Week 9 - Interaction 1 I nterpretation of epi studies II : I nteraction and E ffect- M easure M odification Lydia B. Zablotska, MD, PhD Associate Professor Department of Epidemiology and Biostatistics

Week 9 - Interaction 2 Learning Objectives Biological vs. statistical Multiplicative and additive Evaluation of interaction, presentation of results

Week 9 - Interaction 3 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 9 - Interaction 4 Comparison of absolute and relative effect measures (Rothman 2002) MeasureNumerical RangeDimensionality Risk difference[-1, +1]None Risk ratio [0,  ] None Incidence rate difference [- , +  ] 1/Time Incidence rate ratio [0,  ] None

Week 9 - Interaction 5 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

Week 9 - Interaction 6 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) YesNo YesR 11 R 10 NoR 01 R 00 RG Ch 5

Week 9 - Interaction 7 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

Week 9 - Interaction 8 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

Week 9 - Interaction 9 Additive interaction RD = Risk exposed – Risk unexposed A and B are risk factors with risks R a,- and R -,b and individual risk differences: RD a,- = R a,- – R -,- RD -,b = R -,b – R -,- RD a,b is a RD for those exposed to both A and B and those exposed to neither RD a,b = RD a,- + RD -,b – A and B are non-interacting risk factors RD a,b  RD a,- + RD -,b – Additive interaction between A and B – RD a,b > RD a,- + RD -,b – Additive synergy (positive additive interaction) – RD a,b < RD a,- + RD -,b – Additive antagonism (negative additive interaction)

Week 9 - Interaction 10 Multiplicative interaction RR = Risk exposed / Risk unexposed  Risk exposed = Risk unexposed x RR A and B are risk factors with risks R a,- and R -,b and individual risk ratios: RR a,- = R a,- / R -,- RR -,b = R -,b / R -,- RR a,b is a RR for those exposed to both A and B over those exposed to neither RD a,b = RD a,- x RD -,b – A and B are non-interacting risk factors RD a,b  RD a,- x RD -,b – Multiplicative interaction between A and B – RD a,b > RD a,- + RD -,b – Multiplicative synergy (positive multiplicative interaction) – RD a,b < RD a,- + RD -,b – Multiplicative antagonism (negative multiplicative interaction)

Week 9 - Interaction 11 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 No (-)Yes (E)R -,E 17 Yes (G)No (-)R G,- 10 Yes (G)Yes (E)R G,E 33 A Short allele of the promoter region of the serotonin transporter 5-HTT gene Dunedin Child-Development Study, Caspi et al. 2002, 2003

Week 9 - Interaction 12 Assessing interaction by stratification Effect modification by presence of short allele G on the association between stressful life events E and risk of depression RD E/G is absent = =0.07; RR E/G is absent = 0.17/0.10=1.7 RD E/G is present = =0.23; RR E/G is present = 0.33/0.10=3.3 Both RD and RR are heterogeneous

Week 9 - Interaction 13 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?

Week 9 - Interaction 14 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 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. RD E,- = = RD -,G = =0 3. RD OBSERVED E,G = = RD EXPECTED E,G =0.07+0= RD OBSERVED E,G > RD EXPECTED E,G, additive interaction

Week 9 - Interaction 15 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 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. RD E,- = = RD -,G = =0 3. RD OBSERVED E,G = = RD EXPECTED E,G =0.07+0= RD OBSERVED E,G > RD EXPECTED E,G, additive interaction 6. RD E/ G IS PRESENT – RD E/ G IS ABSENT = =0.16 interaction contrast 1. RR E,- =0.17/0.10= RR -,G =010/0.10= RR OBSERVED E,G =0.33/0.10= RR EXPECTED E,G =1.7x1.0= RR OBSERVED E,G > RR EXPECTED E,G, multiplicative interaction 6. RR E/ G IS PRESENT / RR E/ G IS ABSENT = 3.3 / 1.7 =1.9

Week 9 - Interaction 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 YesNo Yes32%10% No3%1% Brown and Harris 1978 Analysis on the additive scale: Analysis on the multiplicative scale:

Week 9 - Interaction 17 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 9 - Interaction 18 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.

Week 9 - Interaction 19 Possible response types for binary exposure TYPE Outcome (risk) Y for exposure combination Interaction contrast (difference in risk differences) and causal type X=1X=0X=1X=0 Z=1Z=0Z=1Z= =DOOMED (no effect for exposure combination) =PARALLELISM (single + joint causation), factors compete to be INUS component causes in the same sufficient cause =RPEVENTIVE ANTAGONISM (z=1 blocks x=1 effect) =(z=1 is causal, x=1 is ineffective) =RPEVENTIVE ANTAGONISM (x=1 blocks z=1 effect) =(x=1 is causal, z=1 is ineffective) =RPEVENTIVE ANTAGONISM (each factor prevents development of disease when the other is absent) =CAUSAL SYNERGISM (each factor causes disease only if the other is present) =PREVENTIVE SYNERGISM (one factor prevents development of disease if the other is present) =CAUSAL ANTAGONISM (each factor causes disease only if the other is absent) =(x=1 is preventive, z=1 is ineffective) =CAUSAL ANTAGONISM (x=1 blocks z=1 effect) =(z=1 is preventive, x=1 is ineffective) =CAUSAL ANTAGONISM (z=1 blocks x=1 effect) = (single + joint prevention), compete to be INUS partners in the same sufficient cause =IMMUNE (no effect for exposure combination)

Week 9 - Interaction 20 Interaction contrast Interaction contrast=difference in risk differences IC=RD X,- – RD -,Z =(R 11 -R 01 )-(R 10 -R 00 ) = (R 11 -R 10 )-(R 01 -R 00 ) =R 11 -R 10 -R 01 +R 00 Causal additivity=no causal interaction R 11 – R 00 = (R 10 -R 00 ) + (R 01 -R 00 )= (p6+p13-p11-p13) + (p4+p11-p11-p13) =( ) + ( )=0 Main risk factor (X) Effect modifier (Z) YesNo YesR 11 R 10 NoR 01 R 00

Week 9 - Interaction 21 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: RD 11 >RD 10 +RD 01 – type 8 MUST be present Subadditivity: RD 11 <RD 10 +RD 01 – 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

Week 9 - Interaction 22 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 9 - Interaction 23 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

Week 9 - Interaction 24 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 9 - Interaction 25 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

Week 9 - Interaction 26 Dunedin Child-Development Study Caspi et al. 2002, Stressful life eventsGenotype with short allele YesNo Yes33%17% No10% IC= =0.16 >0  synergy

Week 9 - Interaction 27 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=OR both/neither -OR life events/neither -OR short 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/OR both/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 eventsGenotype with short allele YesNo Yes0.33/ /0.83 No0.10/0.90 RG Ch 16

Week 9 - Interaction 28 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

Week 9 - Interaction 29 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 9 - Interaction 30 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 9 - Interaction 31 Arsenic Exposure from Drinking Water and Risk of Premalignant Skin Lesions in Bangladesh, Hasan et al 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.