1. Are You Looking for the Right Interactions? A presentation given 2/28/2012 in the Biostatistics in Psychiatry seminar series at Columbia University by Sharon Schwartz Department of Epidemiology Mailman School of Public Health Columbia University

2. The Problem Interaction is Model Dependent

3. What You See Depends on How you Look at It Forms or word? Musician or Woman? Vases or Faces?

4. MOTIVATING EXAMPLE Tennent and Bebbington NO Do Intimacy problems interact with stressful life events to cause depression? Brown and Harris YES

5. Interaction (aka, Effect Modification) The effect of an exposure differs in the presence or absence of another variable (X)

6. MOTIVATING EXAMPLE Do Intimacy problems interact with stressful life events to cause depression? Stressful Life Events Intimacy Problems YESNO Yes32%10% No3%1% It depends on how you look at it Risk of disease in each cell is displayed

7. MOTIVATING EXAMPLE Is the effect of stressful life events different in the presence vs. absence of Intimacy Problems? Stressful Life Events Intimacy Problems YesNO Yes32%10% No3%1% RD = 32 - 3 = 29RD = 10 - 1 = 9 RR = 32/3 = 10RR = 10/1 = 10 YES NO

8. So who is right – Brown & Harris or Tennet & Bebbington? Is the effect of stressful life events different in the presence vs. absence of Intimacy Problems? Stressful Life Events Intimacy Problems YesNO Yes32%10% No3%1% RD = 32 - 3 = 29RD = 10 - 1 = 9 RR = 32/3 = 10RR = 10/1 = 10 YES NO

9. What is the Causal Question?: Are there some individuals who became depressed because they experienced both stressful life events and intimacy problems

10. Causal Question in Counterfactual Terms Are there some individuals who experienced stressful life and intimacy problems who became depressed, who would not have become depressed had they not experienced both stressful life events and intimacy problems?

11. Stressful Life Events Intimacy Problems Hypothesized Causes of Depression T The exposure has an effect when it is in the presence of its causal partners (here intimacy problems) J

12. Stressful Life Events Intimacy Problems Hypothesized Causes of Depression T J Stressful Life Events Genetic Variant R Intimacy Problems P

13. EFFECT MODIFICATION, INTERACTION AND SYNERGY: WHAT DO WE REALLY WANT TO KNOW? 13

14. 14 Synergy: Who are the Exposure’s Causal Partners?

15. WHAT DO WE MEAN BY INTERACTION ? CONCEPTUALLY: CAUSAL PARTNERS IN THE SAME CAUSAL PIE IMPLICATIONS FOR DISEASE RISK: RISK OF DISEASE FROM CAUSAL PARTNERS WHEN THEY CO-OCCUR IS GREATER THAN WOULD BE EXPECTED BASED ON THEIR INDEPENDENT EFFECTS ALONE 15

16. ROTHMAN’S CAUSAL MODEL SINGLE CAUSE OF INTEREST 16 BEGIN FROM YOUR CAUSAL MODEL

17. CAUSES OF DISEASES AND THE RELATIONSHIP TO “TYPES” EXPOSURE OF INTEREST: X CAUSES OF DISEASE OF INTEREST X A CD NOT X E 17

18. CAUSES OF DISEASES AND THE RELATIONSHIP TO “TYPES” EXPOSURE OF INTEREST: X CAUSES OF DISEASE OF INTEREST X A CD NOT X E “ TYPES” OF PEOPLE REGARDING DISEASE OF INTEREST GIVEN THAT EXPOSURE OF INTEREST IS X C D E NOT A, C&D, OR E SUSCEPTIBLEDOOMEDPROTECTIVE IMMUNE A 18

19. INTEREST IN EXPOSURE X: WHAT CAN WE KNOW ABOUT CAUSATION? TYPE PROPORTION DISEASE EXPERIENCE DISEASE EXPERIENCE IF EXPOSED IF UNEXPOSED 1: DOOMED D+ 2: A CAUSALD+ --- 3: A PROTECTIVE ---D+ 4: IMMUNE --- RISK IF EXPOSED = RISK IF UNEXPOSED = RISK DIFFERENCE = RISK RATIO = P1 P2 P3 P4 P1&P2 P1 & P3 P2 - P3 P1 & P2/ P1 & P3 19

