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Table 2 Enumeration of 16 response types for outcome D and corresponding potential outcomes

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

Dtype Potential outcomes of D
D T( ω) D ce ( ω)
  D 11( ω) D 01( ω) D 10( ω) D 00( ω)
1 1 1 1 1
2 b, c 1 1 1 0
3 a, b, c 1 1 0 1
4 1 1 0 0
5 a, b, c 1 0 1 1
6 1 0 1 0
7 a, b, c 1 0 0 1
8 b 1 0 0 0
9 a, b, c 0 1 1 1
10 a, b, c 0 1 1 0
11 a 0 1 0 1
12 a, b 0 1 0 0
13 a 0 0 1 1
14 a, b 0 0 1 0
15 a, b 0 0 0 1
16 0 0 0 0
  1. a Under the assumption of positive monotonicity (i.e., D ce (ω) ≥ D c ' e '(ω) for ω whenever c ≥ c ' and e ≥ e '), these response types are excluded.
  2. b Given no interaction at the individual level on the additive scale between C and E in the counterfactual framework (i.e., D 11(ω) − D 01(ω) − D 10(ω) + D 00(ω) = 0 for ω), these response types are excluded.
  3. c Given no interaction at the individual level on the multiplicative scale between C and E in the counterfactual framework (i.e., D 11(ω)D 00(ω) = D 01(ω)D 10(ω) for ω), these response types are excluded.