Interaction

CHD Anger Aspirin CHD Anger Interaction = “Effect modification”: The “effect” of the risk factor -- anger – on the outcome – CHD -- differs depending on the presence or absence of a third factor (effect modifier) --aspirin. The third factor (aspirin) modifies the effect of the risk factor (anger) on the outcome (CHD). Note: to assess interaction, a minimum of 3 variables were needed in this study: Aspirin Anger Coronary Heart Disease (CHD)

How is “effect”* measured in epidemiologic studies? If effect is measured on an additive or absolute scale (attributable risks)  additive interaction assessment (Attributable Risk model). If effect is measured on a relative (ratio) scale (relative risks, odds ratios, etc.)  multiplicative interaction assessment (Relative Risk model). *For practical purposes in this lecture, “effect” refers to associations that may or may not be causal.

Two strategies to evaluate interaction based on different, but equivalent definitions: Effect modification (homogeneity/heterogeneity of effects) Comparison between joint expected and joint observed effects

First strategy to assess interaction: Effect Modification ADDITIVE (attributable risk) interaction Hypothetical example of absence of additive interaction ZAIncidence rate (%)AR exp to A (%) No 10.0 Yes20.0 YesNo30.0 Yes40.0 Potential effect modifier Potential risk factor of primary interest

First strategy to assess interaction: Effect Modification ADDITIVE (attributable risk) interaction Hypothetical example of absence of additive interaction ZAIncidence rate (%)AR exp to A (%) No 10.0 Yes20.0 YesNo30.0 Yes40.0 Conclude: Because AR’s associated with A are not modified by exposure to Z, there is no additive interaction. 10.0

Hypothetical example of presence of additive interaction Conclude: Because AR’s associated with A are modified by exposure to Z, additive interaction is present ZAIncidence rate (%)AR exp to A (%) No 5.0 Yes10.0 YesNo10.0 Yes30.0 First strategy to assess interaction: Effect Modification ADDITIVE (attributable risk) interaction

Z+ Z- AR A Example 1 Conclude: -The stratum-specific effects (AR) are homogeneous - Z does not modify the effect of A -There is no (additive) interaction Z+ Z- Example 2 Conclude: -The stratum-specific effects (AR) are heterogeneous - Z modifies the effect of A -There is (additive) interaction AR A Absolute scale

Example of Effect Modification (Interaction) in a Clinical Trial with a Continuous Outcome From: Szklo, Arch Dermatol 2000;136:1546 (Based on Gallagher et al, 2000)

Freckles, % New Nevi, No. Example of Freckling as an Interacting Variable (Effect Modifier) Sunscreen Control Sunscr << Cont Sunscr < Cont From: Szklo, Arch Dermatol 2000;136:1546 (Based on Gallagher et al, 2000)

Hypothetical example of absence of multiplicative interaction ZAIncidence rate (%)RR A No 10.0 Yes20.0 YesNo25.0 Yes50.0 Conclude: Because RR’s associated with A are not modified by exposure to Z, there is no multiplicative interaction. 2.0 First strategy to assess interaction: Effect Modification MULTIPLICATIVE (ratio-based) interaction

Hypothetical example of presence of multiplicative interaction ZAIncidence rate (%)RR A No 10.0 Yes20.0 YesNo25.0 Yes125.0 Conclude: Because RR’s associated with A are modified by exposure to Z, multiplicative interaction is present First strategy to assess interaction: Effect Modification MULTIPLICATIVE (ratio-based) interaction

A-A+ Incidence rate (%) Z+ Z- Example 1 Z+ Z- Example 2 Is this the best way to display the data? NO!

Z+ Z- To assess multiplicative effects, use a log scale: Conclude: -The stratum-specific effects (RR) are homogeneous - Z does not modify the effect of A -There is no (multiplicative) interaction Z+ Z- Conclude: -The stratum-specific effects (RR) are heterogeneous - Z modifies the effect of A -There is (multiplicative) interaction Example 1 Example 2

Two strategies to evaluate interaction based on different, but equivalent definitions: Effect modification (homogeneity/heterogeneity of effects) Comparison between joint expected and joint observed effects

Second strategy to assess interaction: (based on the calculation of “joint effects”) AZ Individual effects Expected joint effect + Observed joint effect A+Z No interaction Observed joint effect A+Z Synergism (Positive Interaction) Observed joint effect A+Z Antagonism (Negative Interaction) +I -I

The two definitions and strategies are completely equivalent. It is impossible to conclude that there is (or there is not) interaction using one strategy, and reach the opposite conclusion upon use of the other strategy! Thus, when there is effect modification, the joint observed and the joint expected effects will be different.

