Comparing two groups

Comparing two groups

This section is based on chapter 8 of Hoff, P. D. (2009).
We can use the following R code to approximate the posterior distribution
p(μ,δ,σ2y1,y2)p(\mu, \delta, \sigma^2\mid \mathbf y_1,\mathbf y_2)
.
#!/usr/bin/env Rscript
## R program for Comparing two groups
## get data
## you can found it on Peter Hoff's website
## https://www.stat.washington.edu/~pdhoff/book.php
Y.school.mathscore<-dget("Y.school.mathscore")
y.school1<-dget("y.school1")
y.school2<-dget("y.school2")
y1 = y.school1; n1 = length(y1)
y2 = y.school2; n2 = length(y2)
## prior parameters
mu0 = 50; g02 = 625
del0 = 0; t02 = 625
s20 = 100; nu0 = 1
## starting value
mu = (mean(y1) + mean(y2)) / 2
del = (mean(y1) - mean(y2)) / 2
## gibbs sampler
MU = DEL = S2 = NULL
set.seed(1)
for (s in 1:5000)
{
# update s2
s2 = 1/rgamma(1, (nu0+n1+n2)/2, (nu0*s20+sum((y1-mu-del)^2)+sum((y2-mu+del)^2))/2)
# update mu
var.mu = 1/(1/g02 + (n1+n2)/s2)
mean.mu = var.mu*(mu0/g02+sum(y1-del)/s2+sum(y2+del)/s2)
mu = rnorm(1, mean.mu, sqrt(var.mu))
# update del
var.del = 1/(1/t02 + (n1+n2)/s2)
mean.del = var.del*(del0/t02 + sum(y1-mu)/s2 - sum(y2 - mu)/s2)
del = rnorm(1, mean.del, sqrt(var.del))
# save parameter values
MU = c(MU, mu)
DEL = c(DEL, del)
S2 = c(S2, s2)
}
# plot
png("reproduce-fig-8-2l.png")
plot(density(MU), main="", xlab=expression(mu), lwd=2, col="black")
lines(density(rnorm(5000, 50, 25)), col="red", lwd=2)
legend("topleft", c("prior", "posterior"), col=c("black","red"), lwd=2)
dev.off()
png("reproduce-fig-8-2r.png")
plot(density(DEL), main="", xlab=expression(delta), lwd=2, col="black")
lines(density(rnorm(5000, 0, 25)), col="red", lwd=2)
legend("topleft", c("prior", "posterior"), col=c("black","red"), lwd=2)
dev.off()
We can reproduce the figures as show in Fig. 8.2.

Comparing multiple groups

Heterogeneity across group means

posterior factorization
full conditional distributions
We can use the following R code to implement this Gibbs sampling.
## comparing multiple groups
Y = Y.school.mathscore
## weakly informative priors
nu0 = 1; s20 = 100
eta0 = 1; t20 = 100
mu0 = 50; g20 = 25
## starting values
m = length(unique(Y[, 1]))
n = sv = ybar = rep(NA, m)
for (j in 1:m)
{
ybar[j] = mean(Y[Y[, 1]==j, 2])
sv[j] = var(Y[Y[, 1]==j, 2])
n[j] = sum(Y[, 1]==j)
}
theta = ybar
sigma2 = mean(sv)
mu = mean(theta)
tau2 = var(theta)
## setup MCMC
set.seed(1)
S = 5000
THETA = matrix(nrow = S, ncol = m)
SMT = matrix(nrow = S, ncol = 3)
## MCMC algorithm
for (s in 1:S)
{
# sample new values of the thetas
for (j in 1:m)
{
vtheta = 1/(n[j]/sigma2+1/tau2)
etheta = vtheta*(ybar[j]*n[j]/sigma2+mu/tau2)
theta[j] = rnorm(1, etheta, sqrt(vtheta))
}
# sample new value of sigma2
nun = nu0 + sum(n)
ss = nu0*s20
for (j in 1:m)
{
ss = ss + sum(Y[Y[,1]==j, 2]-theta[j])^2
}
sigma2 = 1/rgamma(1, nun/2, ss/2)
# sample new value of mu
vmu = 1/(m/tau2+1/g20)
emu = vmu*(m*mean(theta)/tau2 + mu0/g20)
mu = rnorm(1, emu, sqrt(vmu))
# sample a new value of tau2
etam = eta0 + m
ss = eta0*t20 + sum((theta-mu)^2)
tau2 = 1/rgamma(1, etam/2, ss/2)
# store results
THETA[s, ] = theta
SMT[s, ] = c(sigma2, mu, tau2)
}

