are you looking for the right interactions?

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

The Problem

Interaction is

Model Dependent

What You See Depends on How you Look at It

Forms or word?

Musician or Woman?Vases or Faces?

MOTIVATING EXAMPLE

Tennent and BebbingtonNO

Do Intimacy problems interact with stressful life events to cause depression?

Brown and Harris YES

Interaction (aka, Effect Modification)

The effect of an exposure differs in the presence or absence of another variable (X)

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

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

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

What is the Causal Question?:

Are there some individuals who became depressed because they experienced both stressful life events and intimacy problems

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?

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

StressfulLife Events

IntimacyProblems

Hypothesized Causes of Depression

TJ

StressfulLife Events

GeneticVariant

R

IntimacyProblems

P

EFFECT MODIFICATION, INTERACTIONAND SYNERGY:

WHAT DO WE REALLY WANT TO KNOW?

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Synergy: Who are the Exposure’s

Causal Partners?

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

ROTHMAN’S CAUSAL MODEL SINGLE CAUSE OF INTEREST

16

BEGIN FROM YOUR CAUSAL MODEL

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

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

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

X Z

A B

Inevitable Parallelism

Will occur with a probability of A*B

X Z

A A

“Functional Equivalence” Parallelism

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

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

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

CONCLUSION

CAUSAL INTERACTION IS BEST REPRESENTED BY ADDITIVITY

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WITH A TWIST