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