1. October 15H.S.1 Causal inference Hein Stigum Presentation, data and programs at: http://folk.uio.no/heins/talks

2. October 15H.S.2 Contents Background –Error –Bias Define causal effect –Individual –Average Identify causal effect –Exchangeability –Positivity –Consistency

3. Background October 15H.S.3

4. Oct-15H.S.4 Error Random error Source: sampling Expressed as: –p-values –Confidence intervals (precision) Affect –All measures Systematic error Source: design Expressed as bias: 1.Selection bias 2.Information bias 3.Confounding Affect: –Frequency measure –Association measure Causality field: Strong focus on bias at the expense of precision

5. Oct-15H.S.5 Precision and Bias True value Estimate Precision Bias Causal effect Association Precision Bias Bias: association  causal effect Objective: find effects

6. Define Causal Effects October 15H.S.6

7. Individual causal effect Counterfactual outcome Important –Clear definition –Notation  mathematical proofs –Notation  new methods Estimate individual effect? –No, but Crossover design October 15H.S.7 TreatedNot treatedIndividual causal effect ZeusDiedLivedYes HeraLived No

8. October 15H.S.8 Individual causal effects 20 subjects 12 individual causal effects

9. Average causal effect Counterfactual outcome Estimate average effect? –Yes, Randomized controlled trial October 15H.S.9 TreatedNot treatedAverage causal effect Population10/20=0.5 No

10. Identify Causal Effects October 15H.S.10

11. Ideal Randomized Trial Trial –Randomize, Treat, Compare outcomes Features –Exchangeability Comparable groups –Positivity Both treated and untreated –Consistency Well defined intervention and contrast October 15H.S.11 ED C

12. Conditional Randomized Trial Conditional Trial –By sex: Randomize, Treat, Compare outcomes Features –Conditional Exchangeability Comparable groups by sex –Conditional Positivity Both treated and untreated by sex –Consistency Well defined intervention and contrast October 15H.S.12 ED C sex ED C ED C Males Females

13. Observational study Make = conditionally randomized trial Need Features –Conditional Exchangeability Comparable groups by all values of C –Conditional Positivity Both treated and untreated by all values of C –Consistency Well defined intervention and contrast October 15H.S.13 ED C

14. Conditional exchangeability Need to measure all relevant factors October 15H.S.14 Conditional exchangeability = No unmeasured confounding ED C ED C Two ways to remove confounding: Adjust: Balance:

15. Weights –Estimate probability of exposed by C = p i Balance –Weight exposed by 1/ p i, for plot 100/p i –Weight unexposed by 1/(1- p i ), for plot 100/(1-p i ) Effect Balance by Inverse Probability Weights October 15H.S.15 ED C

16. IPW and plots October 15H.S.16 E overweight D Blood pressure C smoke - + Effect of E on D: Crude: 0biased Adjusted: 4 true Balance the data using IPW Result: all plots of D versus E are adjusted Problem: N gets large

17. Conditional positivity example Prior knowledge –Dose response is linear Positivity problem –Estimate dose response for each sex?

18. Conditional positivity October 15H.S.18 Conditional positivity = exposed and unexposed for all values of C Parametric assumption: linear “dose response” ED C

19. Consistency = Well defined intervention and contrast October 15H.S.19

20. Air pollution Excess mortality from air pollution? Standard method: estimate attributable fraction Implicit contrast: current levels versus zero Implicit intervention: not existent October 15H.S.20

21. Body Mass Index Excess mortality from obesity? Standard method: estimate attributable fraction Implicit contrast:  30 versus <25 Exercise Implicit intervention: Diet  Mortality Smoking October 15H.S.21

22. Poorly defined intervention may affect exchangeability Adjust for lung disease? October 15H.S.22 E exercise D mortality C lung disease Adjust E diet D mortality C lung disease Need not adjust E smoking D mortality C lung disease Should not adjust

23. Poorly defined intervention may affect positivity Confounder status unknown –Can not asses positivity October 15H.S.23

24. Summing up Defined bias –Objective: find effects Conditions to find effects –Exchangeability:comparable E+ and E- –Positivity:E+ and E- in all strata –Consistency:well defined intervention and contrast October 15H.S.24

25. October 15H.S.25 Litterature Hernan and Robins, Causal Inference