Skip to main content

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

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

Stype Potential outcomes of S
S T( ω) S ed ( ω)
  S 11( ω) S 01( ω) S 10( ω) S 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., S ed (ω) ≥ S e ' d '(ω) for ω whenever e ≥ e ' and d ≥ d '), these response types are excluded.
  2. 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.
  3. 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.