## DIC is an estimate of expected predictive error (lower deviance is better).
Bayesian analysis with brms
With brms, we write:
bayes.brms <- brm(nbchicks | trials(nbpairs) ~ rain,
family = binomial("logit"),
data = dat,
chains = 2, # nb of chains
iter = 5000, # nb of iterations, including burnin
warmup = 1000, # burnin
thin = 1)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/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/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.1/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 '70a68f0c53365768853c2791038e6352' NOW (CHAIN 1).
## Chain 1:
## Chain 1: Gradient evaluation took 4.7e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.47 seconds.
## Chain 1: Adjust your expectations accordingly!
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## Chain 1:
##
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Display results:
bayes.brms
## Family: binomial
## Links: mu = logit
## Formula: nbchicks | trials(nbpairs) ~ rain
## Data: dat (Number of observations: 23)
## Draws: 2 chains, each with iter = 5000; warmup = 1000; thin = 1;
## total post-warmup draws = 8000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 1.56 0.06 1.45 1.67 1.00 6907 5043
## rain -0.15 0.06 -0.27 -0.03 1.00 7193 5067
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Visualize:
plot(bayes.brms)
We can also calculate the posterior probability of the rainfall effect being below zero with the hypothesis() function:
hypothesis(bayes.brms, 'rain < 0')
## Hypothesis Tests for class b:
## Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
## 1 (rain) < 0 -0.15 0.06 -0.25 -0.05 116.65 0.99 *
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.
These results confirm the important negative effect of rainfall on breeding success.
Linear mixed model
In this example, we have several measurements for 33 Mediterranean plant species, specifically number of seeds and biomass. We ask whether there is a linear relationship between these two variables that would hold for all species.
Read in data, directly from the course website:
df <- read_csv2("https://raw.githubusercontent.com/oliviergimenez/bayesian-stats-with-R/master/slides/dat/VMG.csv")
df
Use relevant format for columns species Sp and biomass Vm:
df$Sp <- as_factor(df$Sp)
df$Vm <- as.numeric(df$Vm)
Define numeric vector of species, for nested indexing in Jags:
species <- as.numeric(df$Sp)
Define response variable, number of seeds:
y <- log(df$NGrTotest)
Standardize explanatory variable, biomass
x <- (df$Vm - mean(df$Vm))/sd(df$Vm)
Now build dataset:
dat <- data.frame(y = y, x = x, species = species)
Maximum-likelihood estimation
You can get maximum likelihood estimates for a linear regression of number of seeds on biomass with a species random effect on the intercept (partial pooling):
mle <- lmer(y ~ x + (1 | species), data = dat)
summary(mle)
## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + (1 | species)
## Data: dat
##
## REML criterion at convergence: 2642.5
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -4.6761 -0.2704 -0.0120 0.2900 7.3163
##
## Random effects:
## Groups Name Variance Std.Dev.
## species (Intercept) 113.146 10.637
## Residual 9.373 3.062
## Number of obs: 488, groups: species, 33
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 14.5258 1.8574 7.821
## x 0.4779 0.2421 1.974
##
## Correlation of Fixed Effects:
## (Intr)
## x 0.002
Bayesian analysis with Jags
In Jags, you would use:
# partial pooling model
partial_pooling <- function(){
# likelihood
for(i in 1:n){
y[i] ~ dnorm(mu[i], tau.y)
mu[i] <- a[species[i]] + b * x[i]
}
for (j in 1:nbspecies){
a[j] ~ dnorm(mu.a, tau.a)
}
# priors
tau.y <- 1 / (sigma.y * sigma.y)
sigma.y ~ dunif(0,100)
mu.a ~ dnorm(0,0.01)
tau.a <- 1 / (sigma.a * sigma.a)
sigma.a ~ dunif(0,100)
b ~ dnorm(0,0.01)
}
# data
mydata <- list(y = y,
x = x,
n = length(y),
species = species,
nbspecies = length(levels(df$Sp)))
# initial values
inits <- function() list(mu.a = rnorm(1,0,5),
sigma.a = runif(0,0,10),
b = rnorm(1,0,5),
sigma.y = runif(0,0,10))
# parameters to monitor
params <- c("mu.a", "sigma.a", "b", "sigma.y")
# call jags to fit model
bayes.jags <- jags(data = mydata,
inits = inits,
