Web - based Supplementary Materials for “ Quantification of the Dose - Response Relationship for a Continuous Treatment in the Presence of Confounding or Informative Non - Compliance

where u ∼ N(0, G) and 2 ∼ N(0, R) with u and 2 independent. This model can be interpreted as Y |β, u ∼ N(Xβ+Zu,R), u ∼ N(0, G), yielding (on integrating out u) the marginal model Y |β ∼ N(Xβ,ZGZ + R). Let V = ZGZ + R. Then the maximum penalized likelihood estimates of β and u given G and R are given by θ̂ = [ β̂ û ] = (CTR−1C + B)−1CTR−1y (A.2) where C = [X Z] and B is the block diagonal matrix with blocks 0 and G−1. The variance of the estimators are given by Var[θ̂] = (CTR−1C + B)−1. Fitted values are obtained routinely as