Table 2.

Consequence of the sequential ignorability and no interaction assumptions on parameters within the CA framework.

	Assumption
	1. α2 = α3 = 0	2. β3 = β5 = 0
	No unmeasured confounders for S	No unmeasured confounders for T
ρs	0	$\frac{{\alpha }_2({\alpha }_2+{\alpha }_3)}{\sqrt{({\alpha }_2^2+{\delta }_{S0}^2)({({\alpha }_2+{\alpha }_3)}^2+{\delta }_{S1}^2)}}$
ρ00	$\frac{{\beta }_2{\delta }_{S0}^2}{\sqrt{{\delta }_{S0}^2({\beta }_3^2+{\beta }_2^2{\delta }_{S0}^2+{\delta }_{T0}^2)}}$	$\frac{{\beta }_2({\alpha }_2^2+{\delta }_{S0}^2)}{\sqrt{({\alpha }_2^2+{\delta }_{S0}^2)({({\beta }_2{\alpha }_2)}^2+{\beta }_2^2{\delta }_{S0}^2+{\delta }_{T0}^2)}}$
ρ01	0	$\frac{{\alpha }_2({\beta }_2+{\beta }_4)({\alpha }_2+{\alpha }_3)}{\sqrt{({\alpha }_2^2+{\delta }_{S0}^2)({[({\beta }_2+{\beta }_4)({\alpha }_2+{\alpha }_3)]}^2+{({\beta }_2+{\beta }_4)}^2{\delta }_{S1}^2+{\delta }_{T1}^2)}}$
ρ10	0	$\frac{({\alpha }_2+{\alpha }_3){\beta }_2{\alpha }_2}{\sqrt{({({\alpha }_2+{\alpha }_3)}^2+{\delta }_{S1}^2)({({\beta }_2{\alpha }_2)}^2+{\beta }_2^2{\delta }_{S0}^2+{\delta }_{T0}^2)}}$
ρ11	$\frac{({\beta }_2+{\beta }_4){\delta }_{S1}^2}{\sqrt{{\delta }_{S1}^2({({\beta }_3+{\beta }_5)}^2+{({\beta }_2+{\beta }_4)}^2{\delta }_{S1}^2+{\delta }_{T1}^2)}}$	$\frac{({\beta }_2+{\beta }_4)({({\alpha }_2+{\alpha }_3)}^2+{\delta }_{S1}^2)}{\sqrt{({({\alpha }_2+{\alpha }_3)}^2+{\delta }_{S1}^2)({[({\beta }_2+{\beta }_4)({\alpha }_2+{\alpha }_3)]}^2+{({\beta }_2+{\beta }_4)}^2{\delta }_{S1}^2+{\delta }_{T1}^2)}}$
ρt	$\frac{{\beta }_3({\beta }_3+{\beta }_5)}{\sqrt{({\beta }_3^2+{\beta }_2^2{\delta }_{S0}^2+{\delta }_{T0}^2)({({\beta }_3+{\beta }_5)}^2+{({\beta }_2+{\beta }_4)}^2{\delta }_{S1}^2+{\delta }_{T1}^2)}}$	$\frac{{\beta }_2{\alpha }_2({\beta }_2+{\beta }_4)({\alpha }_2+{\alpha }_3)}{\sqrt{({({\beta }_2{\alpha }_2)}^2+{\beta }_2^2{\delta }_{S0}^2+{\delta }_{T0}^2)({[({\beta }_2+{\beta }_4)({\alpha }_2+{\alpha }_3)]}^2+{({\beta }_2+{\beta }_4)}^2{\delta }_{S1}^2+{\delta }_{T1}^2)}}$
γ0	$({\beta }_1+{\beta }_4({\alpha }_0+{\alpha }_1))-{\alpha }_1{\beta }_4\left(\frac{{\delta }_{S1}^2}{{\delta }_{S0}^2+{\delta }_{S1}^2}\right)$	$({\beta }_1+{\beta }_4({\alpha }_0+{\alpha }_1))-{\alpha }_1{\beta }_4\left(\frac{{\alpha }_3^2+{\alpha }_3{\alpha }_2+{\delta }_{S1}^2}{{\alpha }_3^2+{\delta }_{S0}^2+{\delta }_{S1}^2}\right)$