Heterogeneity across group means and variances

Similarly, we can get the posterior factorization and full conditional distributions.
We can use the following Julia code to implement this Gibbs sampler.
## Julia program for Hierarchical Modeling of means and variances
## author: weiya <[email protected]>
## date: 27 August, 2018
using Distributions
using SpecialFunctions
using StatsBase
using DelimitedFiles
function higibbs(Y, T, mu0 = 50.0, gamma20 = 25.0, nu0 = 1.0, sigma20 = 100.0, eta0 = 1.0, tau20 = 100.0, a = 1.0, b = 1/100.0, alpha = 1.0, NUMAX = 40)
m = size(unique(Y[:,1]), 1)
# starting value
ybar = ones(m)
sv = ones(m)
n = ones(m)
for j = 1:m
yj = Y[ [Y[i,1] == j for i = 1:end], 2]
ybar[j] = mean(yj)
sv[j] = var(yj)
n[j] = size(yj, 1)
end
theta = ybar
sigma2 = copy(sv)
# sigma20 = 1 / mean(sigma2)
# nu0 = 2 * mean(sigma2)^2 / var(sigma2)
mu = mean(theta)
tau2 = var(theta)
THETA = ones(T, m)
SIGMA2 = ones(T, m)
# mu tau2 sigma20 nu0
MTSN = ones(T, 4)
for t = 1:T
# sample mu
varmu = 1 / (m / tau2 + 1 / gamma20)
meanmu = varmu * (m * mean(theta) / tau2 + mu0 / gamma20)
rnorm = Normal(meanmu, sqrt(varmu))
mu = rand(rnorm, 1)[1]
# sample 1/tau2
shapetau = (eta0 + m) / 2
ratetau = ( eta0 * tau20 + sum((theta .- mu).^2) ) / 2
rgamma = Gamma(shapetau, 1/ratetau)
tau2 = 1 / rand(rgamma, 1)[1]
# sample theta
for j = 1:m
vartheta = 1 / (n[j] / sigma2[j] + 1 / tau2)
meantheta = vartheta * ( n[j] * mean(Y[ [Y[i,1] == j for i = 1:end], 2]) / sigma2[j] + mu / tau2)
rnorm = Normal(meantheta, sqrt(vartheta))
theta[j] = rand(rnorm, 1)[1]
end
THETA[t, :] .= theta
# sample sigma2
for j = 1:m
shapesig = (nu0 + n[j])/2
yj = Y[ [Y[i,1] == j for i = 1:end], 2]
ratesig = ( nu0*sigma20 + sum( (yj .- theta[j]).^2 ) )/2
rgamma = Gamma(shapesig, 1/ratesig)
sigma2[j] = 1 / rand(rgamma, 1)[1]
end
SIGMA2[t, :] .= sigma2
# sample sigma20
shapesig = a + 0.5 * m * nu0
ratesig = b + 0.5 * nu0 * sum(1 ./ sigma2)
rgamma = Gamma(shapesig, 1/ratesig)
sigma20 = rand(rgamma, 1)[1]
# sample nu0
x = 1:NUMAX
lpnu0 = ones(NUMAX)
lpnu0 .= m * ( .5 * x .* log.(sigma20 * x / 2) .- lgamma.(x/2) ) .+ (x / 2 .+ 1) * sum(log.(1 ./ sigma2)) .- x .* (alpha + .5 * sigma20 * sum(1 ./ sigma2))
#println(lpnu0)
nu0 = sample(x, pweights(exp.(lpnu0 .- maximum(lpnu0))))
# println(pweights(exp.(lpnu0 .- maximum(lpnu0))))
# store results
MTSN[t, :] .= [mu, tau2, sigma20, nu0]
end
return THETA, SIGMA2, MTSN, sv, n
end
# run
Y = readdlm("math-score-Y.csv")
THETA, SIGMA2, MTSN, sv, n = higibbs(Y, 5000)
using PyPlot
# histogram
plt[:subplot](221)
plt[:hist](MTSN[:,2])
ylabel(L"$\muquot;)
plt[:subplot](222)
plt[:hist](MTSN[:,2])
ylabel(L"$\tau^2quot;)
plt[:subplot](223)
plt[:hist](MTSN[:,3])
ylabel(L"$\nu_0quot;)
plt[:subplot](224)
plt[:hist](MTSN[:,4])
ylabel(L"$\sigma_0^2quot;)
plt[:tight_layout]()
show()
# shrinkage
f, (ax1, ax2) = plt[:subplots](1, 2)
ax1[:scatter](sv, SIGMA2[end,:])
ax1[:plot](sv, sv)
ax1[:set_xlabel](L"$s^2quot;)
ax1[:set_ylabel](L"$\hat \sigma^2quot;)
ax2[:scatter](n, sv-SIGMA2[end,:])
ax2[:plot](n, zeros(size(n, 1)))
ax2[:set_xlabel]("sample size")
ax2[:set_ylabel](L"$s^2-\hat \sigma^2quot;)
plt[:tight_layout]()
show()
And the results are:
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