Interaction and Effect-Measure Modification Lydia B. Zablotska, MD, PhD Associate Professor Department of Epidemiology and Biostatistics Week 7 - Interaction

Learning Objectives Statistical interaction Multiplicative and additive interaction Biologic interaction Evaluation of interaction, presentation of results Attributable fraction estimation Week 7 - Interaction

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) Week 7 - Interaction

Comparison of absolute and relative effect measures Numerical Range Dimensionality Risk difference [-1, +1] None Risk ratio [0, ] Incidence rate difference [- , + ] 1/Time Incidence rate ratio Week 7 - Interaction Rothman 2002

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) Week 7 - Interaction 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 R11 R10 R01 R00 Week 7 - Interaction 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) https://www.ctspedia.org/do/view/CTSpedia/InterMultipl Week 7 - Interaction 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 Deviation from a specified model form (additive or multiplicative). Week 7 - Interaction 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) Week 7 - Interaction

Multiplicative interaction RR = Riskexposed / Riskunexposed Riskexposed = 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) Week 7 - Interaction

Assessment of interaction for binary data Week 7 - 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 (-) R-,- 10 Yes (E) R-,E 17 Yes (G) RG,- RG,E 33 a Short allele of the promoter region of the serotonin transporter 5-HTT gene Exposure – four or more stressful life events between ages 21 and 26 Outcome – risk of past-year depression at age 26 Dunedin Child-Development Study, Caspi et al. 2002, 2003 Week 7 - Interaction

Ways to assess interaction By stratification By comparing expected and observed joint effects By graphical assessment By estimating interaction magnitude By using statistical tests of interaction: Test of homogeneity LRT comparing test and nested models Week 7 - Interaction

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: for both measures of association, the effect of E when G is present is larger than the effect of E when G is absent Quantitative (ordinal) interaction – stratum-specific effects differ by magnitude but not by direction Qualitative (disordinal) interaction – the same exposure could be protective in one stratum of the other factor but deleterious in the other Week 7 - Interaction

Assessing interaction by 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? Assume that A and B do not interact. Week 7 - Interaction

Comparing expected and observed joint effects 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 (superadditivity, additive synergism) 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? Week 7 - Interaction

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 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 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 Week 7 - Interaction

Assessing interaction by graphical representation Week 7 - Interaction Zablotska et al. 2013

Assessing interaction by graphical representation Week 7 - Interaction Preston et al. 2007

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 32% 10% 3% 1% Analysis on the additive scale: Analysis on the multiplicative scale: Week 7 - Interaction Brown and Harris 1978

Trouble with assessment of synergy Interaction of vulnerability factors (e.g., fear of intimacy) and stressful life events in causing depression Stressful life events (E) Intimacy problems (M) Yes No 32% 10% 3% 1% Debate in Psych Epi about vulnerability factors for depression Analysis on the additive scale: 1. RDE,-=0.10-0.01=0.09; 2. RD-,M=0.03-0.01=0.02; 3. RDOBSERVED E,M=0.32-0.01=0.31; 4. RDEXPECTED E,M=0.09+0.02=0.11; 5. RDOBSERVED E,M>RDEXPECTED E,M; additive interaction Analysis on the multiplicative scale: RRE,-=0.10/0.01=10; 2. RR-,M=0.03/0.01=3; 3. RROBSERVED E,M=0.32/0.01=32; 4. RREXPECTED E,M=10x3=30; 5. RROBSERVED E,M~RREXPECTED E,M no multiplicative interaction Brown and Harris 1978 Week 7 - Interaction Tenent and Bebbington 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? Week 7 - Interaction

Biological interaction Terms: Biological interaction Causal interaction Definition: Modification of potential-response types A process that explains 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 Week 7 - Interaction

Biological 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 Two possible types of interaction between component causes: Synergism (biological interaction) the two component causes work together as INUS partners in the same sufficient cause, they act in synergy. The individual would get disease if they are exposed to both A and B but not get disease if exposed to only A or B. Parallelism 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 and/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. Week 7 - Interaction

4 possible types: parallelism A or B or AB U Week 7 - Interaction Parallelism could be pies 2, 3 or 4 (presented jointly in pie #5) Synergism could only be pie #4 Week 7 - Interaction

Biological 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 Week 7 - Interaction RG Ch 5