parameters.to.save = params,
model.file = partial_pooling,
n.chains = 2,
n.iter = 5000,
n.burnin = 1000,
n.thin = 1)
## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 488
## Unobserved stochastic nodes: 37
## Total graph size: 2484
##
## Initializing model
bayes.jags
## Inference for Bugs model at "/var/folders/r7/j0wqj1k95vz8w44sdxzm986c0000gn/T//RtmpP7MZ1A/modela81d7639dc1c.txt", fit using jags,
## 2 chains, each with 5000 iterations (first 1000 discarded)
## n.sims = 8000 iterations saved
## mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat
## b 0.476 0.244 -0.005 0.314 0.472 0.638 0.964 1.001
## mu.a 14.025 1.916 10.146 12.774 14.009 15.299 17.760 1.001
## sigma.a 11.094 1.465 8.696 10.039 10.934 11.952 14.330 1.001
## sigma.y 3.072 0.101 2.884 3.002 3.069 3.138 3.277 1.001
## deviance 2478.206 8.704 2463.096 2472.022 2477.521 2483.650 2497.107 1.001
## n.eff
## b 8000
## mu.a 8000
## sigma.a 8000
## sigma.y 8000
## deviance 8000
##
## For each parameter, n.eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor (at convergence, Rhat=1).
##
## DIC info (using the rule, pD = var(deviance)/2)
## pD = 37.9 and DIC = 2516.1
## DIC is an estimate of expected predictive error (lower deviance is better).
Bayesian analysis with brms
In brms, you write:
bayes.brms <- brm(y ~ x + (1 | species),
data = dat,
chains = 2, # nb of chains
iter = 5000, # nb of iterations, including burnin
warmup = 1000, # burnin
thin = 1)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/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/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.1/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 'c7dcb37f6d55803a5216866e78a4a476' NOW (CHAIN 1).
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Display the results:
bayes.brms
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: y ~ x + (1 | species)
## Data: dat (Number of observations: 488)
## Draws: 2 chains, each with iter = 5000; warmup = 1000; thin = 1;
## total post-warmup draws = 8000
##
## Group-Level Effects:
## ~species (Number of levels: 33)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 10.74 1.39 8.45 13.78 1.00 427 943
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 14.13 1.90 10.31 17.86 1.01 331 580
## x 0.47 0.25 -0.00 0.97 1.00 2317 3382
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 3.07 0.10 2.88 3.28 1.00 2667 3408
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
You can extract a block of fixed effects:
summary(bayes.brms)$fixed
And a block of random effects:
summary(bayes.brms)$random
## $species
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 10.74273 1.388061 8.454543 13.776 1.001916 426.9568 943.1178
And visualize:
plot(bayes.brms)
GLMM with Poisson distribution
This example is from Jason Matthiopoulos’ excellent book How to be a quantitative ecologist. A survey of a coral reef uses 10 predefined linear transects covered by divers once every week. The response variable of interest is the abundance of a particular species of anemone as a function of water temperature. Counts of anemones are recorded at 20 regular line segments along the transect.
Let’s simulate some data:
set.seed(666)
transects <- 10
data <- NULL
for (tr in 1:transects){
# random effect (intercept)
ref <- rnorm(1,0,.5)
# water temperature gradient
t <- runif(1, 18,22) + runif(1,-.2,0.2)*1:20
# Anemone gradient (expected response)
ans <- exp(ref -14 + 1.8 * t - 0.045 * t^2)
# actual counts on 20 segments of the current transect
an <- rpois(20, ans)
data <- rbind(data, cbind(rep(tr, 20), t, an))
}
data <- data.frame(Transect = data[,1],
Temperature = data[,2],
Anemones = data[,3])
Standardize temperature:
boo <- data$Temperature
data$Temp <- (boo - mean(boo)) / sd(boo)
head(data)
Maximum-likelihood estimation
With lme4, you write:
fit_lme4 <- glmer(Anemones ~ Temp + I(Temp^2) + (1 | Transect),
data = data,
family = poisson)
summary(fit_lme4)
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: poisson ( log )
## Formula: Anemones ~ Temp + I(Temp^2) + (1 | Transect)
## Data: data
##
## AIC BIC logLik deviance df.resid
## 1380.2 1393.4 -686.1 1372.2 196
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.32356 -0.66154 -0.03391 0.63964 2.67283
