Sensitivity Analysis Without Assumptions

1. almost Sensitivity Analysis Without Assumptions Hein Stigum http://folk.uio.no/heins/ courses

2. Agenda • Motivating example • Sensitivity (confounding) • Bounding factor • E-value • Stata: episens (confounding) (Ding and VanderWeele 2016) HS

3. Highlights • Guess at confounder strength, get bounds on true RR RREU RRUD U D E RRtrue RRobs • Assume , to explain away observed RR, confounding must be • E-value HS

4. Smoking and lung cancer 1958 • Hammond: RRobs=10.7 • Fisher: confounding by gene U D E • Cornfield: RREU>RRobs lung cancer smoke gene • Schlesselman: RRUD>RRobs Conclusion: Confounding is unlikely to explain away a RR of 10.7 >10.7 >10.7 10.7 HS

5. Sensitivity analysis • Names • Sensitivity analysis or bias analysis • Sensitivity towards uncontrolled: • Confounding (C) • Selection (S) • Measurement error (M) • Other violations • Methods • Simulation in own data some assumptions (S,M) • 2*2 table corrections some assumptions (C,S,M) • Sensitivity without assumptions (only C) HS

6. Ding and VanderWeele, 2016 Sensitivity Analysis Without Assumptions HS

7. Strength of confounder Variable type: general, cat Observed: True: RREU RRUD U D E RRtrue general, bi RRobs general, bi (RREU, RRUD) represent the max strength of the confounder U (after adjusting for observed confounders) -- “ -- , (Ding and VanderWeele 2016) HS

8. E-U association RREUmeasuresthe E-U association, as theeffectof E on U RREU Sensitivity for binary U: U2 U D E prevalence U exposed RREU= prevalence U unexposed OREU=ORUE RREU≈OREU=ORUE≈RRUE If E is rare If U is rare HS

9. Bounding factor • Strength of confounder (RREU, RRUD) • Bounding factor • Inequality • Use in different ways: • Guess RREU, RRUD ,calculate bf and the “corrected” RRtrue • How large must RREU and RRUD be to “explain away“ • Assume RREU=RRUD, how large must they be to “explain away“ HS

10. Bounding factor, use 1 • Guess at (RREU, RRUD) • Calculate • Calculate U D E SES RR CI Observed 2.0 (1.5, 2.7) 3 2 bf=3*2/(3+2-1) 1.5 1.5 RRtrue>1.3 smoke CVD “True”≥ 1.33 (1.0, 1.8) RRobs=2.0 HS

11. Bounding factor table If the confounder strength is RRU=2 and 3, the bounding factor =1.5 HS

12. Bounding factor, use 2 Rearange: “Explain away”=1 Look up in table U D E SES RR CI Observed 2.0 (1.5, 2.7) 4 3 ? ? Lookup 2.0: (RREU, RRUD)=(4,3) RRtrue=1 smoke CVD RRobs=2.0 HS

13. Bounding factor table To explain away a RR=2.0 we need confounder strength of RRU=4 and 3 HS

14. Simplified bounding factor, use 3 If RREU and RRUD have the same magnitude, then to explain away and observed effect=RR: U D E other dis. >2.4 >2.4 no alcohol hip fract. RRobs=1.5 A confounder of strength 2.4(on both sides) could completely explain away an observed RR=1.5 but a weaker confounder could not. HS

15. P-value and e-value p-value association robust against random error e-value causal conclusion robust against unmeasured confounding Observed RR=1.5, p=0.03 A confounder of strength 2.4(on both sides) could completely explain away an observed RR=1.5 but a weaker confounder could not. HS

16. Smoking and lung cancer 1958 Hammond: RR=10.7 (8.0,14.4) Cornfield: RREU>10.7, RRUD>10.7 U New bounds 10.7: D E lung cancer smoke gene New bounds 8.0: Conclusion: Confounding is unlikely to explain away a RR of 10.7 >21.1 >21.1 >10.7 >10.7 10.7 HS

17. Summing up Sensitivity without assumptions • New bounding factor for the effect of unmeasured confounder • Any type: binary, categorical, cont., multiple • Max strength of the confounder • shows robustness of a causal conclusion on RR against unmeasured confounding HS

18. Sensitivity analysis with assumptions HS

19. Motivation • The bounding factor (bf) and the e-value are based on worst case confounding (yet surprisingly useful) • If we know the confounder is rare, can we do better? • Use “episens” in Stata • Only binary confounder • Also selection bias and measurement error • Or use equations HS

20. Stata: RR from 2 by 2 table csi 50 225 450 4275 cohortstudy, immediate HS

21. Stata: episens • Install • sscinstall episens, replace/* Statistical Software Components */ • helpepisens • Run episensi 50 225 450 4275, st(cs) dpunexp(c(0.1)) dpexp(c(0.2)) drrcd(c(3)) C D E 2*2 table cohort study prevalence unexposed =10% prevalence exposed =20% RRcd=3 RRec=0.20/0.10=2 Result: RRec=2 RRcd=3 RRtrue=1.65 RRobs=2.0 (Orsini, Bellocco et al. 2008) HS

22. Bounding factor (bf): Effect of prevalence of U (or C): episens bf HS

23. Summing up Sensitivity with assumptions • Assumptions about the prevalence of the unmeasured (binary) confounder can lead to sharper bounds than the bf • The episens command can also be used for bias from selection and measurement error HS

24. Conclusion The effect of unmeasured confounding is not as bad as you might think! HS

25. References • Ding P, VanderWeele TJ. Sensitivity Analysis Without Assumptions. Epidemiology. 2016;27(3):368-77. • Orsini N, Bellocco R, Bottai M, Wolk A, Greenland S. A tool for deterministic and probabilisticsensitivityanalysisofepidemiologic studies. Stata Journal. 2008;8(1):29-48. HS

26. Extra material HS

27. Relation to Cornfield conditions From earlier For confounding to change RRobs to RR For confounding to change RRobs to 1 Similar for RREU Recover the Cornfield conditions in the limit The new bounding factor shows joint conditions for RREU and RRUD HS

28. Bounding factor, use Guess at RRUD “Explain away”=1 Calculate RREU U D E RR CI Observed 2.0 (1.5, 2.7) ? 4 3 RREU≥2*(3-1)/(3-2) 4 RRtrue=1 RRobs=2.0 HS