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### Interaction and Effect-Measure Modification

Lydia B. Zablotska, MD, PhD

Associate Professor

Department of Epidemiology and Biostatistics

Learning Objectives

- Statistical interaction
- Multiplicative and additive interaction
- Biologic interaction
- Evaluation of interaction, presentation of results
- Attributable fraction estimation

Review of measures of association

- Effect measures vs. measures of association:
- Can never achieve counterfactual ideal
- Logically impossible to observe the population under both conditions and to estimate true effect measures
- Measures of association
- Compares what happens in two distinct populations
- Constructed to equal the effect measure of interest
- Absolute: differences in occurrence measures (rate or risk difference)
- Relative: ratios of occurrence measures (rate or risk ratio, relative risk, odds ratio)

Concepts of interaction

- Terms:
- statistical interaction
- effect modification or effect measure modification
- synergy (joint action of causal partners)
- heterogeneity of effect
- departure from additivity of effects on the chosen outcome scale
- Definition:
- heterogeneity of effect measures across strata of a third variable
- Problems:
- Scale-dependence, i.e. can be measured on an additive or multiplicative scale
- Ambiguity of terms
- Types:
- Statistical
- Biological
- Public health interaction (public health costs or benefits from altering one factor must take into account the prevalence of other factors and effects of their reduction)

RG Ch 5

Types of interaction:Statistical interaction

- If statistical interaction is being described on an additive scale then the measure of effect is the risk difference
- R11 - R00 = (R10 - R00) + (R01 - R00). If the 2 sides of the equation are equal the relationship is perfectly additive
- If statistical interaction is being described on a multiplicative scale then the measure of effect is the odds ratio or relative risk
- R11 / R00 = (R10/R00 )(R01/R00). If the 2 sides of the equation are equal the relationship is perfectly multiplicative

RG Ch 5

Types of statistical interaction

- Effect modification of the risk difference (absolute effect) corresponds with additive interaction
- Effect modification on the risk ratio or odds ratio (relative effect) corresponds with multiplicative interaction
- If there is no evidence of interaction on the multiplicative scale (i.e, heterogeneity of RR or OR if OR is a good approximation of RR) there will be evidence of interaction on the additive scale (i.e., heterogeneity of RD)

RG Ch 5

Statistical interaction

- Heterogeneity of effects always refers to a specific type of effect: risk ratios, odds ratios, risk differences
- Absence of interaction for one measure does not imply absence of interaction for the other measures of association:
- Homogeneity of risk differences implies heterogeneity of risk ratios and vice-versa
- Most estimates of effect are based on multiplicative models; specify measures of effect when describing effect modification

RG Ch 5

Additive interaction

RD = Riskexposed – Riskunexposed

A and B are risk factors with risks Ra,- and R-,b and individual risk differences:

RDa,- = Ra,- – R-,-

RD-,b = R-,b – R-,-

RDa,b is a RD for those exposed to both A and B and those exposed

to neither

- RDa,b = RDa,- + RD-,b – A and B are non-interacting risk factors
- RDa,b RDa,- + RD-,b – Additive interaction between A and B
- RDa,b > RDa,- + RD-,b – Additive synergy

(positive additive interaction)

- RDa,b < RDa,- + RD-,b – Additive antagonism

(negative additive interaction)

Multiplicative interaction

RR = Riskexposed / RiskunexposedRiskexposed = Riskunexposed x RR

A and B are risk factors with risks Ra,- and R-,b and individual risk ratios:

RRa,- = Ra,- / R-,-

RR-,b = R-,b / R-,-

RRa,b is a RR for those exposed to both A and B over those exposed

to neither

- RDa,b = RDa,- x RD-,b – A and B are non-interacting risk factors
- RDa,b RDa,- x RD-,b – Multiplicative interaction between A and B
- RDa,b > RDa,- + RD-,b – Multiplicative synergy

(positive multiplicative interaction)

- RDa,b < RDa,- + RD-,b – Multiplicative antagonism

(negative multiplicative interaction)

Assessment of interaction for binary data

Risk of past-year depression at age 26 according to genotype

and stressful life events

Dunedin Child-Development Study, Caspi et al. 2002, 2003

Assessing interaction by stratification

- Effect modification by presence of short allele G on the association between stressful life events E and risk of depression

RDE/G is absent = 0.17-0.10=0.07; RRE/G is absent = 0.17/0.10=1.7

RDE/G is present = 0.33-0.10=0.23; RRE/G is present = 0.33/0.10=3.3

- Both RD and RR are heterogeneous

Comparing expected and observed joint effects

- What is the individual effect of cause A in the absence of exposure to cause B?
- What is the individual effect of cause B in the absence of exposure to cause A?
- What is the observed joint effect of A and B?
- What is the expected joint effect of A and B in the absence of interaction?
- Is the observed joint effect similar to the expected joint effect in the absence of interaction?

