Table 1.

Overview of (non-parametric) identification results for case-control studies without matching

Sampling scheme	Estimand	Assumptions	Identification strategy
Case-base	Risk ratio for intention-to-treat effect $\frac{Pr\left(Y_K\right(1)=1)}{Pr\left(Y_K\right(0)=1)}$	∙Control selection S independent of baseline covariates L0 and exposure A0 ∙Consistency ∙Baseline exchangeability given L0 ∙Positivity (Theorem 1, Supplementary Appendix B)	1. Derive time-fixed IP weights W from control data 2. Compute the baseline exposure odds among cases, weighted by W 3. Compute the baseline exposure odds among controls, weighted by W 4. Take the ratio of the results of steps 2 and 3
Survivor	Odds ratio for intention-to-treat effect $\frac{\text{Odds}\left(Y_K\right(1)=1|L_0)}{\text{Odds}\left(Y_K\right(0)=1|L_0)}$	∙Control selection S independent of baseline exposure A0 given baseline covariates L0 and survival until tK (YK=0) ∙Consistency ∙Baseline exchangeability given L0 ∙Positivity (Theorem 3, Supplementary Appendix B)	1. Derive the conditional baseline exposure odds given L0 among cases 2. Derive the conditional baseline exposure odds given L0 among controls 3. Take the ratio of the results of steps 1 and 2
Risk-set	Hazard ratio for intention-to-treat effect $\frac{Pr\left(Y_{k+1}\right(1)=1|Y_k\left(1\right)=0)}{Pr\left(Y_{k+1}\right(0)=1|Y_k\left(0\right)=0)}$	∙Control selection Sk independent of baseline covariates L0 and exposure A0 given eligibility at tk (Yk=0) with constant sampling probability among those eligible † ∙Consistency ∙Baseline exchangeability given L0 ∙Positivity ∙Constant counterfactual hazards (Theorem 4, Supplementary Appendix B)	1. Derive time-fixed IP weights W from control data 2. Compute baseline exposure odds among cases, weighted by W 3. Compute baseline exposure odds among controls, weighted by W times $\sum _{k=0}^{K-1}{S}_k$, the number of times selected as a control 4. Take the ratio of the results of steps 2 and 3
	Hazard ratio for per-protocol effect $\frac{Pr\left(Y_{k+1}\right(\overline{1})=1|Y_k\left(\overline{1}\right)=0)}{Pr\left(Y_{k+1}\right(\overline{0})=1|Y_k\left(\overline{0}\right)=0)}$	∙Control selection Sk independent of covariate and exposure history up to tk given eligibility at tk (Yk=0) with constant sampling probability among those eligible † ∙Consistency ∙Sequential conditional exchangeability ∙Positivity ∙Constant counterfactual hazards (Theorem 6, Supplementary Appendix B)	1. Derive time-varying IP weights Wk from control data 2. Censor from time of protocol deviation 3. Compute (baseline) exposure odds among cases, weighted by those weights Wk such that Yk=0 and Yk+1=1 4. Compute (baseline) exposure odds among all controls, weighted by $\sum _{k=0}^{K-1}{W}_k{S}_k$, the weighted number of times selected as a control 5. Take the ratio of the results of steps 3 and 4

See text or Supplementary material for elaboration on assumptions. ^†Weaker/alternative control selection assumptions are given in the Supplementary material