Download presentation
Presentation is loading. Please wait.
1
Sensitivity Analysis Without Assumptions
almost Sensitivity Analysis Without Assumptions Hein Stigum courses
2
Agenda Motivating example Sensitivity (confounding) Bounding factor
E-value Stata: episens (confounding) (Ding and VanderWeele 2016) Feb-19 HS
3
Highlights U E D Guess at confounder strength, get bounds on true RR
RREU RRUD E D RRtrue RRobs Assume 𝑅𝑅 𝐸𝑈 = 𝑅𝑅 𝑈𝐷 , to explain away observed RR, confounding must be ≥𝑅𝑅+ 𝑅𝑅(𝑅𝑅−1) E-value Feb-19 HS
4
Smoking and lung cancer 1958
Hammond: RRobs=10.7 Fisher: confounding by gene Cornfield: RREU>RRobs Schlesselman: RRUD>RRobs gene U Conclusion: Confounding is unlikely to explain away a RR of 10.7 >10.7 >10.7 smoke E lung cancer D 10.7 Feb-19 HS
5
Sensitivity analysis Names Sensitivity towards uncontrolled: Methods
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) Feb-19 HS
6
Sensitivity Analysis Without Assumptions
Ding and VanderWeele, 2016 Sensitivity Analysis Without Assumptions Feb-19 HS
7
Strength of confounder
Variable type: Observed: 𝑅𝑅 𝑜𝑏𝑠 general, cat U True: 𝑅𝑅 𝑡𝑟𝑢𝑒 RREU RRUD 𝑅𝑅 𝐸𝑈 = max 𝑘 𝑃(𝑈= 𝑘|𝐸=1) 𝑃(𝑈= 𝑘|𝐸=0) E D RRtrue general, bi RRobs general, bi 𝑅𝑅 𝑈𝐷|𝐸=0 = max 𝑘 𝑃(𝐷=1|𝐸=0, 𝑈=𝑘) min 𝑘 𝑃(𝐷=1|𝐸=0, 𝑈=𝑘) (RREU, RRUD) represent the max strength of the confounder U (after adjusting for observed confounders) 𝑅𝑅 𝑈𝐷|𝐸=1 = “ -- 𝑅𝑅 𝑈𝐷 =max( 𝑅𝑅 𝑈𝐷|𝐸=0 , 𝑅𝑅 𝑈𝐷|𝐸=1 ) (Ding and VanderWeele 2016) Feb-19 HS
8
E-U association U2 U E D RREU measures the E-U association,
as the effect of E on U RREU Sensitivity for binary U: prevalence U exposed E D RREU= prevalence U unexposed OREU=ORUE RREU≈OREU=ORUE≈RRUE If U is rare If E is rare Feb-19 HS
9
Bounding factor Strength of confounder (RREU, RRUD)
Inequality 𝑅𝑅 𝑡𝑟𝑢𝑒 ≥ 𝑅𝑅 𝑜𝑏𝑠 /𝑏𝑓 Use in different ways: Guess RREU, RRUD ,calculate bf and the “corrected” RRtrue How large must RREU and RRUD be to “explain away“ 𝑅𝑅 𝑜𝑏𝑠 Assume RREU=RRUD, how large must they be to “explain away“ 𝑅𝑅 𝑜𝑏𝑠 Feb-19 HS
10
Bounding factor, use 1 U E D Guess at (RREU, RRUD)
Calculate 𝑏𝑓= 𝑅𝑅 𝐸𝑈 ∙𝑅𝑅 𝑈𝐷 𝑅𝑅 𝐸𝑈 + 𝑅𝑅 𝑈𝐷 −1 Calculate 𝑅𝑅 𝑡𝑟𝑢𝑒 ≥ 𝑅𝑅 𝑜𝑏𝑠 /𝑏𝑓 SES U RR CI Observed 2.0 (1.5, 2.7) 3 2 bf=3*2/(3+2-1) E D RRtrue>1.3 smoke CVD “True”≥ (1.0, 1.8) RRobs=2.0 Feb-19 HS
11
Bounding factor table If the confounder strength is RRU=2 and 3, the bounding factor =1.5 Feb-19 HS
12
Bounding factor, use 2 U E D
Rearange: 𝑏𝑓≥ 𝑅𝑅 𝑜𝑏𝑠 / 𝑅𝑅 𝑡𝑟𝑢𝑒 “Explain away” 𝑅𝑅 𝑡𝑟𝑢𝑒 =1 𝑏𝑓≥ 𝑅𝑅 𝑜𝑏𝑠 /1 Look up in table 𝑏𝑓= 𝑅𝑅 𝑜𝑏𝑠 SES U RR CI Observed 2.0 (1.5, 2.7) 4 ? 3 ? Lookup 2.0: (RREU, RRUD)=(4,3) E D RRtrue=1 smoke CVD RRobs=2.0 Feb-19 HS
13
Bounding factor table To explain away a RR=2.0 we need confounder strength of RRU=4 and 3 Feb-19 HS
14
Simplified bounding factor, use 3
If RREU and RRUD have the same magnitude, then to explain away and observed effect=RR: 𝑅𝑅 𝐸𝑈 = 𝑅𝑅 𝑈𝐷 ≥𝑅𝑅+ 𝑅𝑅(𝑅𝑅−1) other dis. Observed RR 3 2 1,5 1,3 Confounder 5.4 3.4 2.4 1.9 U >2.4 >2.4 E D 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. Feb-19 HS
