Unified Approach for Causal Inference and Censored Data

Abstract

Let A(t) represent a time-dependent action process as observed on a randomly sampled subject. A general example can be A(t) =(A₁(t),A₂(t),A₃(t),A₄(t)), where A₁ (t) = I(C ≤ t) jumps to 1 when the subject is right-censored, A₂ (t) is a counting process that jumps when the subject is monitored, A₃(t) indicates which of the covariates are measured at time t, and A₄(t) denotes the treatment that the subject receives at time t. In other words, A(t) can denote both censoring/missingness and treatment actions. Let A be the set of possible sample paths of A(·). For technical reasons and simplicity, we will assume that A(t) only changes values at given time points t = 1, 2, ... , p and that A is finite. We note that this does not diminish the practical applicability of the methods presented in this chapter since one can approximate a continuous action process with an arbitrarily fine discrete approximation. In addition, we will present the formulas in such a way that they can be applied to continuous data as well, although we do not present regularity conditions and proofs