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Effect modification (Interaction)

• Goals of stratification of data

– Evaluate and reduce/remove confounding

– Evaluate and describe effect modification

• Description of effect modification

– A change in the magnitude of an effect measure

(between exposure and disease) according to the level

of some third variable

– What two “classes” of effect measures have we used so

far in the course?

exampleEffect modification:#1

• Disease incidence by exposure and age

– Does the relationship between exposure and disease change

over the value of the potential confounder (age)? How?

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2014Page2

Effect modification: example #2

• Disease incidence by exposure and age

• Does the relationship between exposure and disease

change over the value of the potential confounder

(age)? How?

Rothman ’86 (p 178)

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2014Page4

contrastEffect modification: with confounding

• Confounding

– A bias that an investigator hopes to remove

– A nuisance that may or may not be present in a given

study design

• Properties of a confounding variable: (Rothman, p123):

– a) be a risk factor for disease among the non-exposed;

– b) be associated with the exposure variable; and

– c) not be an intermediate step in the “causal pathway”

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2014Page5

contrastEffect modification: with confounding

• Effect modification

– A more detailed description of the “true” relationship

between the exposure and the outcome

– Effect modification is a finding to be reported (even

celebrated), not a bias to be eliminated

– Effect modification is a “natural phenomenon” that

exists independently of the study design

– The presence and interpretation of effect modification

depends upon the choice of effect measure (ratio vs.

difference)

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Some lingo

• Covariate

– Confounder, potential confounder

– Effect modification, interaction

– Intermediate variable

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Effect modification: contrast

with confounding

• Note that for any association under study, a given factor

may be:

– Both a confounder and an effect modifier or

– A confounder but not an effect modifier or An effect

modifier but not a confounder or

– neither

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2014Page8 Examples of confounding/effect

modification

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Level 1 Level 2 Crude/

collapsed/

Combined

“unadjusted

”

Uniform

estimate

(ORMH)

/

“adjusted”

Confounding

present

Interaction

present

4.0 4.0 4.0 4.0 NO NO

4.0 0.25 1.0 1.0 NO YES

1.0 1.0 8.4 1.0 YES NO

4.0 0.25 1.0 2.0 YES

(?relevance)

YES

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2014 Page 90

Effect modification: test ofhomogeneity

• Null hypothesis: The individual stratified estimates of the effect do not

differ from some uniform estimate of effect (such as a Mantel Haenszel

estimator)

• Notation:

– N is the number of strata (N=2 in our smoking/matches example);

–

MH

ln^Ri is the natural logarithm of the estimated (hence the “^”) effect

measure for each stratum (ORi in our example);

– ln^R is the natural logarithm of the uniform effect estimate (e.g. OR in

– X2

(N-1)is chi-square with (N-1) degrees of freedom;

our example—the computer will use the maximum likelihood estimate)

• One formula to test homogeneity:

X2

(N-1)

= ∑ [ln(^ Ri) – ln(RMH)]2

Var[ln(^

Ri)]

N

i= 1

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JC: Comment on choice of signifciance level for test of homogeneity

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Paradox

• If effect modification is present, a uniform estimator of

effect (such as ORMH

) cannot (or at least should not) be

reported.

• However, in order to determine if effect modification is

present, it is necessary to calculate the value of a uniform

estimator of effect (such as ORMH

) because it is needed in

the calculation of the test of homogeneity.

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Effect modification: test of homogeneity(or

is heterogeneity?)

• Comments

– If the test of homogeneity is “significant” (=“reject homogeneity”)

this is evidence that there is heterogeneity (i.e. no homogeneity)

and that effect modification may be present.

• (Null hypothesis: The individual stratified estimates of the

effect do not differ from some uniform estimate of effect)

– The choice of a significance level (e.g. p < 0.05) is somewhat open

to interpretation.

• One “conservative” approach, because of inherent limitations in

the power of the test of homogeneity, is to treat the data as if

interaction is present for p < 0.20).

• In other words, one would rather err on the side of assuming

that interaction is present (and reporting the stratified estimates

of effect) than on reporting a uniform estimate that may not be

true across strata.80

UC Berkeley

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Additive versus multiplicative scale effect modification

● Notation: RXZ

● No additive interaction if (R11 – R01) = (R10 – R00)

○ Rewrite as: (R11-R01)-(R10-R00)=0

● In words: Difference in risk for (X=1 vs. X=0) when Z=1 is

equal to difference in risk for (X=1 vs. X=0) when Z=0

● Note: the values R11, R10, etc. are risks (not counts)

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Additive versus multiplicative scale effect modification

● Notation: RXZ

● No multiplicative interaction if (R11/R01)=(R10/R00)

Rewrite as: (R11/R01)/(R10/R00)=1

● In words: Ratio of risks/rates when X=1 vs. X=0 when

Z=1 is equal to ratio of risks/rates when X=1 vs. X=0

when Z=0

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2014Page17 Effect modification is scale-dependent

•Evidence for effect modification/statistical interaction if the RR or the AR differs between two groups

•However, effect modification/statistical interaction is scale-dependent

–If you do not have interaction on the additive scale (AR is homogenous) then you will have interaction on the multiplicative scale (RR must be heterogeneous)

–If you do not have interaction on the multiplicative scale (RR ishomogenous) then you will have interaction on the additive scale(AR must be heterogeneous)

–Note: It is common to have evidence of interaction on both scales.

Example● No additive scale interaction if (R11-R01)-(R10-R00)=0

● No relative scale interaction if (R11/R01)/(R10/R00)=1

● Additive scale: (60-20) - (50-10) = 0

○ Interaction not present on the additive scale

● Relative scale: (60/20) / (50/10)=0.6

○ Interaction present on the relative scale

Z=1 Z=0

X=1 60 50

X=0 20 10

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Example● No additive scale interaction if (R11-R01)-(R10-R00)=0

● No relative scale interaction if (R11/R01)/(R10/R00)=1

● Additive scale: (60-20) - (30-10) = 20

○ Interaction present on the additive scale

● Relative scale: (60/20) / (30/10)=1

○ Interaction not present on the relative scale

Z=1 Z=0

X=1 60 30

X=0 20 10

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