1. 1 PH 240A: Chapter 8 Mark van der Laan University of California Berkeley (Slides by Nick Jewell)

2. 2 Counterfactual Pancreatic Cancer Responses to Coffee Drinking Conditions GroupCoffee (E ) No Coffee ( ) # 1DD 2D 3D 4

3. 3 Counterfactual Pancreatic Cancer Responses to Coffee Drinking Conditions GroupCoffee (E ) No Coffee ( ) # 1DD 2D 3D 4 Suppose Group 1 and 3 individuals are all coffee drinkers and Group 2 and 4 individuals abstain, then...

4. 4 Population Data Pancreatic Cancer DNot D Coffee Drinking (cups/day) (E) 0 (not E)0

5. 5 Counterfactual Pancreatic Cancer Responses to Coffee Drinking Conditions GroupCoffee (E ) No Coffee ( ) # 1DD 2D 3D 4 Suppose Group 1 and 2 individuals are all abstainers and Group 3 and 4 individuals drink coffee, then...

6. 6 Population Data Pancreatic Cancer DNot D Coffee Drinking (cups/day) (E)0 0 (not E)

7. 7 All is Lost?  Let’s quit—the first four weeks of the class have been a total waste of time...?

8. 8 Counterfactual Pancreatic Cancer Responses to Coffee Drinking Conditions GroupCoffee (E ) No Coffee ( ) # 1DD 2D 3D 4 Suppose individuals choose whether to drink coffee or not at random, (say, toss a coin) then...

9. 9 Population Data Under Random Counterfactual Observation Pancreatic Cancer DNot D Coffee Drinking (cups/day) (E) 0 (not E)

10. 10 Confounding Variables  Randomization assumption  Probability of observing a specific exposure condition (eg coffee drinking or not) must not depend on counterfactual outcome pattern (i.e. vary across groups)  Failure of randomization assumption  Group 1 individuals are more likely to be males than say Group 4 individuals  If males are also more likely to drink coffee, then we are more likely to observe the coffee drinking counterfactual in Group 1 than Group 4

11. 11 Confounding Variables  For example, imagine a world where all males, and only males, drank coffee Pancreatic Cancer DNot D Coffee Drinking (cups/day) males 0females Even if coffee had no effect we would observe an association (due to sex) sex is a confounder

12. 12 Confounding Variables  Conditions for confounding  C must cause D  C must cause E C E D ?

13. 13 Stratification to Control Confounding  Divide the population into strata defined by different levels of C  Within a fixed stratum there can be no confounding due to C  New issue: causal effects of E may vary across levels of C Interaction or effect modification  When no interaction, need methods to combine common causal effects across C strata (Chapter 9)

14. 14 Causal Graph Approach to Confounding  Possible causal effects of childhood vaccination on autism  access to general medical care may affect autism incidence and/or diagnosis  Access to medical care increases vaccination  Family SES influences access to medical care and also ability to pay for vaccination  Family medical history may affect risk for autism and may also influence access to medical care  Which of medical care access, SES, and family history are confounders? Do we need to stratify on all three?

15. 15 Directed Acyclic Graphs  Nodes, directed graphs (edges have direction)  Directed paths: B-A-D B-D-A, C-B-D B A D C

16. 16 Directed Acyclic Graphs  Acyclic  No loops: A cannot cause itself  Mother’s Smoking Status Child’s Respiratory Condition Mother’s Smoking Status (t=0) Mother’s Smoking Status (t=1) Child’s Respiratory Condition (t=0)  node A at the end of a directed path starting at B is a descendant of B (B is an ancestor of A)

17. 17 Directed Acyclic Graphs  A node A can be a collider on a specific pathway if the path entering and leaving A both have arrows pointing into A. A path is blocked if it contains a collider.  D is a collider on the pathway C-D-A-F-B; this path is blocked B F D C A

18. 18 Using Causal Graphs to Detect Confounding  Delete all arrows from E that point to any other node  Is there now any unblocked backdoor pathway from E to D?  Yes—confounding exists  No—no confounding

19. 19 Vaccination & Autism Example Medical Care Access Vaccination Autism Family History SES

20. 20 Using Causal Graphs to Detect Confounding FC DE FC DE FC DE FC DE

21. 21 Checking for Residual Confounding  After stratification on one or more factors, has confounding been removed?  Cannot simply remove stratification factors and relevant arrows and check residual DAG  Have to worry about colliders

22. 22 Controlling for Colliders  Stratification on a collider can induce an association that did not exist previously Rain SprinklerWet Pavement Diet sugar (B) Fluoridation (A)Tooth Decay (D)

23. 23 Hypothetical Data on Water Flouridation, High-Sugar Diet, and Tooth Decay Tooth Decay (D) FlouridationDORER AHigh-Sugar DietB16040 2.670.2 12080 High-Sugar DietB80120 2.670.2 40160 Tooth Decay (D) High-Sugar DietD BFluoridationA160406.000.4 80120 FluoridationA120806.000.4 40160 Pooled tableFluoridation A High-Sugar DietB200 1.000.00 200

24. 24 Hypothetical Data on Water Flouridation, High-Sugar Diet, and Tooth Decay Tooth Decay (D) Fluoridation AORER B160800.67-0.083 12040 No Tooth Decay ( ) Fluoridation AORER B401200.67-0.083 800160

25. 25 Checking for Residual Confounding  Delete all arrows from E that point to any other node  Add in new undirected edges for any pair of nodes that have a common descendant in the set of stratification factors S  Is there still any unblocked backdoor path from E to D that doesn’t pass through S ? If so there is still residual confounding, not accounted for by S.

26. 26 Vaccination & Autism Example Medical Care Access Vaccination Autism Family History SES

27. 27 Vaccination & Autism Example: Stratification on Medical Care Access Vaccination Autism Family History SES Still confounding: need to stratify additionally on SES or Family History, or both

28. 28 Caution: Stratification Can Introduce Confounding! C E D F No Confounding Stratification on C introduces confounding!

29. 29 Collapsibility  No Confounding and No Interaction 500 1202804001486100134366 6034040079310067433500 Pooled

30. 30 Collapsibility  Confounding and No Interaction 820 2165047201486100230590 126880793100191611800 Pooled

31. 31 Collapsibility and Confounding  With the assumption of the causal graph below, and in the absence of interaction, the conditions for collapsibility wrt RR (and ER) are the same as for no confounding (i.i either C & E are independent, or C & E are independent, given E, or both) C E D ? But note that collapsibility cannot distinguish directions of the arrows C E D ?

32. 32 Collapsibility  No Confounding and Interaction 500 60340400148610074426 120280400793100127373500 Pooled has causal interpretation

33. 33 Collapsibility  Confounding and Interaction 820 1086127201486100122 69 8 24568079310031 14 9 180 Pooled has no causal interpretation

34. 34 Collapsibility and OR  Collapsibility and “No Confounding” not quite the same thing for OR