Second strategy to assess interaction: comparison of joint expected and joint observed effects Additive interaction Reference

Second strategy to assess interaction: comparison of joint expected and joint observed effects Additive interaction Reference Independent effects of:  A  Z  A + Z

Second strategy to assess interaction: comparison of joint expected and joint observed effects Additive interaction Conclude: Because the observed joint AR is the same as that expected by adding the individual AR’s, there is no additive interaction (that is, the same conclusion as when looking at the stratified AR’s) observed Joint observed AR A+Z+ = 30% expected Joint expected AR A+Z+ = AR A+Z- + AR A-Z+ = 30% Reference

Expected Second strategy to assess interaction: comparison of joint expected and joint observed effects Additive interaction Conclude: Because the observed joint AR is different from that expected by adding the individual AR’s, additive interaction is present (that is, the same conclusion as when looking at the stratified AR’s) observed Joint observed AR = 25% expected Joint expected AR = AR A+Z- + AR A-Z+ = 10% Reference

Second strategy to assess interaction: comparison of joint expected and joint observed effects Multiplicative interaction (that is, the same conclusion as when looking at the stratified RR’s) Conclude: Because the observed joint RR is same as that expected by adding the individual RR’s in a multiplicative scale (equivalent to multiplying the individual RR’s), multiplicative interaction is not present observed Joint observed RR A+Z+ = 5.0 expected Joint expected RR A+Z+ = RR A+Z-  RR A-Z+ = 2.0  2.5 = 5.0 Reference

Second strategy to assess interaction: comparison of joint expected and joint observed effects Multiplicative interaction Conclude: Since the observed joint RR is different from that expected by multiplying the individual RR’s, there is multiplicative interaction (that is, the same conclusion as when looking at the stratified RR’s) observed Joint observed RR A+Z+ = 12.5 expected Joint expected RR A+Z+ = RR A+Z-  RR A-Z+ = 2.0  2.5 = 5.0 Reference

How can we assess interaction in case-control studies?

First strategy to assess interaction: Effect Modification Case-control study Prospective Study ZAIncidence rate (%)AR exp to A (%) No 5.0 Yes10.0 YesNo Yes30.0 Additive interaction cannot be assessed in case-control studies by using the effect modification (homogeneity/heterogeneity) strategy, as no incidence rates are available to calculate attributable risks in the exposed

First strategy to assess interaction: Effect Modification Layout of table to assess MULTIPLICATIVE interaction Case-control study

Family history of clubfoot Maternal smoking CasesControls Yes 147 No1120 NoYes No2032,143 Honein et al. Family history, maternal smoking, and clubfoot: an indication of gene-environment interaction. Am J Epidemiol 2000;152: Odds Ratios for the association among isolated clubfoot, maternal smoking, and a family history of clubfoot, Atlanta, Georgia, Hypothesis: Family History is a potential effect modifier of the association between Maternal Smoking and clubfoot Use the first strategy (homogeneity/heterogeneity) to evaluate the presence of multiplicative interaction

Family history of clubfoot Maternal smoking CasesControls Stratified OR maternal smk Yes No1120 NoYes No2032,143 Honein et al. Family history, maternal smoking, and clubfoot: an indication of gene- environment interaction. Am J Epidemiol 2000;152: Odds Ratios for the association among isolated clubfoot, maternal smoking, and a family history of clubfoot, Atlanta, Georgia, Conclude: Since the stratified ORs are different (heterogeneous), there is multiplicative interaction. Now evaluate the same hypothesis (that there is an interaction between family history of clubfoot and maternal smoking) using the second strategy: comparison between joint observed and joint expected “effects”. reference

Second strategy to assess interaction: comparison of joint observed and expected effects Layout of table to assess both ADDITIVE and MULTIPLICATIVE interaction Case-control study Under ADDITIVE MODEL: Exp’d OR ++ = OR +- + OR OR -+ OR +- OR ++ Note common reference category

If disease is “rare” (e.g., <5%): Derivation of formula for expected joint OR observed RR ++ RR RR expected

Derivation of formula: Exp ected OR ++ = OR +- + OR Intuitive graphical derivation:* OR Baseline+Excess due to A Baseline+Excess due to Z OR -- OR -+ OR +- Exp’d OR ++ EXC Z Baseline BL EXC A BL EXC Z BL EXC A [EXC A +BL] + [EXC Z +BL] - BL = OR -+ + OR +- – 1.0 *For a more formal derivation, see Szklo & Nieto, pp (not required). BL

OR OR -- OR -+ OR +- Exp’d OR ++ Observed OR ++ Conclude: If the observed joint OR is the same as the expected under the additive model, there is no additive interaction

OR OR -- OR -+ OR +- Exp’d OR ++ Observed OR ++ Conclude: If the observed joint OR is different than the expected under the additive model, there is additive interaction Excess due to interaction (“interaction term”) Excess due to the joint effects of A and Z

