Web-based Supplementary Materials for “Improving efficiency of inferences on treatment effect in two-arm randomized trials using auxiliary covariates”

At the end of Section 2 of the main paper, the semiparametric model framework within which we derive the proposed methods is stated, and is repeated here for convenience. The data from a clinical trial are denoted by (Yi, Xi, Zi), i = 1, . . . , n, assumed iid across i, where Y denotes the response of interest; X is a vector of baseline auxiliary covariates; and Z = 1, . . . , k depending on to which of k possible treatment groups subject i was randomized, with randomization probabilities P (Z = g) = πg, g = 1, . . . , k, ∑k g=1 πg = 1. Randomization guarantees that Z⊥⊥X. We assume that interest focuses on a parameter β involved in characterizing treatment comparisons, defined in the context of a model for the conditional density of Y given Z, pY |Z(y|z; θ, η), where θ = (β T , γ ) . Here, γ represents a finite-dimensional vector of possible additional parameters, and η is a finiteor infinite-dimensional nuisance parameter required to describe fully the class of models under consideration. Examples of such models are discussed in Section 2 of the main paper. The semiparametric model introduced in Section 2 of the main paper consists of all joint densities pY,X,Z(y, x, z ; θ, η, ψ, π) = pY,X|Z(y, x | z; θ, η, ψ)pZ(z;π) such that the conditional densities for (Y,X) given Z, denoted pY,X|Z(y, x | z; θ, η, ψ), satisfy