A counterfactual approach to bias and effect modification in terms of response types

Table 3 Enumeration of 16 response types for selection variable S and corresponding potential outcomes

^a Under the assumption of positive monotonicity (i.e., S _ed(ω) ≥ S _{e ' d '}(ω) for ∀ω whenever e ≥ e ' and d ≥ d '), these response types are excluded.
^b Given no interaction at the individual level on the additive scale between E and D in the counterfactual framework (i.e., S ₁₁(ω) − S ₀₁(ω) − S ₁₀(ω) + S ₀₀(ω) = 0 for ∀ω), these response types are excluded.
^c Given no interaction at the individual level on the multiplicative scale between E and D in the counterfactual framework (i.e., S ₁₁(ω)S ₀₀(ω) = S ₀₁(ω)S ₁₀(ω) for ∀ω), these response types are excluded.

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