What is the individual effect of cause A in the absence of exposure to cause B?

What is the individual effect of cause A in the absence of exposure to cause A?

What is the observed joint effect of A and B?

What is the expected joint effect of A and B in the absence of interaction?

Is the observed joint effect similar to the expected joint effect in the absence of interaction?

RDE,-=0.17-0.10=0.07

RD-,G=0.10-0.10=0

RDOBSERVED E,G=0.33-0.10=0.23

RDEXPECTED E,G=0.07+0=0.07

RDOBSERVED E,G > RDEXPECTED E,G, additive interaction

Comparing expected and observed joint effectsWhat is the individual effect of cause A in the absence of exposure to cause B?

What is the individual effect of cause A in the absence of exposure to cause A?

What is the observed joint effect of A and B?

What is the expected joint effect of A and B in the absence of interaction?

Is the observed joint effect similar to the expected joint effect in the absence of interaction?

What is the interaction magnitude

RDE,-=0.17-0.10=0.07

RD-,G=0.10-0.10=0

RDOBSERVED E,G=0.33-0.10=0.23

RDEXPECTED E,G=0.07+0=0.07

RDOBSERVED E,G > RDEXPECTED E,G,

additive interaction

RDE/ G IS PRESENT – RDE/ G IS ABSENT

= 0.23 - 0.07 =0.16

interaction contrast

Comparing expected and observed joint effects- RRE,-=0.17/0.10=1.7
- RR-,G=010/0.10=1.0
- RROBSERVED E,G=0.33/0.10=3.3
- RREXPECTED E,G=1.7x1.0=1.7
- RROBSERVED E,G > RREXPECTED E,G,

multiplicative interaction

- RRE/G IS PRESENT / RRE/G IS ABSENT

= 3.3 / 1.7 =1.9

7. Trouble with assessment of synergy

Interaction of vulnerability factors (e.g., fear of

intimacy) and stressful life events in causing depression

- Analysis on the additive scale:
- Analysis on the multiplicative scale:

Brown and Harris 1978

The conundrum

- Each of these alternative interpretations is consistent with the premises of the mathematical models that were used:
- Brown and Harris assumed that, absent interaction, risk factors add in their effects
- Tennet and Bebbington assumed that, absent interaction, risk factors multiply in their effects
- What is the answer and what could be done to elucidate one correct answer?

Biological interaction

- Terms:
- Biological interaction
- Causal interaction
- Definition:
- Modification of potential-response types
- A process that explain potential mechanisms that can account for observed cases of disease
- Exchangeability (i.e., the same data pattern would result if exposure status was switched or the rate in E would be equal to not E if E were not exposed) is required to test for interaction

Biologic interaction

- Biological interaction can be defined under the counterfactual approach and the sufficient cause approach
- Sufficient cause approach
- 2 exposures are 2 component causes in a sufficient cause for the disease where the presence of both exposures is required to complete the sufficient cause ie., they are insufficient but necessary component causes of a unnecessary but sufficient cause (INUS partners)
- interaction between component causes is implicit in the sufficient cause model
- each component cause requires the presence of the others to act, their action is interdependent
- Parallelism (type 2) in terms of the sufficient cause approach indicates that both A and B can complete the sufficient cause, the result depending on which gets there first.
- The two component causes compete to be INUS partners in the same sufficient cause, they act in parallel. The individual would get disease if they are exposed to either A or B but not get disease if exposed to neither.
- Synergy and parallelism have different component causes i.e, A and B, A or B.

Biologic vs. statistical interaction

- When two factors have effects but risk ratios within the strata of the second factor are homogeneous, there is no interaction on the multiplicative scale
- This implies that there is heterogeneity of the corresponding risk differences
- The non-additivity of risk differences implies the presence of some type of biologic interaction

RG Ch 5

Biological interaction

- Biological interaction can be defined under the counterfactual approach and the sufficient cause approach
- Counterfactual approach (potential outcome)
- 4 exposure categories for 2 binary variables=16 possible patterns of response types (given disease or no disease)
- 10 categories can be considered interaction (interdependence) of some type (i.e., both of the 2 exposure types have an effect) and interaction contrast not equal 0
- If it is assumed the effect is causal, Type 8 in the counterfactual approach is equivalent to causal or biological synergy. Each exposure only causes disease if the other is present.

