ANOVA – part 2
Petr Keil
January 2016
The aim of this lesson is to (1) leave the participants to come up with their code for simple one-way ANOVA, and (2) to experiment with random effects ANOVA.
1 The Data
We will use modified data from the example from Marc Kery’s Introduction to WinBUGS for Ecologists, page 119 (Chapter 9 - ANOVA). The data describe snout-vent lengths in 5 populations of Smooth snake (Coronella austriaca) (Uzovka hladka in CZ).
Loading the data from the web:
snakes <- read.csv("http://www.petrkeil.com/wp-content/uploads/2014/02/snakes.csv")
# we will artificially delete 9 data points in the first population
snakes <- snakes[-(1:9),]
summary(snakes)## population snout.vent
## Min. :1.000 Min. :36.56
## 1st Qu.:2.000 1st Qu.:43.02
## Median :3.000 Median :49.24
## Mean :3.439 Mean :50.07
## 3rd Qu.:4.000 3rd Qu.:57.60
## Max. :5.000 Max. :61.37Plotting the data:
par(mfrow=c(1,2))
plot(snout.vent ~ population, data=snakes,
ylab="Snout-vent length [cm]")
boxplot(snout.vent ~ population, data=snakes,
ylab="Snout-vent length [cm]",
xlab="population",
col="grey")
2 Fixed-effects ANOVA
For a given snake \(i\) in population \(j\) the model can be written as:
\(y_{ij} \sim Normal(\alpha_j, \sigma)\)
Here is how we prepare the data:
snake.data <- list(y=snakes$snout.vent,
x=snakes$population,
N=nrow(snakes),
N.pop=5)Loading the library that communicates with JAGS
library(R2jags)JAGS Model definition:
cat("
model
{
# priors
sigma ~ dunif(0,100)
tau <- 1/(sigma*sigma)
for(j in 1:N.pop)
{
alpha[j] ~ dnorm(0, 0.001)
}
# likelihood
for(i in 1:N)
{
y[i] ~ dnorm(alpha[x[i]], tau)
}
}
", file="fixed_anova.txt")And we will fit the model:
model.fit.fix <- jags(data=snake.data,
model.file="fixed_anova.txt",
parameters.to.save=c("alpha"),
n.chains=3,
n.iter=2000,
n.burnin=1000,
DIC=FALSE)## module glm loaded
## module dic loaded## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 41
## Unobserved stochastic nodes: 6
## Total graph size: 106
##
## Initializing modelplot(as.mcmc(model.fit.fix))
model.fit.fix## Inference for Bugs model at "fixed_anova.txt", fit using jags,
## 3 chains, each with 2000 iterations (first 1000 discarded)
## n.sims = 3000 iterations saved
## mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat n.eff
## alpha[1] 45.244 3.235 38.930 43.124 45.288 47.376 51.577 1.001 2000
## alpha[2] 41.238 1.017 39.257 40.587 41.241 41.912 43.236 1.001 3000
## alpha[3] 45.868 1.026 43.784 45.209 45.884 46.537 47.878 1.002 1700
## alpha[4] 54.409 1.057 52.311 53.693 54.424 55.137 56.476 1.001 3000
## alpha[5] 59.019 1.031 57.018 58.357 59.026 59.695 61.065 1.001 3000
##
## 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).3 Random-effects ANOVA
For a given snake \(i\) in population \(j\) the model can be written in a similar way as for the fixed-effects ANOVA above:
\(y_{ij} \sim Normal(\alpha_j, \sigma)\)
But now we will also add a random effect:
\(\alpha_j \sim Normal(\mu_{grand}, \sigma_{grand})\)
In short, a random effect means that the parameters itself come from (are outcomes of) a given distribution, here it is the Normal.
The data stay the same as in the fixed-effect example above.
Loading the library that communicates with JAGS
library(R2jags)JAGS Model definition:
cat("
model
{
# priors
grand.mean ~ dnorm(0, 0.001)
grand.sigma ~ dunif(0,100)
grand.tau <- 1/(grand.sigma*grand.sigma)
group.sigma ~ dunif(0, 100)
group.tau <- 1/(group.sigma*group.sigma)
for(j in 1:N.pop)
{
alpha[j] ~ dnorm(grand.mean, grand.tau)
}
# likelihood
for(i in 1:N)
{
y[i] ~ dnorm(alpha[x[i]], group.tau)
}
}
", file="random_anova.txt")And we will fit the model:
model.fit.rnd <- jags(data=snake.data,
model.file="random_anova.txt",
parameters.to.save=c("alpha"),
n.chains=3,
n.iter=2000,
n.burnin=1000,
DIC=FALSE)## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 41
## Unobserved stochastic nodes: 8
## Total graph size: 106
##
## Initializing modelplot(as.mcmc(model.fit.rnd))
model.fit.rnd## Inference for Bugs model at "random_anova.txt", fit using jags,
## 3 chains, each with 2000 iterations (first 1000 discarded)
## n.sims = 3000 iterations saved
## mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat n.eff
## alpha[1] 45.968 3.126 39.757 43.801 45.957 48.033 51.975 1.003 950
## alpha[2] 41.408 1.037 39.370 40.719 41.412 42.092 43.411 1.003 970
## alpha[3] 45.977 1.027 43.954 45.312 45.957 46.657 48.018 1.001 3000
## alpha[4] 54.436 1.022 52.459 53.746 54.454 55.098 56.449 1.001 3000
## alpha[5] 58.930 1.028 56.892 58.245 58.931 59.618 60.902 1.001 3000
##
## 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).4 Plotting the posteriors from both models
Let’s extract the medians posterior distributions of the expected values of \(\alpha_j\) and their 95% credible intervals:
rnd.alphas <- model.fit.rnd$BUGSoutput$summary
fix.alphas <- model.fit.fix$BUGSoutput$summary
plot(snout.vent ~ population, data=snakes,
ylab="Snout-vent length [cm]", col="grey", pch=19)
points(rnd.alphas[,'2.5%'], col="red", pch="-", cex=1.5)
points(fix.alphas[,'2.5%'], col="blue", pch="-", cex=1.5)
points(rnd.alphas[,'97.5%'], col="red", pch="-", cex=1.5)
points(fix.alphas[,'97.5%'], col="blue", pch="-", cex=1.5)
points(rnd.alphas[,'50%'], col="red", pch="+", cex=1.5)
points(fix.alphas[,'50%'], col="blue", pch="+", cex=1.5)
abline(h=mean(snakes$snout.vent), col="grey")
legend("bottomright", pch=c(19,19), col=c("blue","red"),
legend=c("classical","random effects"))
Note the shrinkage effect!
Also, how would you plot the grand.mean estimated in the random effects model? How would you extract the between- and within- group variances?