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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).