γ1	${\beta }_2+{\beta }_4\left(\frac{{\delta }_{S1}^2}{{\delta }_{S0}^2+{\delta }_{S1}^2}\right)$	${\beta }_2+{\beta }_4\left(\frac{{\alpha }_3^2+{\alpha }_3{\alpha }_2+{\delta }_{S1}^2}{{\alpha }_3^2+{\delta }_{S0}^2+{\delta }_{S1}^2}\right)$
	Assumption
	3. β4 = α2 = α3 = 0	4. β4 = β3 = β5 = 0
	No interaction, no unmeasured confounders for S	No interaction, no unmeasured confounders for T
ρs	0	$\frac{{\alpha }_2({\alpha }_2+{\alpha }_3)}{\sqrt{({\alpha }_2^2+{\delta }_{S0}^2)({({\alpha }_2+{\alpha }_3)}^2+{\delta }_{S1}^2)}}$
ρ00	$\frac{{\beta }_2{\delta }_{S0}^2}{\sqrt{{\delta }_{S0}^2({\beta }_3^2+{\beta }_2^2{\delta }_{S0}^2+{\delta }_{T0}^2)}}$	$\frac{{\beta }_2({\alpha }_2^2+{\delta }_{S0}^2)}{\sqrt{({\alpha }_2^2+{\delta }_{S0}^2)({({\beta }_2{\alpha }_2)}^2+{\beta }_2^2{\delta }_{S0}^2+{\delta }_{T0}^2)}}$
ρ01	0	$\frac{{\alpha }_2{\beta }_2({\alpha }_2+{\alpha }_3)}{\sqrt{({\alpha }_2^2+{\delta }_{S0}^2)({[{\beta }_2({\alpha }_2+{\alpha }_3)]}^2+{\beta }_2^2{\delta }_{S1}^2+{\delta }_{T1}^2)}}$
ρ10	0	$\frac{({\alpha }_2+{\alpha }_3){\beta }_2{\alpha }_2}{\sqrt{({({\alpha }_2+{\alpha }_3)}^2+{\delta }_{S1}^2)({({\beta }_2{\alpha }_2)}^2+{\beta }_2^2{\delta }_{S0}^2+{\delta }_{T0}^2)}}$
ρ11	$\frac{{\beta }_2{\delta }_{S1}^2}{\sqrt{{\delta }_{S1}^2({({\beta }_3+{\beta }_5)}^2+{\beta }_2^2{\delta }_{S1}^2+{\delta }_{T1}^2)}}$	$\frac{{\beta }_2({({\alpha }_2+{\alpha }_3)}^2+{\delta }_{S1}^2)}{\sqrt{({({\alpha }_2+{\alpha }_3)}^2+{\delta }_{S1}^2)({[{\beta }_2({\alpha }_2+{\alpha }_3)]}^2+{\beta }_2^2{\delta }_{S1}^2+{\delta }_{T1}^2)}}$
ρt	$\frac{{\beta }_3({\beta }_3+{\beta }_5)}{\sqrt{({\beta }_3^2+{\beta }_2^2{\delta }_{S0}^2+{\delta }_{T0}^2)({({\beta }_3+{\beta }_5)}^2+{\beta }_2^2{\delta }_{S1}^2+{\delta }_{T1}^2)}}$	$\frac{{\alpha }_2{\beta }_2^2({\alpha }_2+{\alpha }_3)}{\sqrt{({({\beta }_2{\alpha }_2)}^2+{\beta }_2^2{\delta }_{S0}^2+{\delta }_{T0}^2)({[{\beta }_2({\alpha }_2+{\alpha }_3)]}^2+{\beta }_2^2{\delta }_{S1}^2+{\delta }_{T1}^2)}}$
γ0	β1	β1
γ1	β2	β2