##
## Random effects:
## Groups Name Variance Std.Dev.
## Transect (Intercept) 0.1497 0.3869
## Number of obs: 200, groups: Transect, 10
##
## Fixed effects:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 3.95436 0.12368 31.971 <2e-16 ***
## Temp -0.04608 0.02435 -1.892 0.0584 .
## I(Temp^2) -0.11122 0.01560 -7.130 1e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) Temp
## Temp 0.037
## I(Temp^2) -0.118 -0.294
Bayesian analysis with Jags
In Jags, we fit the corresponding GLMM with:
model <- function() {
for (i in 1:n){
count[i] ~ dpois(lambda[i])
log(lambda[i]) <- a[transect[i]] + b[1] * x[i] + b[2] * pow(x[i],2)
}
for (j in 1:nbtransects){
a[j] ~ dnorm (mu.a, tau.a)
}
mu.a ~ dnorm (0, 0.01)
tau.a <- pow(sigma.a, -2)
sigma.a ~ dunif (0, 100)
b[1] ~ dnorm (0, 0.01)
b[2] ~ dnorm (0, 0.01)
}
dat <- list(n = nrow(data),
nbtransects = transects,
x = data$Temp,
count = data$Anemones,
transect = data$Transect)
inits <- function() list(a = rnorm(transects),
b = rnorm(2),
mu.a = rnorm(1),
sigma.a = runif(1))
par <- c ("a", "b", "mu.a", "sigma.a")
fit <- jags(data = dat,
inits = inits,
parameters.to.save = par,
model.file = model,
n.chains = 2,
n.iter = 5000,
n.burn = 1000)
## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 200
## Unobserved stochastic nodes: 14
## Total graph size: 1622
##
## Initializing model
round(fit$BUGSoutput$summary[, -c(4,6)], 3)
## mean sd 2.5% 50% 97.5% Rhat n.eff
## a[1] 4.383 0.026 4.332 4.384 4.435 1.001 2000
## a[2] 3.223 0.050 3.120 3.224 3.322 1.001 2000
## a[3] 3.584 0.058 3.471 3.584 3.696 1.001 2000
## a[4] 3.620 0.054 3.513 3.622 3.721 1.001 2000
## a[5] 4.092 0.050 3.990 4.092 4.190 1.001 2000
## a[6] 3.848 0.038 3.774 3.849 3.921 1.000 2000
## a[7] 4.476 0.029 4.419 4.475 4.531 1.002 1300
## a[8] 4.398 0.055 4.288 4.397 4.499 1.003 550
## a[9] 3.835 0.033 3.771 3.835 3.900 1.001 1700
## a[10] 4.076 0.029 4.018 4.076 4.132 1.001 2000
## b[1] -0.047 0.024 -0.093 -0.047 0.000 1.001 2000
## b[2] -0.111 0.016 -0.142 -0.111 -0.079 1.004 450
## deviance 1324.697 4.651 1317.432 1324.185 1335.440 1.001 2000
## mu.a 3.950 0.160 3.617 3.957 4.272 1.005 310
## sigma.a 0.482 0.140 0.296 0.457 0.826 1.003 1000
Bayesian analysis with brms
What about with brms?
bayes.brms <- brm(Anemones ~ Temp + I(Temp^2) + (1 | Transect),
data = data,
family = poisson("log"),
chains = 2, # nb of chains
iter = 5000, # nb of iterations, including burnin
warmup = 1000, # burnin
thin = 1)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/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/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.1/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 '9326e0aa5a2dca4008c9e5b2ee4e86bf' NOW (CHAIN 1).
