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Consider a situation with two treatments, the first of which is randomizedbut the second is not, and the multifactor version of this. Interest is intreatment effects, defined using standard factorial notation. We defineestimators for the treatment effects and explore their properties when there isinformation about the nonrandomized treatment assignment and when there is noinformation on the assignment of the nonrandomized treatment. We show when andhow hidden treatments can bias estimators and inflate their sampling variances.

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