1. Oct-15H.S.1Oct-15H.S.1Oct-15H.S.1 Directed Acyclic Graphs DAGs Hein Stigum http://folk.uio.no/heins/

2. Oct-15H.S.2Oct-15H.S.2Oct-15H.S.2 Graphical Models Concepts –Confounder, Collider –Conditional independence –Undirected graphs, separation Directed Acyclic Graphs (DAGs) –Graphic tools: d-separation –Examples –Selection bias examples

3. Oct-15H.S.3Oct-15H.S.3Oct-15H.S.3 Concepts “Close to the edge”

4. Oct-15H.S.4Oct-15H.S.4Oct-15H.S.4 Confounder, Collider –C is a common cause of E and D –C is a common effect of E and D (bias if we condition on C) Oct-15H.S.4 ED C ED C Confounder Collider Condition on: adjust stratify Bias: Selection Information Confounding Confounder Collider Hernan et al, A structural approach to selection bias, Epidemiology 2004

5. Oct-15H.S.5Oct-15H.S.5 Collider idea Conditioning on a collider induces an association between the causes Condition = (restrict, stratify, adjust) Bias direction? Hip arthritis Limp Knee injury Two causes for limping - ++ Hip arthritis Limp Knee injury Select limping subjects ++

6. Oct-15H.S.6Oct-15H.S.6Oct-15H.S.6 Marginal and conditional dependence E and D are: Marginally dependent Conditionally Independent | C

7. Oct-15H.S.7Oct-15H.S.7Oct-15H.S.7 Undirected Graph, separation 1 23 45 Are 1 and 5 separated by 2 ? Are 1 and 5 separated by 3 ? Are 1 and 5 separated by 3 and 4 ? Yes If 1 and 5 are separated by 2, then 1 and 5 are conditionally independent given 2 Definition: “1” and “5” are separated by “2” if all paths from “1” to “5” pass thru “2” If 1 and 5 are separated by 2, path from 1 and 5 is blocked by 2 No Yes

8. Oct-15H.S.8Oct-15H.S.8Oct-15H.S.8 Graphic tools: d-separation “We have the moral edge!”

9. Oct-15H.S.9Oct-15H.S.9Oct-15H.S.9 No bias No E-D bias if E and D are conditionally independent under H 0, given the variables we adjust for (C) ED C ED C OK Not OK Need graphic tools: “d-separation”

10. Oct-15H.S.10Oct-15H.S.10Oct-15H.S.10 D-separation 1 24 35 Are 1 and 4 separated by 2 ?No Steps: 1.Take ancestral graph of {1,2,4} 2.Moralize the graph 3.Look for separation

11. Oct-15H.S.11Oct-15H.S.11Oct-15H.S.11 D-separation, cont 1 24 35 Are 1 and 4 separated by 2 ?No Take ancestral graph of {1,2,4} 1 24 3 Moralize the graph 1 24 3 Are 1 and 4 separated by 2 ?No

12. Oct-15H.S.12Oct-15H.S.12Oct-15H.S.12 Examples

13. Oct-15H.S.13 1.Ancestral graph of E,D,C 2.Moralize 3.Separated? 1.Ancestral graph of E,D 2.Moralize 3.Separated? Vitamine example Bias in E-D: E vitamine D birth defects C age U obesity E ╨ Dno E vitamine D birth defects C age U obesity Bias in E-D? Adjust for C? Adjust for C : E ╨ D|Cyes E vitamine D birth defects C age U obesity

14. Oct-15H.S.14 Conditions giving confounding E vitamine D birth defects C age U obesity E vitamine D birth defects C age U obesity Direct C-D effect? Path still blocked undirected graph! Unmeasured U1 leads to age and vitamine Not logical (age), path still blocked! Unmeasured U1 leads to obesity and vitamine New path via U1 and U, Bias! U1

15. Oct-15H.S.15Oct-15H.S.15Oct-15H.S.15 Selection bias examples

16. Oct-15H.S.16H.S. HCB and SGA SGA HCB SBreast SGA HCB S Breast Directed Acyclic Graph Moralized undirected graph Conclusion: Selection bias: HCB and SGA are not independent under selection. Adjusting for breastfeeding blocks the bias. Breastfeeding leads to higher participation SGA leads to lower participation Selection bias? Adjust for breastfeeding?

17. Oct-15H.S.17H.S. Conditions giving selection bias SGA HCB SBreast U1 SGA HCB S Breast HCB leads to higher participation? Unlikely! U1 U2 Unmeasured U1 leads to HCB and S No new path in UDG! Unmeasured U2 leads to SGA and S New path via U1 and U2, Bias!

18. Oct-15H.S.18 Cohort: Differential loss to follow up Loss to follow up: Severe symptoms Side effects of therapy Questions: Bias in the E-D effect? Adjust for symptoms? E therapy D AIDS S follow up U immuno sup V sympt 1. Take ancestral graph of {E,D,S} 2. Moralize 3. Separation by S? 4. Separation by S,V? No Yes E therapy D AIDS S follow up U immuno sup V sympt

19. Oct-15H.S.19 Self-selection bias Self-selection: Awareness of disease Awareness of smoke effect Bias: Biased in the sample May adjust for awareness E smoke D CHD S Self selection U family hist V awareness

20. Oct-15H.S.20 Berkson’s bias Selection: Cases from hospital Controls also from hospital Bias: Any cause of D2 will be associated with D in the sample E exposure D cases S hospital D2 controls ED S not D Proper control selection: Controls sampled independent of exposure Problem: E-D bias for linear models???

21. Oct-15H.S.21 Healthy worker effect Working: Good health (proven by test) Exposed with symp. leave work Bias: Negative bias in the sample E chemical D mortality S working U health V test E chemical D mortality S working U health V test + - - + + biased true Negative bias 0

22. Oct-15H.S.22 Exercise, Firefighters Show: No E-D bias in the pop. E ╨ D under H 0 E-D bias in the sample. E ╨ D|S under H 0 Adj for V does help. E ╨ D|S,V under H 0 Draw DAG: Study exercise on CHD among firefighters High socioeconomic status have less CHD and less firefighters Unmeasured personality have more exercise and more firefighters

23. Oct-15H.S.23 DAG No E-D bias in the pop. E ╨ D under H 0 E-D bias in the sample. E ╨ D|S under H 0 Adj for V does help. E ╨ D|S,V under H 0 E exercise D CHD S firefighter V socio stat. U personality E exercise D CHD V socio stat. U personality E exercise D CHD S firefighter V socio stat. U personality E exercise D CHD S firefighter V socio stat. U personality

24. Oct-15H.S.24 Inverse probability weighting Idea: Each subject not lost should weigh up for those lost (with the same E,V values) Method: In (E=0, V=1), 4 subject, 3 lost, 1 left: s=1/4, w=4 4 org. pop.  4 copies pseudo pop. Result Pseudo pop. measure is unbiased given the DAG, (w|V,E) More general than adjusting for V E therapy D AIDS S follow up U immuno sup V sympt

25. Oct-15H.S.25 Exercise, AIDS Show: No E-D bias in the pop. E ╨ D E-D bias in the sample. E ╨ D|S Adj for V does not help. E ╨ D|S,V E therapy D AIDS S follow up U immuno sup V sympt Inverse probability weighting would still work here (w|V)

26. Oct-15H.S.26H.S. Lifestyle, diet SGA HCB SLifestyle Diet Directed Acyclic Graph Moralized undirected graph SGA HCB SLifestyle Diet Conclusion: HCB and SGA are not independent under selection unless we condition (adjust) on lifestyle or diet