20. BOTTOM LINE OF CAUSAL INFERENCE WITH SINGLE RISK FACTOR RR = P1 & P2/ Q1 & Q3 >1 MORE TYPE 2’S THAN 3’S (ASSUMING EXCHANGEABILITY) THERE ARE MORE PEOPLE FOR WHOM THE EXPOSURE IS CAUSAL THAN PEOPLE FOR WHOM THE EXPOSURE IS PROTECTIVE RR = P1 & P2/Q1 & Q3 = 1 TYPE 2’S = TYPE 3’S OR NO TYPE 2’S OR 3’S EITHER THE EXPOSURE HAS NO EFFECT OR THERE ARE EQUAL NUMBERS OF PEOPLE FOR WHOM IT IS CAUSAL AND FOR WHOM IT IS PROTECTIVE THERE ARE MORE PEOPLE FOR WHOM THE EXPOSURE IS PROTECTIVE THAN PEOPLE FOR WHOM THE EXPOSURE IS CAUSAL RR = P1& P2/Q1+Q3 < 1 MORE TYPE 3’S THAN 2’S NOTE: SINCE THIS IS A STUDY WE USED AN UNEXPOSED AS A COUNTERFACTUAL FOR THE EXPOSED - ERGO Q’S NOW ENTER 20

21. Using this Causal Model: How do we know if There is Synergy Out There? 21

22. IF WE ASSUME ALL ACTIVE EFFECTS ARE CAUSAL AND NONE ARE PROTECTIVE, THEN THERE ARE 6 POSSIBLE OUTCOMES FROM THE COMBINATION OF ANY TWO RISK FACTORS OF INTEREST DARROCH, ROTHMAN, GREENLAND USE THIS ASSUMPTION IN DEVELOPING THEIR MODEL 22

23. ASSESSING INTERACTION BETWEEN VARIABLES X AND Z B C D NONE CAUSETYPE DISEASE EXPERIENCE IF EXPOSED TO: XZ X&ZNEITHER X A Z A B X OR Z X Z E A B C DC D A B D+----D+--- D+ --- D+ --- D+ --- D+--- D =100% 6 - X susceptible 4 - Z susceptible 1 - doomed 16 - immune 2 - parallelism 8 - synergy E 23

24. WHEN TWO RISK FACTORS ARE PARTNERS IN THE SAME SUFFICIENT CAUSE BOTH ARE NECESSARY FOR THE COMPLETION OF THE CAUSAL PIE WHAT IS SYNERGY? (TYPE 8) X Z HERE X&Z ARE SYNERGISTIC FOR INDIVIDUALS WITH E E

25. WHAT IS PARALLELISM? (TYPE 2) WHEN INDIVIDUALS HAVE COMPONENTS TO COMPLETE TWO DIFFERENT PIES EACH WITH A RISK FACTOR UNDER STUDY X HERE X&Z ARE PARALLEL FOR INDIVIDUALS WITH A & B ZAB

26. T0 T End B = 1 Exposures of interest = X,Z John is exposed to X, Z John’s perfect proxies are Nhoj X, Z - John Nhoj X Did John get the disease? What caused John’s disease? What is the causal effect of z? What is the causal effect of x? X Z A B T2 Effect of XEffect of Z Nhoj Z Nhoj - B = 1 D D D -- A = 1 T1 A = 1

27. X Z A B Inevitable Parallelism Will occur with a probability of A*B

28. X Z A A “Functional Equivalence” Parallelism

29. ASSESSING INTERACTION BETWEEN VARIABLES X AND Z B C D NONE CAUSETYPE DISEASE EXPERIENCE IF EXPOSED TO: XZ X&ZNEITHER X A Z A B X OR Z X Z E A B C DC D A B E D+----D+--- D+ --- D+ --- D+ --- D+--- 100% 6 - X susceptible 4 - Z susceptible 1 - doomed 16 - immune 2 - parallelism 8 - synergy 29