15
P-value and e-value 𝐸−𝑣𝑎𝑙𝑢𝑒=𝑅𝑅+ 𝑅𝑅(𝑅𝑅−1)
𝐸−𝑣𝑎𝑙𝑢𝑒=𝑅𝑅+ 𝑅𝑅(𝑅𝑅−1) p-value association robust against random error e-value causal conclusion robust against unmeasured confounding Observed RR=1.5, p=0.03 𝐸−𝑣𝑎𝑙𝑢𝑒=𝑅𝑅+ 𝑅𝑅 𝑅𝑅−1 = −1 =2.4 A confounder of strength 2.4 (on both sides) could completely explain away an observed RR=1.5 but a weaker confounder could not. Feb-19 HS
16
Smoking and lung cancer 1958
Hammond: RR=10.7 (8.0,14.4) Cornfield: RREU>10.7, RRUD>10.7 New bounds 10.7: 𝑅𝑅 𝐸𝑈 = 𝑅𝑅 𝑈𝐷 ≥𝑅𝑅+ 𝑅𝑅 𝑅𝑅−1 =21.1 New bounds 8.0: 𝑅𝑅 𝐸𝑈 = 𝑅𝑅 𝑈𝐷 ≥𝑅𝑅+ 𝑅𝑅 𝑅𝑅−1 =15.5 gene U Conclusion: Confounding is unlikely to explain away a RR of 10.7 >10.7 >21.1 >10.7 >21.1 smoke E lung cancer D 10.7 Feb-19 HS
17
Summing up Sensitivity without assumptions
New bounding factor for the effect of unmeasured confounder 𝑏𝑓= 𝑅𝑅 𝐸𝑈 ∙𝑅𝑅 𝑈𝐷 𝑅𝑅 𝐸𝑈 + 𝑅𝑅 𝑈𝐷 −1 Any type: binary, categorical, cont., multiple Max strength of the confounder 𝐸−𝑣𝑎𝑙𝑢𝑒 =𝑅𝑅+ 𝑅𝑅(𝑅𝑅−1) shows robustness of a causal conclusion on RR against unmeasured confounding Feb-19 HS
18
Sensitivity analysis with assumptions
Feb-19 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 Feb-19 HS
20
Stata: RR from 2 by 2 table csi cohort study, immediate Feb-19 HS
21
Stata: episens C E D Install Run Result:
ssc install episens, replace /* Statistical Software Components */ help episens Run episensi , st(cs) dpunexp(c(0.1)) dpexp(c(0.2)) drrcd(c(3)) 2*2 table cohort study prevalence unexposed =10% prevalence exposed =20% RRcd=3 RRec=0.20/0.10=2 C Result: RRec=2 RRcd=3 E D RRtrue=1.65 RRobs=2.0 (Orsini, Bellocco et al. 2008) Feb-19 HS
22
Bounding factor (bf): Effect of prevalence of U (or C): episens bf
Feb-19 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 Feb-19 HS
24
The effect of unmeasured confounding is
Conclusion 𝐸−𝑣𝑎𝑙𝑢𝑒 =𝑅𝑅+ 𝑅𝑅(𝑅𝑅−1) The effect of unmeasured confounding is not as bad as you might think! Feb-19 HS
25
References Ding P, VanderWeele TJ. Sensitivity Analysis Without Assumptions. Epidemiology. 2016;27(3): Orsini N, Bellocco R, Bottai M, Wolk A, Greenland S. A tool for deterministic and probabilistic sensitivity analysis of epidemiologic studies. Stata Journal. 2008;8(1):29-48. Feb-19 HS
26
Extra material Feb-19 HS
27
Relation to Cornfield conditions
From earlier 𝑅𝑅 𝑡𝑟𝑢𝑒 ≥ 𝑅𝑅 𝑜𝑏𝑠 /𝑏𝑓 For confounding to change RRobs to RR 𝑏𝑓≥ 𝑅𝑅 𝑜𝑏𝑠 / 𝑅𝑅 For confounding to change RRobs to 1 𝑏𝑓≥ 𝑅𝑅 𝑜𝑏𝑠 lim 𝑏𝑓 = lim 𝑅𝑅 𝐸𝑈 →∞ 𝑅𝑅 𝐸𝑈 ∙𝑅𝑅 𝑈𝐷 𝑅𝑅 𝐸𝑈 + 𝑅𝑅 𝑈𝐷 −1 = 𝑅𝑅 𝑈𝐷 l`Hopitals rule Similar for RREU Recover the Cornfield conditions in the limit The new bounding factor shows joint conditions for RREU and RRUD Feb-19 HS
28
Bounding factor, use U E D
Guess at RRUD “Explain away” 𝑅𝑅 𝑡𝑟𝑢𝑒 =1 𝑏𝑓≥ 𝑅𝑅 𝑜𝑏𝑠 /1 Calculate RREU≥ 𝑅𝑅 𝑜𝑏𝑠 𝑅𝑅 𝑈𝐷 −1 𝑅𝑅 𝑈𝐷 − 𝑅𝑅 𝑜𝑏𝑠 U RR CI Observed 2.0 (1.5, 2.7) 4 ? 3 RREU≥2*(3-1)/(3-2) 4 E D RRtrue=1 RRobs=2.0 Feb-19 HS
Similar presentations
© 2025 SlidePlayer.com Inc.
All rights reserved.