Are You Looking for the Right Interactions?
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Transcript of Are You Looking for the Right Interactions?
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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 SchwartzDepartment of EpidemiologyMailman School of Public HealthColumbia University
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The Problem
Interaction is
Model Dependent
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What You See Depends on How you Look at It
Forms or word?
Musician or Woman?Vases or Faces?
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MOTIVATING EXAMPLE
Tennent and BebbingtonNO
Do Intimacy problems interact with stressful life events to cause depression?
Brown and Harris YES
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Interaction (aka, Effect Modification)
The effect of an exposure differs in the presence or absence of another variable (X)
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MOTIVATING EXAMPLE
Do Intimacy problems interact with stressful life events to cause depression?
Stressful Life Events
Intimacy Problems
YES NO
Yes 32% 10%
No 3% 1%
It depends on how you look at it
Risk of disease in each cell is displayed
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MOTIVATING EXAMPLE
Is the effect of stressful life events different in the presence vs. absence of Intimacy Problems?
Stressful Life Events
Intimacy Problems
Yes NO
Yes 32% 10%
No 3% 1%
RD = 32 - 3 = 29 RD = 10 - 1 = 9
RR = 32/3 = 10 RR = 10/1 = 10
YESNO
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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
EventsIntimacy Problems
Yes NO
Yes 32% 10%
No 3% 1%
RD = 32 - 3 = 29 RD = 10 - 1 = 9
RR = 32/3 = 10 RR = 10/1 = 10YESNO
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What is the Causal Question?:
Are there some individuals who became depressed because they experienced both stressful life events and intimacy problems
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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?
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StressfulLife Events
IntimacyProblems
Hypothesized Causes of Depression
T
The exposure has an effect when itis in the presence of its causal partners
(here intimacy problems)
J
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StressfulLife Events
IntimacyProblems
Hypothesized Causes of Depression
TJ
StressfulLife Events
GeneticVariant
R
IntimacyProblems
P
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EFFECT MODIFICATION, INTERACTIONAND SYNERGY:
WHAT DO WE REALLY WANT TO KNOW?
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Synergy: Who are the Exposure’s
Causal Partners?
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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
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ROTHMAN’S CAUSAL MODEL SINGLE CAUSE OF INTEREST
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BEGIN FROM YOUR CAUSAL MODEL
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CAUSES OF DISEASES AND THE RELATIONSHIP TO “TYPES” EXPOSURE OF INTEREST: X
CAUSES OF DISEASE OF INTEREST
X A C D NOT X E
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CAUSES OF DISEASES AND THE RELATIONSHIP TO “TYPES” EXPOSURE OF INTEREST: X
CAUSES OF DISEASE OF INTEREST
X A C D NOT X E
“TYPES” OF PEOPLE REGARDING DISEASE OF INTEREST GIVEN THAT EXPOSURE OF INTEREST IS X
C D ENOTA, C&D, OR E
SUSCEPTIBLE DOOMED PROTECTIVE IMMUNE
A
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INTEREST IN EXPOSURE X: WHAT CAN WE KNOW ABOUT CAUSATION?
TYPE PROPORTION DISEASE EXPERIENCE DISEASE EXPERIENCE IF EXPOSED IF UNEXPOSED
1: DOOMED D+ D+
2: A CAUSAL D+ ---
3: A PROTECTIVE --- D+
4: IMMUNE --- ---
RISK IF EXPOSED = RISK IF UNEXPOSED = RISK DIFFERENCE = RISK RATIO =
P1
P2
P3
P4
P1&P2P1 & P3P2 - P3 P1 & P2/ P1 & P3
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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 EQUALNUMBERS OF PEOPLE FOR WHOM IT IS CAUSAL AND FOR WHOM IT IS PROTECTIVE
THERE ARE MORE PEOPLE FOR WHOM THE EXPOSURE IS PROTECTIVETHAN 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
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Using this Causal Model:
How do we know if There is Synergy Out There?