Interaction vs. effect modification Effect modification by Q of the effect of E on D without interaction between the effects of E and Q on D Potential interaction between the effects of E and Q on D without effect modification by Q of the effect of E on D E - drug in an RCT, D - hypertension, X - person’s genotype, and Q - person’s hair color. In this example, Q might serve as an effect modifier for the effect of E on D since Q is a proxy for X. We will thus likely have a situation in which EDe 1Q q EDe 0Q q is not constant over q. However, because Q is not itself a cause of D, there will be no interaction between the effects of E and Q on D. Because Q has no effect whatsoever on D, we will have EDe 1,q - EDe 0,q =EDe 1= EDe 0 for all values of q. Interventions on Q do nothing to D, so there can be no interaction between E and Q. A variable Q can thus be an effect modifier for the effect of E on D without there being an interaction between the effects of E and Q on D. In the second RCT, sugar intake mightaffect exercise by making children more hyperactive. Suppose further that the effects of the diet drug E and exercise Q interact, if we were to intervene to give children the drug and also to force exercise, then the effect of these 2 interventions together, compared with the baseline of no drug and no exercise, would be greater than the sum of the effects of each intervention considered separately. We would then say that there is an interaction between the effects of E and Q on D. However, it does not follow from this that Q is an effect modifier for the effect of E on D. Suppose that sugar intake had 2 effects on weight; first, high levels of sugar intake within the diet will likely increase weight gain; second, sugar intake might give children more energy, making them hyperactive and thus more likely to exercise, and thereby lowering their weight through exercise. It is furthermore possible that the indirect effect of sugar intake that lowers weight through exercise essentially cancels out the direct effect of sugar intake which increases weight. In such cases, exercise Q may not be an effect modifier for the effect of drug E on weight D. E - drug in an RCT for weight loss for obese children, D – weight at 6 mos, X – sugar intake, and Q – some measure of exercise. Week 7 - Interaction VanderWeele 2009

Biological interaction, cont’ed 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. Week 7 - Interaction

Week 7 - Interaction

Possible response types for binary exposure Person TYPE Outcome (risk) Y for exposure combination Interaction contrast (difference in risk differences) and causal type IC = R11 – R01 – R10 + R00 X=1 X=0 Z=1 Z=0 R11 R01 RR10 R00 1 0=DOOMED (no effect for exposure combination) 2 -1=PARALLELISM (single + joint causation), factors compete to be INUS component causes in the same sufficient cause 3 1=RPEVENTIVE ANTAGONISM (z=1 blocks x=1 effect) 4 0=Z ONLY TYPE (z=1 is causal, x=1 is ineffective) 5 1=RPEVENTIVE ANTAGONISM (x=1 blocks z=1 effect) 6 0=X ONLY TYPE (x=1 is causal, z=1 is ineffective) 7 2=RPEVENTIVE ANTAGONISM (each factor prevents development of disease when the other is absent) 8 1=CAUSAL SYNERGISM (each factor causes disease only if the other is present) 9 -1=PREVENTIVE SYNERGISM (one factor prevents development of disease if the other is present) 10 -2=CAUSAL ANTAGONISM (each factor causes disease only if the other is absent) 11 0=(x=1 is preventive, z=1 is ineffective) 12 -1=CAUSAL ANTAGONISM (x=1 blocks z=1 effect) 13 0=(z=1 is preventive, x=1 is ineffective) 14 -1=CAUSAL ANTAGONISM (z=1 blocks x=1 effect) 15 1= (single + joint prevention), compete to be INUS partners in the same sufficient cause 16 0=IMMUNE (no effect for exposure combination) Darroch 1997 R11, R10, R01, and R00 are average risks (i.e., incidence proportions, population proportions or probabilities) Week 7 - Interaction

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) Main risk factor (X) Effect modifier (Z) Yes No R11 R10 R01 R00 Week 7 - Interaction 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) Week 7 - Interaction 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 Week 7 - Interaction

This is all good, but how do we know the response types? 16 1 6 8 R Week 7 - Interaction R R R

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 Week 7 - Interaction

Summary of R&G scheme under counterfactual theory 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 Week 7 - Interaction

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. Week 7 - Interaction

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. Week 7 - Interaction

16 1 6 8 2 R R R R Week 7 - Interaction

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. Week 7 - Interaction

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 Week 7 - Interaction

Example from Darroch 1997 Week 7 - Interaction

Darroch vs. R&G p8 = (R11 – R01) – (R10 – R00) R R R R R R R R R R 6 2 R R R R R R Week 7 - Interaction

Darroch vs. R&G p8 = (20.7 – 5.1) – (7.2 – 1) = 9.4 > 0 - superadditivity p8 = (R11 – R01) – (R10 – R00) R R R R R R 8 R R 6 2 R R R R R R 8 Week 7 - Interaction