Family history of clubfoot Maternal smoking CasesControlsStratified ORs ORs using No/No as the reference category Expected under the ADDITIVE model Yes No NoYes No2032, (reference) Honein et al. Family history, maternal smoking, and clubfoot: an indication of gene- environment interaction. Am J Epidemiol 2000;152: Odds Ratios for the association among isolated clubfoot, maternal smoking, and a family history of clubfoot, Atlanta, Georgia, Conclude: Since the observed joint OR(20.3) is different from the joint OR expected under the additive model (6.26), there is additive interaction Effect of Maternal Smoking only, i.e., in the absence of Family History Effect of Family History only, i.e., in the absence of Maternal Smoking – 1.0= Second Strategy: Comparison between joint expected and joint observed effects - - allows assessment of both ADDITIVE and MULTIPLICATIVE interactions--

Second strategy to assess interaction: comparison of joint observed and expected effects Layout of table to assess both ADDITIVE and MULTIPLICATIVE interaction Case-control study Under ADDITIVE MODEL: Exp’d OR ++ = OR +- + OR OR -+ OR +- OR ++ Under MULTIPLICATIVE MODEL: Exp’d OR ++ = OR +-  OR -+

Family history of clubfoot Maternal smoking CasesControlsStratified ORs ORs using No/No as the reference category Expected under the MULT. model Yes No NoYes No2032, (reference) Honein et al. Family history, maternal smoking, and clubfoot: an indication of gene- environment interaction. Am J Epidemiol 2000;152: Odds Ratios for the association among isolated clubfoot, maternal smoking, and a family history of clubfoot, Atlanta, Georgia, Conclude: Since the observed joint OR(20.3) is different from the joint OR expected under the multiplicative model(8.42), there is multiplicative interaction Effect of Maternal Smoking only, i.e., in the absence of Family History Effect of Family History only, i.e., in the absence of Maternal Smoking  5.81 =

Back to the terms... Synergism or Synergy: The observed joint “effect” is greater than that expected from the individual “effects”. Which is equivalent to saying that the “effect” of A in the presence of Z is stronger than the “effect” of A when Z is absent. Antagonism: The observed joint “effect” is smaller than that expected from the individual “effects”. Which is equivalent to saying that the “effect” of A in the presence of Z is weaker than the “effect” of A when Z is absent Note: the expressions “synergism/antagonism” and “effect modification” should ideally be reserved for situations in which one is sure of a causal connection. In the absence of evidence supporting causality, it is preferable to use terms such as “heterogeneity” or “positive/negative interaction”.

Terminology Positive interaction = Synergism = “More than additive effect” (for the additive model) or “More than multiplicative effect” (for the multiplicative model) Negative interaction = Antagonism = “Less than additive/multiplicative effect” Some investigators reserve the term “synergy” to define a biologically plausible interaction

Quantitative vs. qualitative interaction

Family history of clubfoot Maternal smoking CasesControls Stratified OR maternal smk Yes No1120 NoYes No2032,143 Honein et al. Family history, maternal smoking, and clubfoot: an indication of gene- environment interaction. Am J Epidemiol 2000;152: Odds Ratios for the association among isolated clubfoot, maternal smoking, and a family history of clubfoot, Atlanta, Georgia, Quantitative interaction: both ORs are in the same direction(>1.0), but they are heterogeneous

Am J Epidemiol 1995;142: Reproductive Health Study, retrospective study of 1,430 non-contraceptive parous women, Fishkill, NY, Burlington, VT, Odds ratios are not only different; they have different directions (>1, and <1). Smoking modifies the effect of caffeine on delayed conception in a qualitative manner, i.e., there is qualitative interaction.

When there is qualitative interaction in one scale (additive or multiplicative), it must also be present in the other A- A+ Risk of outcome Z- Z+ Qualitative Interaction: Effect ModifierRisk FactorIncidence/1000AR A RR A Z+A / A-5.0Reference1.0 Z-A+3.0-3/ A-6.0Reference1.0 Interaction in both scales

When there is qualitative interaction in one scale (additive or multiplicative), it must also be present in the other A- A+ Risk of outcome Z- Z+ Qualitative Interaction: Effect ModifierRisk FactorIncidence/1000AR A RR A Z+A / A-5.0Reference1.0 Z-A+3.0-3/ A-6.0Reference1.0

When there is qualitative interaction in one scale (additive or multiplicative), it must also be present in the other A- A+ Risk of outcome Z- Z+ “cross-over” A- A+ Risk of outcome Z- Z+ Another type of qualitative interaction: “effect”of A is flat in one stratum of the effect modifier; in the other stratum, an association is observed

Circulation 2000;101: Age-adjusted HR of CHD: Normotensive persons Anger score LowModerateHigh Hypertensive persons Anger score LowModerateHigh Example of qualitative interaction (CHD) normotensive CHD event-free survival probabilities among normotensive individuals by trait anger scores Days of follow-up hypertensive CHD event-free survival probabilities among hypertensive individuals by trait anger scores Days of follow-up Low Moderate High Anger score: Low (10-14) Moderate (15-21 High (22-40) CHD-free cumulative probabilities