30. KNOWN: UNKNOWN: PARTIAL SOLUTION: PROPORTION OF PEOPLE WITH DISEASE IN EACH EXPOSURE CATEGORY PROPORTION OF “TYPES” OF PEOPLE IN EACH EXPOSURE CATEGORY RELATIONSHIP BETWEEN THE KNOWN AND UNKNOWN 30

31. ASSESSING INTERACTION BETWEEN VARIABLES X AND Z B C D NONE CAUSETYPE DISEASE EXPERIENCE IF EXPOSED TO: XZ X&ZNEITHER X A Z A B X OR Z X Z E A B C DC D A B E D+----D+--- D+ --- D+ --- D+ --- D+--- 100% 6 - X susceptible 4 - Z susceptible 1 - doomed 16 - immune 2 - parallelism 8 - synergy --- 31

32. PROPORTION DISEASE AMONG X ONLY: PROPORTION DISEASE AMONG Z ONLY: PROPORTION DISEASE AMONG X&Z: PROPORTION DISEASE (I.E., DISEASE RISK) AMONG UNEXPOSED: KNOWN: UNKNOWN: PARTIAL SOLUTION: PROPORTION OF PEOPLE WITH DISEASE IN EACH EXPOSURE CATEGORY PROPORTION OF “TYPES” OF PEOPLE IN EACH EXPOSURE CATEGORY RELATIONSHIP BETWEEN THE KNOWN AND UNKNOWN DOOMED + X SUSCEPTIBLE + PARALLE L DOOMED + Z SUSCEPTIBLE + PARALLEL DOOMED + Z SUSCEPTIBLE + X SUSCEPTIBLE + PARALLEL & SYNERGISTIC DOOMED 32

33. R(XZ) = RX = RZ = DARROCH’S TABLE R = R(XZ) - R R(XZ)= R(X) = R(Z) = R = PROPORTION DISEASE AMONG THOSE EXPOSED TO X & Z PROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY PROPORTION DISEASE AMONG UNEXPOSED TO X AND Z PROPORTION DISEASE AMONG THOESE EXPOSED TO Z ONLY RISKS IN TERMS OF OBSERVED PROPORTIONS RISKS IN TERMS OF UNOBSERVABLE TYPES DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC DOOMED, SUS X, PARALLEL DOOMED, SUS Z, PARALLEL DOOMED DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC R(XZ)R(X) - - DOOMED, SUS X, PARALLEL = SUS Z + SYNERGISTIC 33

34. R(XZ) = RX = RZ = DARROCH’S TABLE SYNERGISM SUS Z R(XZ) - R(X) R = R(XZ)= R(X) = R(Z) = R = PROPORTION DISEASE AMONG THOSE EXPOSED TO X & Z PROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY PROPORTION DISEASE AMONG UNEXPOSED TO X AND Z PROPORTION DISEASE AMONG THOESE EXPOSED TO Z ONLY RISKS IN TERMS OF OBSERVED PROPORTIONS RISKS IN TERMS OF UNOBSERVABLE TYPES DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC DOOMED, SUS X, PARALLEL DOOMED, SUS Z, PARALLEL DOOMED 34

35. R(XZ) = RX = RZ = DARROCH’S TABLE R = R(XZ) - R R(XZ)= R(X) = R(Z) = R = PROPORTION DISEASE AMONG THOSE EXPOSED TO X & Z PROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY PROPORTION DISEASE AMONG UNEXPOSED TO X AND Z PROPORTION DISEASE AMONG THOESE EXPOSED TO Z ONLY RISKS IN TERMS OF OBSERVED PROPORTIONS RISKS IN TERMS OF UNOBSERVABLE TYPES DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC DOOMED, SUS X, PARALLEL DOOMED, SUS Z, PARALLEL DOOMED DOOMED, SUS X, PARALLEL R(X)R - - DOOMED = SUS X + PARALLEL 35

