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

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

			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)

* Unit-level interaction (interdependence) present because (a-b) ≠ (c-d)
^† Potential response type representation indices are: 1 = "doomed", 2 = "causal" and 4 = "immune"
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).