Bayesian mixed model regression with a between subjects factor

Question

I'm trying to specify a model in JAGS/rjags with one between subjects factor (a, with two levels - Y, N) interacting with one repeated measures continuous variable x plus subject varying slopes and intercepts that correlate. I can specify this model simply enough with the lmer function:

lmer(y ~ a + x + a:x + (1 + a | id))

My JAGS/rjags is very rusty (or very fresh). The below seems to me to be fitting a model with subject varying intercepts and subject varying slopes while estimating the slope for both levels of a, but I'm not sure I'm doing what I think I'm doing. There's also no correlation specified between the two.

modelstring = "
model {
  for ( i in 1:Ntotal ) {
    y[i] ~ dnorm( mu[i] , tau )
    mu[i] <- a1[aLvl[i]] + s1[sLvl[i]] + a2[aLvl[i]] * x[i] + s2[sLvl[i]] * x[i]
  }
  # Prior:
  tau <- pow( sigma , -2 )
  sigma ~ dunif(0,1000)
  for ( j in 1:2 ) { 
      a1[j] ~ dnorm( 0.0 , aTau ) 
      a2[j] ~ dnorm( 0.0 , aTau ) 
  }
  aTau <- 1 / pow( aSD , 2 )
  aSD <- abs( aSDunabs ) + .1
  aSDunabs ~ dt( 0 , 1.0E-7 , 2 )
  #
  for ( j in 1:NsLvl ) { 
    s1[j] ~ dnorm( 0.0 , sTau ) 
    s2[j] ~ dnorm( 0.0 , sTau )
  }
  sTau <- 1 / pow( sSD , 2 )
  sSD <- abs( sSDunabs ) + .1
  sSDunabs ~ dt( 0 , 1.0E-7 , 2 )
}
"

The framework for this comes from Kruschke and this has been of some help too. I would appreciate some pointers or links to examples of similar analyses.

Answer 1

I eventually figured this one out with much help from Doing Bayesian Data Analysis (Kruschke) and Data Analysis Using Regression and Multilevel/Hierarchical Models (Gelman). This model gives varying intercepts and slopes and the correlation between them.

y = dependent variable
sLvl = participant id at each data point
aLvlx = between subjects factor for each id
NaLvl = Number of levels for the between subject's factor
Ntotal = total length of data in long form

modelstring = "
model {
for( r in 1 : Ntotal ) {
    y[r] ~ dnorm( mu[r] , tau )
    mu[r] <- b0[ sLvl[r] ] + b1[ sLvl[r] ] * x[r]
}
#General priors
tau ~ dgamma( sG , rG )
sG <- pow(m,2)/pow(d,2)
rG <- m/pow(d,2)
m ~ dgamma(1, 0.001)
d ~ dgamma(1, 0.001)
#Subject level priors
for ( s in 1 : NsLvl ) {
    b0[s] <- B[s,1]
    b1[s] <- B[s,2]
    B[s, 1:2] ~ dmnorm( B.hat[s, ], Tau.B[ , ] )
    B.hat[s,1] <- hix1[aLvlx[s]]
    B.hat[s,2] <- hix2[aLvlx[s]]        
}
Tau.B[1:2 , 1:2] <- inverse(Sigma.B[,])
Sigma.B[1,1] <- pow(tau0G, 2)
Sigma.B[2,2] <- pow(tau1G, 2)
Sigma.B[1,2] <- rho * tau0G * tau1G
Sigma.B[2,1] <- Sigma.B[1,2]

tau0G ~ dunif(0.001,100)
tau1G ~ dunif(0.001,100)
rho ~ dunif(-1,1)
#Between subjects level priors
for ( k in 1:NaLvl ) { 
    hix1[k] ~ dnorm( 0 , 0.0000001 )
    hix2[k] ~ dnorm( 0 , 0.0000001 ) 
}
}"
writeLines(modelstring,con="model.txt")