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12_bonus_mundlak.r
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12_bonus_mundlak.r
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# endogenous group confound example
set.seed(8672)
N_groups <- 30
N_id <- 200
a0 <- (-2)
bZY <- (-0.5)
g <- sample(1:N_groups,size=N_id,replace=TRUE) # sample into groups
Ug <- rnorm(N_groups,1.5) # group confounds
X <- rnorm(N_id, Ug[g] ) # individual varying trait
Z <- rnorm(N_groups) # group varying trait (observed)
Y <- rbern(N_id, p=inv_logit( a0 + X + Ug[g] + bZY*Z[g] ) )
table(g)
# confounded by correlation
precis(glm(Y~X+Z[g],family=binomial),2)
# fixed effects
# X deconfounded, but Z unidentified now!
precis(glm(Y~X+Z[g]+as.factor(g),family=binomial),pars=c("X","Z"),2)
dat <- list(Y=Y,X=X,g=g,Ng=N_groups,Z=Z)
# naive model
m0 <- ulam(
alist(
Y ~ bernoulli(p),
logit(p) <- a + bxy*X + bzy*Z[g],
a ~ dnorm(0,10),
c(bxy,bzy) ~ dnorm(0,1)
) , data=dat , chains=4 , cores=4 )
# fixed effects
mf <- ulam(
alist(
Y ~ bernoulli(p),
logit(p) <- a[g] + bxy*X + bzy*Z[g],
a[g] ~ dnorm(0,10),
c(bxy,bzy) ~ dnorm(0,1)
) , data=dat , chains=4 , cores=4 )
# random effects
mr <- ulam(
alist(
Y ~ bernoulli(p),
logit(p) <- a[g] + bxy*X + bzy*Z[g],
transpars> vector[Ng]:a <<- abar + z*tau,
z[g] ~ dnorm(0,1),
c(bxy,bzy) ~ dnorm(0,1),
abar ~ dnorm(0,1),
tau ~ dexp(1)
) , data=dat , chains=4 , cores=4 , sample=TRUE )
# random effects + Xbar
# The Mundlak Machine
xbar <- sapply( 1:N_groups , function(j) mean(X[g==j]) )
dat$Xbar <- xbar
mrx <- ulam(
alist(
Y ~ bernoulli(p),
logit(p) <- a[g] + bxy*X + bzy*Z[g] + buy*Xbar[g],
transpars> vector[Ng]:a <<- abar + z*tau,
z[g] ~ dnorm(0,1),
c(bxy,buy,bzy) ~ dnorm(0,1),
abar ~ dnorm(0,1),
tau ~ dexp(1)
) , data=dat , chains=4 , cores=4 , sample=TRUE )
# random effects + latent U
# The Latent Mundlak Machine
mru <- ulam(
alist(
# Y model
Y ~ bernoulli(p),
logit(p) <- a[g] + bxy*X + bzy*Z[g] + buy*u[g],
transpars> vector[Ng]:a <<- abar + z*tau,
# X model
X ~ normal(mu,sigma),
mu <- aX + bux*u[g],
vector[Ng]:u ~ normal(0,1),
# priors
z[g] ~ dnorm(0,1),
c(aX,bxy,buy,bzy) ~ dnorm(0,1),
bux ~ dexp(1),
abar ~ dnorm(0,1),
tau ~ dexp(1),
sigma ~ dexp(1)
) , data=dat , chains=4 , cores=4 , sample=TRUE )
precis(mf)
precis(mr)
precis(mrx)
precis(mru)
# density plots
# bxy
post <- extract.samples(mf)
dens(post$bxy,lwd=3,col=1,xlab="b_XY",ylim=c(0,2))
abline(v=1,lty=2)
post <- extract.samples(m0)
dens(post$bxy,lwd=3,col=grau(),add=TRUE)
post <- extract.samples(mr)
dens(post$bxy,lwd=3,col=2,add=TRUE)
post <- extract.samples(mrx)
dens(post$bxy,lwd=3,col=4,add=TRUE)
post <- extract.samples(mru)
dens(post$bxy,lwd=8,col="white",add=TRUE)
dens(post$bxy,lwd=4,col=3,add=TRUE)
# bzy
post <- extract.samples(mf)
dens(post$bzy,lwd=3,col=1,xlab="b_ZY",ylim=c(0,2))
abline(v=-0.5,lty=2)
post <- extract.samples(m0)
dens(post$bzy,lwd=3,col=grau(),add=TRUE)
post <- extract.samples(mr)
dens(post$bzy,lwd=3,col=2,add=TRUE)
post <- extract.samples(mrx)
dens(post$bzy,lwd=3,col=4,add=TRUE)
post <- extract.samples(mru)
dens(post$bzy,lwd=8,col="white",add=TRUE)
dens(post$bzy,lwd=4,col=3,add=TRUE)
##########
# show better estimates of intercepts
af <- coef(mf)[1:N_groups]
ar <- coef(mr)[1:N_groups]
plot( af , col=4 )
points( 1:N_groups , ar , col=2 )
points( 1:N_groups , a0+Ug , col=1 )
# treatment effect in each group now
# counterfactual increase of X at individual level, stratified by each group
# fixed estimates
pf0 <- link(mf,data=list(g=1:N_groups,X=rep(0,N_groups)))
pf1 <- link(mf,data=list(g=1:N_groups,X=rep(1,N_groups)))
cf <- apply( pf1 - pf0 , 2 , mean )
# random estimates
pr0 <- link(mr,data=list(g=1:N_groups,X=rep(0,N_groups)))
pr1 <- link(mr,data=list(g=1:N_groups,X=rep(1,N_groups)))
cr <- apply( pr1 - pr0 , 2 , mean )
# true
ctrue <- inv_logit( a0 + Ug + 1 ) - inv_logit( a0 + Ug )
plot( ctrue , ylim=c(0,0.3) )
points( 1:N_groups , cf , col=4 )
points( 1:N_groups , cr , col=2 )
plot( cf - ctrue , col=4 , ylim=c(-0.1,0.1) )
points( 1:N_groups , cr - ctrue , col=2 )
abline(h=0,lty=2)
mean((cf - ctrue)^2)
mean((cr - ctrue)^2)