Interaction contrast

- Causal additivity = no causal interaction

R11– R00= (R10 – R00) + (R01 – R00)=(p6+p13-p11-p13) + (p4+p11-p11-p13)

=(0+0-0-0) + (0+0-0-0)=0

- Interaction contrast=difference in risk differences

IC = RDX,-– RD-,Z = (R11 – R01)-(R10 – R00) = (R11 – R10)-(R01 – R00)

= R11 – R10 – R01 + R00 = (p3+p5+2p7+p8+p15) – (p2+p9+2p10+p12+p14)

RG Ch 5, p. 77

Necessary conditions for interaction

- Departures from additivity can only occur when interaction causal types are present in the cohort
- Absence of interaction does not imply absence of interaction types because sometimes different interaction types counterbalance each other’s effect on the average risk
- Definitions of response types depend on the definition of the outcome under study (if it changes, then response type can change too)

RG Ch 5

Departures from additivity

- Superadditivity: RD11>RD10+RD01 – type 8 MUST be present
- Subadditivity: RD11<RD10+RD01 – type 2 MUST be present
- However, presence of synergistic responders (type 8) or competitive responders (type 2) does not imply departures from additivity
- If neither factor is ever preventive: IC = p8 –p2,
- i.e. synergism – parallelism = additive interaction

Simplified assessment of synergy based on 5 response types

- p8 = (R11 – R01) – (R10 – R00)
- Effect of Z (effect modifier) when X=1 – Effect of Z when X=0
- Assumptions when only 5 types are used
- Effect measure is the Risk Difference, biologic interaction is then interaction for risk differences
- p5 > 0, biologic interaction must be positive (although one can reparameterise the exposures X and Z to get a negative interaction)
- Huge reduction of person types, from 16 to 5!
- Keep in mind that this is a "biologic“ model

Summary of R&G scheme

- The reduction from 16 person types to 5 makes it possible to get the p’s for the 5 types, by using the 4 observed probabilities, and the fact that the 4 R’s sum to 1.
- By solving the equations we get that the person type “synergy” is equal to additive interaction, with risk differences as measure of effect

Critique of R&G scheme

- Rothman and Greenland\'s model is simplistic.
- One reasonable person type is missing!

p2 -Parallelism

- If A and B are both causal, then it is reasonable to think that some individuals in the population will develop the disease when exposed to only A, only B or both A and B.

Darroch, J. “Biologic Synergism and Parallelism”, AmJEpi 1997; 145:7 page 661-668

- John Darroch discusses an expansion of the ideas by Rothman and Greenland. He assumes 6 person types, including "parallelism".
- By using 6 person types he covers all the possible person types if A and B are directly causal in their effect on disease.

Simplified assessment of synergy based on 6 response types

- p8 – p2 = (R11 – R01) – (R10 – R00)
- Effect of Z (effect modifier) when X=1 – Effect of Z when X=0
- This means you will not be able to specify the biologic interaction (p8) exactly from the 4 known probabilities, but you can find the boundaries.

Summary notes on synergy and parallelism

- Can only be partially determined from the data at hand
- Example of synergy (assuming the factors are causal ): if the gene and environment factors acted together, infants would only get the congenital disorder if exposed to both gene and environment
- Example of parallelism (assuming the factors are causal ): infants would only get the congenital disorder if exposed to either gene or environment but would not get the congenital disorder if exposed to neither.
- If synergy - parallelism or R(AB) - R(AB) - R(A) - R(B) + R is a positive number the result is consistent with the presence of more synergy than parallelism in the population studied
- The public health approach would be to prevent exposure to either genes or environment
- Greater than an additive relationship is consistent with superadditivity and multiplicativity but inconsistent with the single hit model of disease causation
- If synergy – parallelism or R(AB) - R(A) - R(B) + R is a negative number it is an indication that there is more parallelism than synergy in the population
- Less than an additive relationship is consistent with subaddivitity and inconsistent with the no hit and multistage models of disease
- The public health approach would be to prevent exposure to both genes and environment.
- If there is no additive interaction there may be no synergism or the proportion of individuals for whom the exposures work synergistically may be the same for whom the exposures work in a parallel manner

An additive model with a “twist”

