week 9 - interaction 1 i nterpretation of epi studies ii : i nteraction and e ffect- m easure m...

Week 9 - Interaction 1

Interpretation of epi studies II: Interaction and Effect-Measure

Modification

Lydia B. Zablotska, MD, PhDAssociate ProfessorDepartment of Epidemiology and Biostatistics


Learning Objectives

• Biological vs. statistical

• Multiplicative and additive

• Evaluation of interaction, presentation of results


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)


Comparison of absolute and relative effect measures (Rothman 2002)

Measure Numerical Range Dimensionality

Risk difference [-1, +1] None

Risk ratio [0, ] None

Incidence rate difference

[- , + ] 1/Time

Incidence rate ratio [0, ] None


Concepts of interaction

Terms:– statistical interaction– effect modification or effect measure modification– synergy– 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

Main risk factor (X)

Effect modifier (Z)

Yes No

Yes R11 R10

No R01 R00RG 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 / Riskunexposed Riskexposed = Riskunexposed x RRA 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

Short allele (G)a

Life events

(E)

Risk Stratum

(R)

Risk (%)

No (-) No (-) R-,- 10

No (-) Yes (E) R-,E 17

Yes (G) No (-) RG,- 10

Yes (G) Yes (E) RG,E 33

A Short allele of the promoter region of the serotonin

transporter 5-HTT geneDunedin 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

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

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

3. What is the observed joint effect of A and B?4. What is the expected joint effect of A and B in

the absence of interaction?5. Is the observed joint effect similar to the

expected joint effect in the absence of interaction?




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

3. What is the observed joint effect of A and B?

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

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

1. RDE,-=0.17-0.10=0.07

2. RD-,G=0.10-0.10=0

3. RDOBSERVED E,G=0.33-0.10=0.23

4. RDEXPECTED E,G=0.07+0=0.07

5. RDOBSERVED E,G > RDEXPECTED E,G,

additive interaction




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

3. What is the observed joint effect of A and B?

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

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

6. What is the interaction magnitude

1. RDE,-=0.17-0.10=0.07

2. RD-,G=0.10-0.10=0

3. RDOBSERVED E,G=0.33-0.10=0.23

4. RDEXPECTED E,G=0.07+0=0.07

5. RDOBSERVED E,G > RDEXPECTED E,G,

additive interaction

6. RDE/ G IS PRESENT – RDE/ G IS ABSENT

= 0.23 - 0.07 =0.16interaction contrast

1. RRE,-=0.17/0.10=1.7

2. RR-,G=010/0.10=1.0

3. RROBSERVED E,G=0.33/0.10=3.3

4. RREXPECTED E,G=1.7x1.0=1.7

5. RROBSERVED E,G > RREXPECTED E,G,

multiplicative interaction

6. 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

Stressful life events

Intimacy problems

Yes No

Yes 32% 10%

No 3% 1%

Brown and Harris 1978

• Analysis on the additive scale:

• Analysis on the multiplicative scale:


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 Biological interaction can be defined under the counterfactual approach and the sufficient cause

approach– Using the counterfactual approach, there are 4 exposure categories possible with 2 binary variables. There are 16 possible

patterns of response types (given disease or no disease) to those 4 exposure categories. Ten of the categories can be considered interaction of some type (i.e., both of the 2 exposure types have an effect). 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.

– In the sufficient cause approach, the 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.


Possible response types for binary exposure

TYPE

Outcome (risk) Y for exposure combination

Interaction contrast (difference in risk differences) and causal typeX=1 X=0 X=1 X=0Z=1 Z=0 Z=1 Z=0

1 1 1 1 1 0=DOOMED (no effect for exposure combination)

2 1 1 1 0 -1=PARALLELISM (single + joint causation), factors compete to be INUS component causes in the same sufficient cause

3 1 1 0 1 1=RPEVENTIVE ANTAGONISM (z=1 blocks x=1 effect)

4 1 1 0 0 0=(z=1 is causal, x=1 is ineffective)

5 1 0 1 1 1=RPEVENTIVE ANTAGONISM (x=1 blocks z=1 effect)

6 1 0 1 0 0=(x=1 is causal, z=1 is ineffective)

7 1 0 0 1 2=RPEVENTIVE ANTAGONISM (each factor prevents development of disease when the other is absent)

8 1 0 0 0 1=CAUSAL SYNERGISM (each factor causes disease only if the other is present)

9 0 1 1 1 -1=PREVENTIVE SYNERGISM (one factor prevents development of disease if the other is present)

10 0 1 1 0 -2=CAUSAL ANTAGONISM (each factor causes disease only if the other is absent)

11 0 1 0 1 0=(x=1 is preventive, z=1 is ineffective)

12 0 1 0 0 -1=CAUSAL ANTAGONISM (x=1 blocks z=1 effect)

13 0 0 1 1 0=(z=1 is preventive, x=1 is ineffective)

14 0 0 1 0 -1=CAUSAL ANTAGONISM (z=1 blocks x=1 effect)

15 0 0 0 1 1= (single + joint prevention), compete to be INUS partners in the same sufficient cause

16 0 0 0 0 0=IMMUNE (no effect for exposure combination)


Interaction contrast

Interaction contrast=difference in risk differences

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

=R11-R10-R01+R00

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

Main risk factor (X)

Effect modifier (Z)

Yes No

Yes R11 R10

No R01 R00


Necessary conditions for interaction

1. Departures from additivity can only occur when interaction causal types are present in the cohort

2. Absence of interaction does not imply absence of interaction types because sometimes different interaction types counterbalance each other’s effect on the average risk

3. Departures from additivity could be of two kinds: 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

4. 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


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


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


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

4+ Stressful life events Genotype with short allele

Yes No

Yes 33% 17%

No 10% 10%

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

4+ Stressful life events Genotype with short allele

Yes No

Yes 0.33/0.67 0.17/0.83

No 0.10/0.90 0.10/0.90

RG Ch 16


Final notes on interaction

Superadditivity implies synergy, absence of superadditivity does not imply absence of synergy

In epidemiological studies we estimate average effects

In the presence of contravening effects (parallelism, antagonism), synergy will be difficult to detect


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


Arsenic Exposure from Drinking Water and Risk of Premalignant Skin Lesionsin Bangladesh, Hasan et al. 2006

The statistical significance of the joint effect of arsenic exposure and host characteristics was assessed by estimating relative excess risk due to interaction(RERI) and its 95 percent confidence intervals, as suggested by Hosmer and Lemeshow (34). RERI is estimated as follows:RERI~POR1k-POR10-POR0k+1,

where POR1k indicates the POR for skin lesion comparing participants with arsenic exposure at k level and a hypothesized more susceptible attribute (e.g., male gender) with the reference group, that is, participants with the lowest arsenic exposure level and a less susceptible attribute (e.g., female gender);

POR0k indicates the POR for skin lesion comparing participants with arsenic exposure at k level alone with the reference group;

POR10 denotes the POR for skin lesion comparing participants with a more susceptible attribute (e.g., male gender) alone with the reference group.

week 9 - interaction 1 i nterpretation of epi studies ii : i nteraction and e ffect- m easure m...

Documents

measures of effect

measure of effect

risk differencerelative

heterogeneity of risk

odds ratio relative

effect modificationrg

risk differencer11 r00

types of interaction