36. R(XZ) = RX = RZ = DARROCH’S TABLE SYNERGISM SUS Z R(XZ) - R(X) R = SUS X PARALLEL R(X) - R R(XZ)= R(X) = R(Z) = R = PROPORTION DISEASE AMONG THOSE EXPOSED TO X & Z PROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY PROPORTION DISEASE AMONG UNEXPOSED TO X AND Z PROPORTION DISEASE AMONG THOESE EXPOSED TO Z ONLY RISKS IN TERMS OF OBSERVED PROPORTIONS RISKS IN TERMS OF UNOBSERVABLE TYPES DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC DOOMED, SUS X, PARALLEL DOOMED, SUS Z, PARALLEL DOOMED 36

37. R(XZ) = RX = RZ = DARROCH’S TABLE R = R(XZ) - R R(XZ)= R(X) = R(Z) = R = PROPORTION DISEASE AMONG THOSE EXPOSED TO X & Z PROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY PROPORTION DISEASE AMONG UNEXPOSED TO X AND Z PROPORTION DISEASE AMONG THOESE EXPOSED TO Z ONLY RISKS IN TERMS OF OBSERVED PROPORTIONS RISKS IN TERMS OF UNOBSERVABLE TYPES DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC DOOMED, SUS X, PARALLEL DOOMED, SUS Z, PARALLEL DOOMED DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC R(XZ)RZ - - DOOMED, SUS Z, PARALLEL = SUS X + SYNERGISTIC 37

38. R(XZ) = RX = RZ = DARROCH’S TABLE SYNERGISM SUS ZR(XZ) - R(X) R = SUS XPARALLEL R(X) - R R(XZ) - R(Z) R(XZ) - R R(XZ)= R(X) = R(Z) = R = PROPORTION DISEASE AMONG THOSE EXPOSED TO X & Z PROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY PROPORTION DISEASE AMONG UNEXPOSED TO X AND Z PROPORTION DISEASE AMONG THOESE EXPOSED TO Z ONLY RISKS IN TERMS OF OBSERVED PROPORTIONS RISKS IN TERMS OF UNOBSERVABLE TYPES DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC DOOMED, SUS X, PARALLEL DOOMED, SUS Z, PARALLEL DOOMED 38

39. R(XZ) = RX = RZ = DARROCH’S TABLE R = R(XZ) - R R(XZ)= R(X) = R(Z) = R = PROPORTION DISEASE AMONG THOSE EXPOSED TO X & Z PROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY PROPORTION DISEASE AMONG UNEXPOSED TO X AND Z PROPORTION DISEASE AMONG THOESE EXPOSED TO Z ONLY RISKS IN TERMS OF OBSERVED PROPORTIONS RISKS IN TERMS OF UNOBSERVABLE TYPES DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC DOOMED, SUS X, PARALLEL DOOMED, SUS Z, PARALLEL DOOMED DOOMED, SUSZ, PARALLEL R(Z)R - - DOOMED = SUS Z + PARALLEL 39

40. R(XZ) = RX = RZ = DARROCH’S TABLE SYNERGISM SUS ZR(XZ) - R(X) R = SUS XPARALLEL R(X) - R R(XZ) - R(Z)R(Z) - R R(XZ) - R R(XZ)= R(X) = R(Z) = R = PROPORTION DISEASE AMONG THOSE EXPOSED TO X & Z PROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY PROPORTION DISEASE AMONG UNEXPOSED TO X AND Z PROPORTION DISEASE AMONG THOESE EXPOSED TO Z ONLY RISKS IN TERMS OF OBSERVED PROPORTIONS RISKS IN TERMS OF UNOBSERVABLE TYPES DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC DOOMED, SUS X, PARALLEL DOOMED, SUS Z, PARALLEL DOOMED 40

