# Longitudinal model over time
# Emax time-course function
# Fixed treatment effects for ET50 with treatment effects constrained to a single mean effect
# Fixed treatment effects for Emax
# Reference treatment effect for ET50 exchangeable across studies (Equation [12] in manuscript)
# Multivariate likelihood (compound symmetry covariance structure) to account for correlation between time points
# Correlation coefficient (rho) estimated within model - deviance / residual deviance not calculated as saturated model is not uniquely defined

model{ 			# Begin Model Code
 
for(i in 1:NS){ # Run Through all NS trials
    
	mu.Emax[i] ~ dnorm(0,0.0001)
	mu.ET50[i] ~ dnorm(m.mu,tau.mu)	# Exchangeable baseline effect for ET50 (Equation [12] in manuscript)
	delta.Emax[i,1] <- 0
	delta.ET50[i,1] <- 0
	
    for (k in 1:narm[i]){
	
		# Generates covariance matrix upper and lower triangles - compound symmetry covariance structure
		for (c in 1:(fups[i]-1)) {
			for (r in (c+1):fups[i]) {
			
				cov.mat[i,k,r,c] <- se[i,k,c]*se[i,k,r]*rho   # Lower triangle
				cov.mat[i,k,c,r] <- se[i,k,c]*se[i,k,r]*rho   # Upper triangle

			}
		}

		# Generates covariance matrix diagonals
		for (m in 1:fups[i]) {      
			cov.mat[i,k,m,m] <- pow(se[i,k,m],2)
		}

		omega[i,k,1:fups[i], 1:fups[i]] <- inverse(cov.mat[i,k,1:fups[i],1:fups[i]])	# Inverse covariance matrix
		y[i,k,1:fups[i]] ~ dmnorm(theta[i,k,1:fups[i]], omega[i,k,1:fups[i],1:fups[i]])	# Multivariate normal likelihood

	
		for (m in 1:fups[i]) {	# Run through all observations within a study	
			theta[i,k,m] <- exp(l.E0[i,k]) + ((Emax[i,k] * time[i,m]) / (exp(l.ET50[i,k]) + time[i,m]))	# Time-course function
		}
		
		# Relative effects for time-course parameters
		Emax[i,k] <- mu.Emax[i] + delta.Emax[i,k]
		l.ET50[i,k] <- mu.ET50[i] + delta.ET50[i,k]
		
		l.E0[i,k] ~ dnorm(0,0.0001)	# priors for baseline nuisance parameter
		
	}
	
	# Consistency equations
	for (k in 2:narm[i]){
		delta.Emax[i,k] <- d.Emax[treat[i,k]] - d.Emax[treat[i,1]]
		delta.ET50[i,k] <- d.ET50[treat[i,k]] - d.ET50[treat[i,1]]
		}
		
	}
	
	d.Emax[1] <- 0
	d.ET50[1] <- 0
		
	for (k in 2:NT){
		d.Emax[k] ~ dnorm(0,0.001)	# Priors on relative treatment effects	
		d.ET50[k] <- m.ET50	# ET50 treatment effects constrained to a single mean treatment effect
		}

	m.ET50 ~ dnorm(0,0.001) # Prior	for mean ET50 treatment effect
	
	m.mu ~ dnorm(0,0.001)	# Prior for mean reference treatment ET50
	
	# Between-study SD for reference treatment ET50
	tau.mu <- pow(sd.mu, -2)
	sd.mu ~ dunif(0,5)
	
	rho ~ dunif(-0.1,1)	# Prior for correlation between time points - assumed to be positive

	# Model ends
}