Table 1 Overview of (non-parametric) identification results for case-control studies without matching
From: Identification of causal effects in case-control studies
Sampling scheme | Estimand | Assumptions | Identification strategy |
|---|---|---|---|
Case-base | Risk ratio for intention-to-treat effect \(\frac {\Pr (Y_{K}(1)=1)}{\Pr (Y_{K}(0)=1)}\) | ∙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 \(\frac {\text {Odds}(Y_{K}(1)=1|L_{0})}{\text {Odds}(Y_{K}(0)=1|L_{0})}\) | ∙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 \(\frac {\Pr (Y_{k+1}(1)=1|Y_{k}(1)=0)}{\Pr (Y_{k+1}(0)=1|Y_{k}(0)=0)}\) | ∙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 \(\frac {\Pr (Y_{k+1}(\overline {1})=1|Y_{k}(\overline {1})=0)}{\Pr (Y_{k+1}(\overline {0})=1|Y_{k}(\overline {0})=0)}\) | ∙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 |