Table 2 Overview of (non-parametric) identification results for case-control studies with exact pair matching

Risk ratio for intention-to-treat effect \(\frac {\Pr (Y_{K}(1)=1)}{\Pr (Y_{K}(0)=1)}\)

∙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(Y_K=1|L₀=l,A₀=1)/Pr(Y_K=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

Odds ratio for intention-to-treat effect \(\frac {\text {Odds}(Y_{K}(1)=1|L_{0})}{\text {Odds}(Y_{K}(0)=1|L_{0})}\)

∙Matched control exposure A^′ sampled from all the baseline exposure levels of all survivors (Y_K=0) with same value for L₀ as case, independently of the subjects’ baseline exposure ∙Consistency ∙Baseline conditional exchangeability ∙Positivity ∙Odds(Y_K=1|L₀,A₀=1)/Odds(Y_K=1|L₀,A₀=0) constant across levels l (Theorem 8, Supplementary Appendix C)

(Same as identification strategy for case-base sampling)

Hazard ratio for intention-to-treat effect \(\frac {\Pr (Y_{k+1}(1)=1|L_{0},Y_{k}(1)=0)}{\Pr (Y_{k+1}(0)=1|L_{0},Y_{k}(0)=0)}\)

∙For a case with incident event in [t_k,t_k+1) (i.e., Y_k=0,Y_k+1=1), matched control exposure A^′ sampled from the baseline exposure levels of all subjects that are event-free at t_k (Y_k=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(Y_k+1=1|L₀=l,A₀=1,Y_k=0)/Pr(Y_k+1=1|L₀=l,A₀=0,Y_k=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 \(\frac {\Pr (Y_{k+1}(\overline {1})=1|L_{0},...,L_{k},A_{0}=...=A_{k}=1,Y_{k}(\overline {1})=0)}{\Pr (Y_{k+1}(\overline {0})=1|L_{0},...,L_{k},A_{0}=...=A_{k}=0,Y_{k}(\overline {0})=0)}\)

∙For a case with incident event in [t+k,t_k+1) (i.e., Y_k=0,Y_k+1=1), matched control exposure A^′ sampled from the baseline exposure levels A₀ of all individuals who adhered to one of the protocols until t_k (i.e., A₀=...=A_k) and have covariate history up to t_k. Sampling among these individuals is independent of baseline exposure or survival status ∙Consistency ∙Positivity ∙Pr(Y_k+1=1|L₀,...,L_k,A₀=...=A_k=1,Y_k=0)/Pr(Y_k+1=1|L₀,...,L_k,A₀=...=A_k=0,Y_k=0) constant across levels k and independent of L₀,...,L_k(Theorem 10, Supplementary Appendix C)

(Same as identification strategy for case-base sampling)