Table 2.

Overview of (non-parametric) identification results for case-control studies with exact pair matching

Sampling scheme	Estimand	Assumptions	Identification strategy
Case-base	Risk ratio for intention-to-treat effect $\frac{Pr\left(Y_K\right(1)=1)}{Pr\left(Y_K\right(0)=1)}$	∙Matched control exposure A′ sampled from the baseline exposure levels of all subjects with same baseline covariate level L0 as case, independently of the subjects’ baseline exposure or survival status ∙Consistency ∙Baseline conditional exchangeability ∙Positivity ∙Pr(YK=1|L0=l,A0=1)/Pr(YK=1|L0=l,A0=0) constant across levels l (Theorem 7, Supplementary Appendix C)	1. Compute the frequency of discordant case-control pairs with A0=1 and A′=02. Compute the frequency of discordant case-control pairs with A0=0 and A′=1 3. Take the ratio of the results of steps 1 and 2
Survivor	Odds ratio for intention-to-treat effect $\frac{\text{Odds}\left(Y_K\right(1)=1|L_0)}{\text{Odds}\left(Y_K\right(0)=1|L_0)}$	∙Matched control exposure A′ sampled from all the baseline exposure levels of all survivors (YK=0) with same value for L0 as case, independently of the subjects’ baseline exposure ∙Consistency ∙Baseline conditional exchangeability ∙Positivity ∙Odds(YK=1|L0,A0=1)/Odds(YK=1|L0,A0=0) constant across levels l (Theorem 8, Supplementary Appendix C)	(Same as identification strategy for case-base sampling)
Risk-set	Hazard ratio for intention-to-treat effect $\frac{Pr\left(Y_{k+1}\right(1)=1|L_0,Y_k\left(1\right)=0)}{Pr\left(Y_{k+1}\right(0)=1|L_0,Y_k\left(0\right)=0)}$	∙For a case with incident event in [tk,tk+1) (i.e., Yk=0,Yk+1=1), matched control exposure A′ sampled from the baseline exposure levels of all subjects that are event-free at tk (Yk=0) and have the same value for L0 as case. Sampling among these individuals is independent of baseline exposure or survival status ∙Consistency ∙Baseline conditional exchangeability ∙Positivity ∙Pr(Yk+1=1|L0=l,A0=1,Yk=0)/Pr(Yk+1=1|L0=l,A0=0,Yk=0) constant across levels k,l(Theorem 9, Supplementary Appendix C)	(Same as identification strategy for case-base sampling)
	Hazard ratio for per-protocol effect $\frac{Pr\left(Y_{k+1}\right(\overline{1})=1|L_0,\mathrm{...},L_k,A_0=\mathrm{...}=A_k=1,Y_k\left(\overline{1}\right)=0)}{Pr\left(Y_{k+1}\right(\overline{0})=1|L_0,\mathrm{...},L_k,A_0=\mathrm{...}=A_k=0,Y_k\left(\overline{0}\right)=0)}$	∙For a case with incident event in [t+k,tk+1) (i.e., Yk=0,Yk+1=1), matched control exposure A′ sampled from the baseline exposure levels A0 of all individuals who adhered to one of the protocols until tk (i.e., A0=...=Ak) and have covariate history up to tk. Sampling among these individuals is independent of baseline exposure or survival status ∙Consistency ∙Positivity ∙Pr(Yk+1=1|L0,...,Lk,A0=...=Ak=1,Yk=0)/Pr(Yk+1=1|L0,...,Lk,A0=...=Ak=0,Yk=0) constant across levels k and independent of L0,...,Lk(Theorem 10, Supplementary Appendix C)	(Same as identification strategy for case-base sampling)

See text or Supplementary material for elaboration on assumptions