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 Week 7 - Interaction

Estimating synergy in epi studies 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: Interaction contrast and interaction contrast ratio Relative excess risk due to interaction (RERI) – for multivariate models or models with continuous exposures Attributable proportion Synergy index We generally analyze our data according to multiplicative models such as logistic regression. This practice poses a dilemma for the assessment of synergy if we think that synergy should be assessed under an additive model. Most researchers assess interaction by entering a product term into the linear or logistic regression model. However, the interpretation of the regression coefficient of the product term depends on the statistical model. In linear regression analysis the regression coefficient of the product term means departure from additivity, whereas in logistic regression (and in Cox regression) the regression coefficient of the product term estimates departure from multiplicativity (Appendix 1). Biologic interaction means that two causes are both needed to cause disease; the two causes are component causes in the same causal model. Rothman has argued that when biologic interaction is examined, we should focus on interaction as departure from additivity rather than departure from multiplicativity.1 In aetiologic epidemiologic research, we are interested in biologic interaction rather than in statistical interaction. However, by adding a product term to a logistic model, interaction is (unknowingly) estimated as departure from multiplicativity. Week 7 - Interaction

In a 2x2 table 4+ Stressful life events Genotype with short allele Yes 33% 17% 10% IC=0.33-0.17-0.10+0.10=0.16 >0  synergy Dunedin Child-Development Study, Caspi et al. 2002, 2003 Week 7 - Interaction

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 0.33/0.67 0.17/0.83 0.10/0.90 Week 7 - Interaction RG Ch 16

Bringing it all together: From synergy to its mathematical representation Brown and Harris 1978 Week 7 - Interaction

Causes of depression: Theory about life events and their interaction with intimacy problems Week 7 - Interaction

Assessing interaction between life events and intimacy problems Week 7 - Interaction

Relationship between observed risk and unobserved types Week 7 - Interaction

Mathematical model representing conceptual model for interaction Stressful life events Intimacy problems Yes No 32% 10% 3% 1% 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 Week 7 - Interaction

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. Week 7 - Interaction

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 Week 7 - Interaction

Relationship to single-hit (1), no-hit (2) or multistage (3) models of carcinogenesis model 1 incorporates the restriction that there is no synergism and models 2 and 3 the restriction that there is no parallelism. model 1 must be false if the additive interaction is positive, and models 2 and 3 must be false if the additive interaction is negative. Week 7 - Interaction

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% Stressful life events Intimacy problems Yes No 32% 10% 3% 1% Week 7 - Interaction

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) Risk estimate*mean dose of kth class*weight of the kth class in the population (represented by controls) RRkth=1+mean dose in kth class*ERR/Gy ARkth=weight of the kth class*(RRkth-1)/overall RR Week 7 - Interaction RG Ch 16

Week 7 - Interaction Zablotska et al. 2013

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 Week 7 - Interaction

Presentation of results for effect modification Knol and VanderWeele 2012 Presentation of results for effect modification Step 3 Step 1 Step 2 A cohort study by Knol et al. investigated whether the risk of antidepressant use (the exposure of interest A) on diabetes (the outcome D) was modified by the chronic disease score (the potential effect modifier X). The chronic disease score is a measure of the chronic disease status among drug users and can be considered as an indicator of an individual’s morbid- ity and overall health status. The score was dichotomized as 0 and 5 1, where 0 means that no chronic disease is present. Present RRs, ORs or RDs with CIs for each stratum of A and X with a single reference category (possibly taken as the stratum with the lowest risk of D). Present RRs, ORs or RDs with CIs for A within strata of X. Present measures of effect modification on both additive (e.g. RERI) and multiplicative scales with CIs and P-values. List the confounders for which the relation between A and D was adjusted. If A has more than two levels, additional columns could be added

Presentation of results for interaction Knol and VanderWeele 2012 Step 1 Step 2 Step 3 Van Gils et al. investigated the interaction between dietary intake of vitamin E (exposure of interest A) and a polymorphism in a gene coding for proteins in the DNA repair system (XRCC1 Codon 399 genotype) (exposure of interest B) on the risk of prostate cancer(outcome D) in a case–control study. Present RRs, ORs or RDs with CIs and P–values for each stratum of A and B with a single reference category (possibly taken as the stratum with the lowest risk of D). Present RRs, ORs or RDs with CIs and P-values of the effect of A on D in strata of B and of B on D in strata of A. Present measures of effect modification on both additive (e.g. RERI) and multiplicative scales with CIs and P-values. List the confounders for which the relation between A and D and for which the relation between B and D were adjusted