- Additive model with a “twist” allows the best representation of synergy
- An additive model assumes that risks add in their effects
- Positive deviations from additivity (superadditivity) indicates the presence of synergy
- The “twist” is that risks do something slightly less than add (parallelism – some individuals can develop disease from either one of the two exposures under study)
- What we see as the combined effect of two exposures reflects the balance of synergy and parallelism
- In summary, although superadditivity indicates synergy, a failure to find superadditivity does not imply the absence of synergy

Estimating synergy

- If there is positive interaction on the multiplicative scale, there will be positive interaction on the additive scale (supermultiplicativity implies superadditivity)
- We can assess interaction on the additive scale from the multiplicative model by calculating an interaction contrast

Dunedin Child-Development StudyCaspi et al. 2002, 2003

- IC=0.33-0.17-0.10+0.10=0.16 >0 synergy

Estimation of IC and ICR

- Cohort studies
- Intercept provides the baseline odds of disease
- OR for risk factors could be used to obtain the odds of disease under the other conditions
- Odds could be converted to risks (odds=p/ (1-p))
- Case-controls studies
- Intercept may be biased
- Odds for those exposed to both factors: 0.33/0.67; odds for those exposed to life events only: 0.17/0.83; odds for those with short allele only: 0.10/0.90; odds for those exposed to neither: 0.10/0.90
- ICR=ORboth/neither-ORlife events/neither-ORshort allele/neither + baseline

ICR=((0.33/0.67)/(0.10/0.90)) –((0.17/0.83)/(0.10/0.90)) –

–((0.10/0.90)/(0.10/0.90)) +1=2.6

ICR/ORboth/neither=2.6/4.4=0.59 – the proportion of disease

among those with both risk factors that is attributable

to interaction

RG Ch 16

Causes of depression: Theory about life events and their interaction with intimacy problems

Mathematical model representing conceptual model for interaction

- Synergy – parallelism = p8 – p2 = (R11 – R01) – (R10 – R00)
- Synergy – parallelism = 0.32 – 0.10 – 0.03 + 0.01 = 0.20
- Conclusion:
- Stressful life events and intimacy problems work in a synergistic manner to produce depression for at least some people
- The estimate of the proportion of people who developed disease because of synergy is underestimate because of parallelism
- Among the group with both risk factors, there may be some people for whom either risk factor alone would be sufficient to complete a sufficient cause for the disease
- Parallel types are likely to occur when social forces, such as SES, are linked to disease through multiple pathways

Final notes on interaction

- Superadditivity implies synergy, absence of superadditivity does not imply absence of synergy
- In the presence of contravening effects (parallelism, antagonism), synergy will be difficult to detect
- Darroch’s method using an additive model with a twist, through interaction contrasts, helps to detect synergy that usual approaches based on multiplicative models would miss (they can only detect synergy that produces such large deviations from additive effects that they are also greater than multiplicativity)
- Fits into the larger picture of causal theory: identification of causal partners of the exposure under study specifies the conditions under which the exposure will and will not have an effect.

Evaluation of interaction

- Observed heterogeneity within categories of the third variable may be due to:
- Random variability
- Typical scenario: no a priori subgroup analyses were planned and after null overall findings, the researcher decides to pursue subgroup analyses. Sample size inevitably decreases with such testing, making it likely that heterogeneity will be observed due to chance alone.
- Confounding effects
- If confounding is only present in one group of the third variable, it can explain the apparent heterogeneity of effect estimates within strata of the third variable
- Bias
- Differential bias across strata
- Differential intensity of exposure
- Apparent heterogeneity of effects could be due to differential intensity of exposure of some other variable

Presentation of results

- An important assumption when generalizing results from a study is that the study population should have an “average” susceptibility to the exposure under study with regard to a given outcome
- Results cannot be “adjusted”, need to present heterogeneous effect estimates
- When we select a risk factor to study, we can introduce a particular confounder; effect modifiers exist independently of any particular study design or study group

Attributable fraction:Taking the estimation of interaction effects one step further

- What proportion of cases is attributable to the interaction of two factors?
- (0.32 – 0.10 – 0.03 + 0.01) / 0.32 = 0.20 / 0.32 = 62.5%

General principles of attributable fraction estimation

- AF = (RR – 1) / RR
- PAR = population attributable risk
- PAR={ ∑k* Pk* (RRk – 1) } / ( ∑k* Pk* RRk )
- where k = 0, 1, .. 100, and where Pk and RRk are the proportion and relative risk at the kth dose level
- Confidence limits for PAR could be calculated by using the substitution method (Daly 1998)

RG Ch 16

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