## Chain 1:
## Chain 1: Gradient evaluation took 4.5e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.45 seconds.
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##
## SAMPLING FOR MODEL '9326e0aa5a2dca4008c9e5b2ee4e86bf' NOW (CHAIN 2).
## Chain 2:
## Chain 2: Gradient evaluation took 2.4e-05 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.24 seconds.
## Chain 2: Adjust your expectations accordingly!
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## Chain 2:
## Chain 2: Elapsed Time: 1.21769 seconds (Warm-up)
## Chain 2: 4.45281 seconds (Sampling)
## Chain 2: 5.6705 seconds (Total)
## Chain 2:
Display results:
bayes.brms
## Family: poisson
## Links: mu = log
## Formula: Anemones ~ Temp + I(Temp^2) + (1 | Transect)
## Data: data (Number of observations: 200)
## Draws: 2 chains, each with iter = 5000; warmup = 1000; thin = 1;
## total post-warmup draws = 8000
##
## Group-Level Effects:
## ~Transect (Number of levels: 10)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 0.47 0.14 0.29 0.81 1.00 1366 2089
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 3.95 0.16 3.63 4.26 1.00 1359 1751
## Temp -0.05 0.02 -0.09 0.00 1.00 4126 4490
## ITempE2 -0.11 0.02 -0.14 -0.08 1.00 4018 4209
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Visualize:
plot(bayes.brms)
We can assess the quality of fit of this model:
pp_check(bayes.brms, ndraws = 100, type = 'ecdf_overlay')
As expected, the fit is almost perfect because we simulated data from this very model.
What if we’d like to test the effect of temperature using WAIC?
We fit a model with no effect of temperature:
bayes.brms2 <- brm(Anemones ~ 1 + (1 | Transect),
data = data,
family = poisson("log"),
chains = 2, # nb of chains
iter = 5000, # nb of iterations, including burnin
warmup = 1000, # burnin
thin = 1)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.1/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/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.1/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/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.1/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 'fd9804091450c044661531e89123d1dc' NOW (CHAIN 1).
## Chain 1:
## Chain 1: Gradient evaluation took 4.4e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.44 seconds.
## Chain 1: Adjust your expectations accordingly!
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## Chain 1:
## Chain 1: Elapsed Time: 0.704799 seconds (Warm-up)
## Chain 1: 2.58736 seconds (Sampling)
## Chain 1: 3.29216 seconds (Total)
## Chain 1:
##
## SAMPLING FOR MODEL 'fd9804091450c044661531e89123d1dc' NOW (CHAIN 2).
## Chain 2:
## Chain 2: Gradient evaluation took 1.8e-05 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.18 seconds.
## Chain 2: Adjust your expectations accordingly!
## Chain 2:
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## Chain 2:
## Chain 2: Elapsed Time: 0.648008 seconds (Warm-up)
## Chain 2: 2.85026 seconds (Sampling)
## Chain 2: 3.49826 seconds (Total)
## Chain 2:
Then we compare both models, by ranking them with their WAIC or using ELPD:
(waic1 <- waic(bayes.brms)) # waic model w/ tempterature
##
## Computed from 8000 by 200 log-likelihood matrix
##
## Estimate SE
## elpd_waic -668.2 9.8
## p_waic 11.0 1.1
## waic 1336.3 19.7
(waic2 <- waic(bayes.brms2)) # waic model wo/ tempterature
##
## Computed from 8000 by 200 log-likelihood matrix
##
## Estimate SE
## elpd_waic -702.4 13.3
## p_waic 12.3 1.2
## waic 1404.9 26.5
##
## 2 (1.0%) p_waic estimates greater than 0.4. We recommend trying loo instead.
loo_compare(waic1, waic2)
## elpd_diff se_diff
## bayes.brms 0.0 0.0
## bayes.brms2 -34.3 8.9
Conclusions
In all examples, you can check that i) there is little or no difference between Jags and brms numeric summaries of posterior distributions, and ii) maximum likelihood parameter estimates (glm(), lmer() and glmer() functions) and posterior means/medians are very close to each other.
In contrast to Jags, Nimble or Stan, you do not need to code the likelihood yourself in brms, as long as it is available in the package. You can have a look to a list of distributions with ?brms::brmsfamily, and there is also a possibility to define custom distributions with custom_family().
Note I haven’t covered diagnostics of convergence, more
here. You can also integrate brms outputs in a
tidy workflow with tidybayes, and use the ggplot() magic.
As usual, the code is on GitHub, visit https://github.com/oliviergimenez/fit-glmm-with-brms.
Now I guess it’s up to the students to pick their favorite Bayesian tool.