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IF WE ASSUME ALL ACTIVE EFFECTS ARE CAUSAL AND NONE ARE PROTECTIVE, THEN THERE ARE6 POSSIBLE OUTCOMES FROM THE COMBINATION OF
ANY TWO RISK FACTORS OF INTEREST
DARROCH, ROTHMAN, GREENLAND USE THIS ASSUMPTION IN DEVELOPING THEIR MODEL
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ASSESSING INTERACTION BETWEEN VARIABLES X AND Z
B
C D
NONE
CAUSE TYPE TYPE
DISEASE EXPERIENCE IF EXPOSED TO:
X Z X&Z NEITHER
X A
Z
AB
XOR
Z
XZ E
A
B
C D
AB
D+ ---- D+ ---
--- D+ D+ ---
D+ D+ D+ D+
--- --- --- ---
D+ D+ D+ ---
--- --- D+ ---
D =100% D=100% D=100% D=100%
6 - X susceptible
4 - Z susceptible
1 - doomed
16 - immune
2 - parallelism
8 - synergyE
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WHEN TWO RISK FACTORS ARE PARTNERS IN THE SAME SUFFICIENT CAUSEBOTH 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
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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
ZA B
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T0 T End
B = 1
B = 1
Exposures of interest = X,Z
John is exposed to X, ZJohn’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 X Effect of Z
Nhoj Z
Nhoj -
B = 1
B = 1
D
D
D
--
A = 1
A = 1
T1
A = 1
A = 1
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X Z
A B
Inevitable Parallelism
Will occur with a probability of A*B
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X Z
A A
“Functional Equivalence” Parallelism
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ASSESSING INTERACTION BETWEEN VARIABLES X AND Z
B
C D
NONE
CAUSE TYPE TYPE
DISEASE EXPERIENCE IF EXPOSED TO:
X Z X&Z NEITHER
X A
Z
AB
XOR
Z
XZ E
A
B
C D
AB
E
D+ ---- D+ ---
--- D+ D+ ---
D+ D+ D+ D+
--- --- --- ---
D+ D+ D+ ---
--- --- D+ ---
100% 100% 100% 100%
6 - X susceptible
4 - Z susceptible
1 - doomed
16 - immune
2 - parallelism
8 - synergy
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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
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ASSESSING INTERACTION BETWEEN VARIABLES X AND Z
B
C D
NONE
CAUSE TYPE TYPE
DISEASE EXPERIENCE IF EXPOSED TO:
X Z X&Z NEITHER
X A
Z
AB
XOR
Z
XZ E
A
B
C D
AB
E
D+ ---- D+ ---
--- D+ D+ ---
D+ D+ D+ D+
--- --- ---
D+ D+ D+ ---
--- --- D+ ---
100% 100% 100% 100%
6 - X susceptible
4 - Z susceptible
1 - doomed
16 - immune
2 - parallelism
8 - synergy
---
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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
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R(XZ) =RX =RZ =
DARROCH’S TABLE
R =
R(XZ)=R(X) =R(Z) = R =
PROPORTION DISEASE AMONG THOSE EXPOSED TO X & ZPROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY
PROPORTION DISEASE AMONG UNEXPOSED TO X AND ZPROPORTION 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, SYNERGISTICDOOMED, SUS X, PARALLELDOOMED, SUS Z, PARALLELDOOMED
DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC
R(XZ) R(X)-
-
DOOMED, SUS X, PARALLEL = SUS Z + SYNERGISTIC
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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 & ZPROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY
PROPORTION DISEASE AMONG UNEXPOSED TO X AND ZPROPORTION 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, SYNERGISTICDOOMED, SUS X, PARALLELDOOMED, SUS Z, PARALLELDOOMED
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R(XZ) =RX =RZ =
DARROCH’S TABLE
R =
R(XZ)=R(X) =R(Z) = R =
PROPORTION DISEASE AMONG THOSE EXPOSED TO X & ZPROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY
PROPORTION DISEASE AMONG UNEXPOSED TO X AND ZPROPORTION 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, SYNERGISTICDOOMED, SUS X, PARALLELDOOMED, SUS Z, PARALLELDOOMED
DOOMED, SUS X, PARALLEL
R(X) R-
-
DOOMED = SUS X + PARALLEL
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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 & ZPROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY
PROPORTION DISEASE AMONG UNEXPOSED TO X AND ZPROPORTION 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, SYNERGISTICDOOMED, SUS X, PARALLELDOOMED, SUS Z, PARALLELDOOMED
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R(XZ) =RX =RZ =
DARROCH’S TABLE
R =
R(XZ)=R(X) =R(Z) = R =
PROPORTION DISEASE AMONG THOSE EXPOSED TO X & ZPROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY
PROPORTION DISEASE AMONG UNEXPOSED TO X AND ZPROPORTION 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, SYNERGISTICDOOMED, SUS X, PARALLELDOOMED, SUS Z, PARALLELDOOMED
DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTIC
R(XZ) RZ-
-
DOOMED, SUS Z, PARALLEL = SUS X + SYNERGISTIC
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R(XZ) =RX =RZ =
DARROCH’S TABLE
SYNERGISM SUS Z R(XZ) - R(X)
R =
SUS X PARALLEL R(X) - R
R(XZ) - R(Z) R(XZ) - R
R(XZ)=R(X) =R(Z) = R =
PROPORTION DISEASE AMONG THOSE EXPOSED TO X & ZPROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY
PROPORTION DISEASE AMONG UNEXPOSED TO X AND ZPROPORTION 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, SYNERGISTICDOOMED, SUS X, PARALLELDOOMED, SUS Z, PARALLELDOOMED
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R(XZ) =RX =RZ =
DARROCH’S TABLE
R =
R(XZ)=R(X) =R(Z) = R =
PROPORTION DISEASE AMONG THOSE EXPOSED TO X & ZPROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY
PROPORTION DISEASE AMONG UNEXPOSED TO X AND ZPROPORTION 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, SYNERGISTICDOOMED, SUS X, PARALLELDOOMED, SUS Z, PARALLELDOOMED
DOOMED, SUSZ, PARALLEL
R(Z) R-
-
DOOMED = SUS Z + PARALLEL
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R(XZ) =RX =RZ =
DARROCH’S TABLE
SYNERGISM SUS Z R(XZ) - R(X)
R =
SUS X PARALLEL 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 & ZPROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY
PROPORTION DISEASE AMONG UNEXPOSED TO X AND ZPROPORTION 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, SYNERGISTICDOOMED, SUS X, PARALLELDOOMED, SUS Z, PARALLELDOOMED
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R(XZ) =RX =RZ =
DARROCH’S TABLE
SYNERGISM SUS Z R(XZ) - R(X)
R =
SUS X PARALLEL 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 & ZPROPORTION DISEASE AMONG THOSE EXPOSED TO X ONLY
PROPORTION DISEASE AMONG UNEXPOSED TO X AND ZPROPORTION 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, SYNERGISTICDOOMED, SUS X, PARALLELDOOMED, SUS Z, PARALLELDOOMED
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R(XZ) = DOOMED, SUS X, SUS Z, PARALLEL, SYNERGISTICRX = DOOMED, SUS X, PARALLELRZ = DOOMED, SUS Z, PARALLEL
DARROCH’S TABLE: example
SYNERGISM SUS Z R(XZ) - R(X)
R = DOOMED
SUS X PARALLEL 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
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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 SYNERGISMTHE PROPORTION OF PEOPLE FOR WHOM THE EXPOSURES WORK SYNERGISTICALLY MAY BE THE SAME AS THE PROPORTION FOR WHOM THEEXPOSURES 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:
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Synergy Assessment in Practice
Interaction ContrastR11-R10-R01+R00
Interaction Contrast Ratio [IC/R00: (aka RERI: Relative Excess Risk due to Interaction)] RR11-RR10-RR01 + 1
Attributable Proportion due to interactionRR11-RR10-RR01 +1 / RR11
Synergy indexRR11-1/(RR10-1) ( RR01-1)
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SO WHO IS RIGHT?
Tennent and Bebbington
Do Intimacy problems interact with stressful life events to cause depression?
Brown and Harris
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So who is right? Brown and Harris
Stressful Life Events
Intimacy Problems
Yes NO
Yes 32% 10%
No 3% 1%
Interaction Contrast =
R11-R10-R01+R00 = SYNERGY - PARALLEL
32-10-3 +1 = 20
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CONCLUSION
CAUSAL INTERACTION IS BEST REPRESENTED BY ADDITIVITY
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WITH A TWIST