41. R(XZ) = RX = RZ = DARROCH’S TABLE SYNERGISM SUS ZR(XZ) - R(X) R = SUS XPARALLEL R(X) - R R(XZ) - R(Z)R(Z) - R R(XZ) - R [SYNERGISM + SUS Z] - [SUS Z + PARALLEL] = [R(XZ) - R(X)] - [R(Z) - R] = [SYNERGISM - PARALLEL] = R(XZ)= R(X) = R(Z) = R = PROPORTION DISEASE AMONG THOSE EXPOSED TO X & Z PROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY PROPORTION DISEASE AMONG UNEXPOSED TO X AND Z PROPORTION DISEASE AMONG THOESE EXPOSED TO Z ONLY RISKS IN TERMS OF OBSERVED PROPORTIONS RISKS IN TERMS OF UNOBSERVABLE TYPES R(XZ) - R(X) - R(Z) + R DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC DOOMED, SUS X, PARALLEL DOOMED, SUS Z, PARALLEL DOOMED 41

42. R(XZ) = DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC RX = DOOMED, SUS X, PARALLEL RZ = DOOMED, SUS Z, PARALLEL DARROCH’S TABLE: example SYNERGISM SUS ZR(XZ) - R(X) R = DOOMED SUS XPARALLEL R(X) - R R(XZ) - R(Z)R(Z) - R R(XZ) - R [SYNERGISM + SUS Z] - [SUS Z + PARALLEL] = [R(XZ) - R(X)] - [R(Z) - R] = [SYNERGISM - PARALLEL] = R(XZ) - R(X) - R(Z) + R R(XZ) = 20.7R R(X) = 7.2R R(Z) = 5.1R 20.7 - 7.2 - 5.1 + 1 = 9.4 INTERPRETATION:THERE ARE MORE SYNERGISTIC THAN PARALLEL TYPES IN THIS SAMPLE 42

43. BOTTOM LINE CAUSAL INFERENCE RE: INTERACTION(DARROCH MODEL) SYNERGISM EXISTS THERE ARE MORE PEOPLE FOR WHOM THE EXPOSURES WORK SYNERGISTICALLY THAN IN A PARALLEL MANNER IF THERE IS NO ADDITIVE INTERACTION IN YOUR DATA: THERE MAY BE NO SYNERGISM THE PROPORTION OF PEOPLE FOR WHOM THE EXPOSURES WORK SYNERGISTICALLY MAY BE THE SAME AS THE PROPORTION FOR WHOM THE EXPOSURES WORK IN A PARALLEL MANNER PARALLELISM EXISTS THERE ARE MORE PEOPLE FOR WHOM THE EXPOSURES WORK IN A PARALLEL MANNER THAN FOR WHOM THE EXPOSURES WORK SYNERGISTICALLY IF THERE IS EVIDENCE OF NEGATIVE ADDITIVE INTERACTION IN YOUR DATA IF THERE IS EVIDENCE OF POSITIVE ADDITIVE INTERACTION IN YOUR DATA: 43

44. Synergy Assessment in Practice Interaction Contrast R 11 -R 10 -R 01 +R 00 Interaction Contrast Ratio [IC/R 00 : (aka RERI: Relative Excess Risk due to Interaction)] RR 11 -RR 10 -RR 01 + 1 Attributable Proportion due to interaction RR 11 -RR 10 -RR 01 +1 / RR 11 Synergy index RR 11 -1/(RR 10 -1) ( RR 01 -1) 44

45. SO WHO IS RIGHT? Tennent and Bebbington Do Intimacy problems interact with stressful life events to cause depression? Brown and Harris

46. So who is right? Brown and Harris Stressful Life Events Intimacy Problems YesNO Yes32%10% No3%1% Interaction Contrast = R 11 -R 10 -R 01 +R 00 = SYNERGY - PARALLEL 32-10-3 +1 = 20

47. CONCLUSION CAUSAL INTERACTION IS BEST REPRESENTED BY ADDITIVITY 47 WITH A TWIST