Table 1.

Comparisons between assumptions made regarding nonidentifiable associations in the proposed Bayesian approach and those implicit in the previously-used sequential methods. Assumptions common to both methods are: SUTVA, ignorable treatment assignment, equal individual clinical risk, and multivariate normality of {S(0), S(1), X}.

	Sequential Approach
Association	Imputation Stage	Risk Model Stage
Constant Biomarker
Y(0), Y(1)|X, S(0), S(1)	Not Applicable	Y(0) ⫫ Y(1)|X, S(0), S(1)
Y(0), X|S(1)	Not Applicable	No interaction, β02 = β12
S(1), Y(0)|X	S(1) ⫫ Y(0)|X	Possible dependence, β01 ≠ 0

	Bayesian Approach
Association	Imputation Stage	Risk Model Stage
Constant Biomarker
Y(0), Y(1)|X, S(0), S(1)	Y(0) ⫫ Y(1)|X, S(0), S(1)
Y(0), X|S(1)	Prior on β02 – β12
S(1), Y(0)|X	Prior on β01 – β11
No Constant Biomarker
Y(0), Y (1)|X, S(0), S(1)	Y (0) ⫫ Y(1)|X, S(0), S(1)
S(0), S(1)|X	Sensitivity parameter, φ
Y(1), S(0)|X	Prior on β11 – β01 and β12 – β02
Y(1), X|S(0)	Prior on β13 – β03
Y(0), S(1)|X	Prior on β01 and β02
Y(0), X|S(1)	Prior on β03