Causal inference from treatment-control studies having an additional factor with unknown assignment mechanism

TL;DR

This paper investigates the estimation of treatment effects in factorial studies with one randomized and one nonrandomized treatment, analyzing biases and variances under different levels of assignment information.

Contribution

It introduces estimators for treatment effects in multifactor studies with unknown assignment mechanisms and characterizes their properties and potential biases.

Findings

Hidden treatments can bias estimators.

Unknown assignment mechanisms inflate sampling variances.

Proper estimators can mitigate bias under certain conditions.

Abstract

Consider a situation with two treatments, the first of which is randomized but the second is not, and the multifactor version of this. Interest is in treatment effects, defined using standard factorial notation. We define estimators for the treatment effects and explore their properties when there is information about the nonrandomized treatment assignment and when there is no information on the assignment of the nonrandomized treatment. We show when and how hidden treatments can bias estimators and inflate their sampling variances.