Further issues for discussion Quantitative vs. qualitative interaction  Reciprocity of interactionReciprocity of interaction If Z modifies the effect of A on disease Y, then Z will necessarily modify the effect of Z on disease Y

Reciprocity of interaction The decision as to which is the “principal” variable and which is the effect modifier is arbitrary, because if A modifies the effect of Z, then Z modifies the effect of A. Z modifies the effect of A A modifies the effect of Z

Matched case-control study (matching by gender) of the relationship of risk factor X (e.g., alcohol drinking ) and disease Y (e.g., esophageal cancer) Pair No.CaseControlOR by gender 1 (male)+- 2 (male)+- 3 (male)-+ 4 (male)+- 5 (male)++ 6 (female)-- 7 (female)+- 8 (female)-+ 9 (female)++ 10 (female)-- Total (Pooled) Odds Ratio INTERACTION IS NOT CONFOUNDING

Matched case-control study (matching by gender) of the relationship of risk factor X (e.g., alcohol drinking ) and disease Y (e.g., esophageal cancer) Pair No.CaseControlOR by gender 1 (male)+- 2 (male)+- 3 (male)-+ 4 (male)+- 5 (male)++ 6 (female)-- 7 (female)+- 8 (female)-+ 9 (female)++ 10 (female)-- Total (Pooled) Odds Ratio4/2= 2.0 INTERACTION IS NOT CONFOUNDING

Matched case-control study (matching by gender) of the relationship of risk factor X (e.g., alcohol drinking ) and disease Y (e.g., esophageal cancer) Pair No.CaseControlOR by sex 1 (male)+- 3/1 = (male)+- 3 (male)-+ 4 (male)+- 5 (male)++ 6 (female)-- 1/1= (female)+- 8 (female)-+ 9 (female)++ 10 (female)-- Total (Pooled) Odds Ratio4/2= 2.0 INTERACTION IS NOT CONFOUNDING

Further issues for discussion Quantitative vs. qualitative interaction Reciprocity of interaction Interpretation and uses of interactionInterpretation and uses of interaction –Additive interaction as “public health interaction” –Additive interaction as “public health interaction” (term coined by Rothman)

Current Smoking Status Low Vitamin C intake (mg/day) Odds Ratio No 1.0 YesNo6.8 NoYes1.8 Yes 10.6 Joint effects of current cigarette smoking and low consumption of vitamin C (≤ 100 mg/day) with regard to adenocarcinoma of the salivary gland, San Francisco-Monterey Bay area, California, (Horn-Ross et al. Diet and risk of salivary gland cancer. Am J Epidemiol 1997;146:171-6) Additive Model: Expected joint Odds Ratio = – 1.0= 7.6 Positive additive interaction= “Public Health interaction” Multiplicative Model: Expected joint Odds Ratio = 6.8  1.8 = 12.4 Conclude: For Public Health purposes, ignore negative multiplicative interaction, and focus on smokers for prevention of low vitamin C intake Negative multiplicative interaction

Additive interaction as “Public Health interaction” Incidence of disease “Y” by smoking and family history of “Y” Thus, if there are enough subjects who are positive for both variables and if resources are limited, smokers with a positive family history should be regarded as the main “target” for prevention  examine the prevalence of (Fam HIst+ and Smk+ ) and estimate the attributable risk in the population Positive additive interaction (synergism), but negative multiplicative interaction (antagonism) EM- effect modifier RF- risk factor of interest

Further issues for discussion Quantitative vs. qualitative interaction  Reciprocity of interaction  Interpretation and uses of interactionInterpretation and uses of interaction –Additive interaction as “public health interaction”  –Biological interaction

Further issues for discussion Quantitative vs. qualitative interaction  Reciprocity of interaction  Interpretation and uses of interactionInterpretation and uses of interaction –Additive interaction as “public health interaction”  –Biological interaction –Statistical interaction (not causal) Differential confounding across strata of the effect modifier  Differential misclassification across strata of the effect modifier  The dose (amount of exposure) may be higher in one stratum than in the other Biologic interaction: –Consistent with pathophysiologic mechanisms (biologic plausibility) –Confirmed by animal studies –What is best model from the biologic viewpoint?  No one knows for sure… Think about the specific condition under study – Examples: trauma, cancer Problem: Epidemiology usually assesses proximal cause X1  X2  X3  Y

Conclusion If heterogeneity is present… is there interaction? –What is the magnitude of the difference? (p-value?) –Is it qualitative or just quantitative? –Is it biologically plausible? If we conclude that there is interaction, what should we do? –Report the stratified measures of association … The interaction may be the most important finding of the study!