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Table 1 Potential Response Type Characteristics Under Monotonicity Assumption (18 Response Types)

From: A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation

    a b c d    
  Response of Y to fixing X to value: Response of Y to fixing X and Z to values: Contributes to:
Potential Response
Type Representation
X = 1 X = 0 X = 1
Z = 0
X = 0
Z = 0
X = 1
Z = 1
X = 0
Z = 1
Total
Effect
Direct Effect in
(Z-stratum)
Indirect Effect in
(Z-stratum)
{111} 1 1 1 1 1 1    
{141} 1 1 0 0 1 1    
{211} 1 1 1 1 1 1    
{122} 1 0 1 0 1 0 + + (0,1)  
{241} 1 0 0 0 1 1 +   + (0,1)
{222} 1 0 1 0 1 0 + + (0,1)  
{411} 1 1 1 1 1 1    
{422} 1 0 1 0 1 0 + + (0,1)  
{144} 0 0 0 0 0 0    
{244} 0 0 0 0 0 0    
{441} 0 0 0 0 1 1    
{444} 0 0 0 0 0 0    
{121}* 1 1 1 0 1 1   + (0)  
{221}* 1 0 1 0 1 1 + + (0) + (1)
{421}* 1 0 1 0 1 1 + + (0)  
{142}* 1 0 0 0 1 0 + + (1)  
{242}* 1 0 0 0 1 0 + + (1) + (0)
{442}* 0 0 0 0 1 0   + (1)  
  1. * Unit-level interaction (interdependence) present because (a-b) ≠ (c-d)
  2. Potential response type representation indices are: 1 = "doomed", 2 = "causal" and 4 = "immune"
  3. Index i of the {ijk}representation specifies the Z[X = x] response, index j specifies the Y[X = x; Z = 0] response (columns a and b), and index k specifies the Y[X = x; Z = 1] response (columns c and d).