Effect Heterogeneity & Joint Effects – Comprehensive Notes

PAGE 1 – Recap of Effect Measures & Scales

• Effect measures are contrasts between two quantities (risk, rate, hazard, odds, etc.).
• Two principal scales:
– Relative (scale of division) → risk ratio (RR), rate ratio, odds ratio (OR), hazard ratio (HR).
– Absolute (scale of subtraction) → risk difference (RD), rate difference, etc.
• Neither scale is “better”; they answer different questions.
– Absolute (additive) scale is vital for public-health messages: “How many cases will be averted or caused?”
– Relative (multiplicative) scale tends to travel better across settings—more stable when baseline risks differ.

PAGE 2 – Why Absolute Effects Vary Across Populations

• If RR is stable at, say, $2.0$ across countries, RD will still vary because baseline risks differ.
• Population mix (sex, age, comorbidities) changes absolute risk, producing different RDs even when RRs are constant.
• Hence journals often ask for both scales plus baseline risk of target population.

PAGE 3 – Defining Effect Heterogeneity & Effect-Measure Modification (EMM)

• Average causal effect $= E[Y^1 – Y^0]$ in the population P (ATE).
• If effect of exposure $A$ on outcome $Y$ varies across levels of a third variable $M$ → Effect Heterogeneity.
• When heterogeneity is demonstrated on a specific scale it is called Effect-Measure Modification.
– EMM can exist on the additive scale only, multiplicative scale only, or both.
– Always specify the scale (hence the word measure).

PAGE 4 – Illustrative Transplant Example (Women vs Men)

Group	RR	RD
Entire pop (50 ♀ + 50 ♂)	$1.0$	$0$
Women	$1.5$	$+0.20$ (harm)
Men	$0.67$	$-0.20$ (benefit)

• Overall null hides opposite-direction effects within strata—classic qualitative EMM.
• Take-home: “No overall effect” ≠ “No effect in any subgroup.”

PAGE 5 – Formal Definition & Key Properties of EMM

• $M$ is an effect modifier of $A \to Y$ when
$E[Y^{a=1}-Y^{a=0} \mid M=m1] \neq E[Y^{a=1}-Y^{a=0} \mid M=m2]$ .
• M cannot be a mediator (a variable on the causal path $A \to M \to Y$ ).
• Qualitative EMM: direction reverses across strata.
• Quantitative EMM: same direction, different magnitude.

PAGE 6 – Scale Dependence Example

Study 2 findings:

Stratum	RD	RR
$M=1$	$0.10$	$2.0$
$M=0$	$0.10$	$1.1$

• No EMM on additive scale (RD equal).
• Clear EMM on multiplicative scale (RRs differ).
• Always report both if feasible.

PAGE 7 – Visualising EMM & DAG Considerations

• DAGs depict causal structure; EMM is about magnitude, not pathway, so it isn’t directly shown in a DAG.
• Sometimes drawn informally with arrows of varying thickness, but that is heuristic.
• Surrogate modifiers (e.g., nationality) may show EMM yet not be causal themselves—interpret cautiously.

PAGE 8 – Why EMM Matters

Transportability: effect in new population depends on modifier distribution.
Targeting: identify sub-groups that truly benefit or are harmed (precision public health / medicine).
Etiologic clues: heterogeneity can suggest underlying mechanisms.

Caveat: Sample-size; small strata = imprecise estimates.

PAGE 9 – Interaction / Joint Effects vs EMM

• Interaction (Joint Effect): interest lies in the combined exposure $A$ & $E$ both causal for $Y$ .
– Synergistic: joint effect > sum (additive) or product (multiplicative) of individual effects.
– Antagonistic: joint effect < expected.
• Requires 4 cells (factorial):
${A,E}\in{0,1}^2$ → none, $A$ only, $E$ only, both.
• Conceptually distinct: Interaction = causal interplay between two exposures; EMM = heterogeneity of one exposure’s effect across levels of a third variable (not necessarily causal).

PAGE 10 – Additive vs Multiplicative Interaction Metrics

• Additive metric: Relative Excess Risk Due to Interaction (RERI).
$\text{RERI}=RR{11}-RR{10}-RR{01}+1$ – $=0$ → no additive interaction. • Multiplicative metric: interaction term $RR{11} \div (RR{10}\times RR{01})$ .
• Same regression product term estimates both EMM and interaction; interpretation depends on question.

PAGE 11 – Confounding vs Effect Modification

Aspect	Confounding	Effect Modification
Desirability	Unwanted bias (nuisance)	Valued finding
Causal claim	Yes – creates spurious effect	No claim; describes heterogeneity
Remedy	Control (adjust, stratify, weight)	Report, interpret, maybe exploit

• A variable can be both a confounder and an effect modifier (e.g., sex).
• Procedure: first control confounding of $A \to Y$ , then assess EMM on residual causal effect.

PAGE 12 – Regression Implementation

General model (linear form for clarity):
$Y = \beta0 + \beta1 A + \beta2 M + \beta3 (A\times M) + \epsilon$

• Product term $\beta3$ = statistical interaction. – If focus is EMM: interpret $\beta3$ as difference in $\beta_1$ across levels of $M$ .
– If focus is interaction: treat $A\times M$ as joint-exposure cell.
• In logistic, Poisson, Cox models the same logic applies (but scale = log RR or log HR).
• Alternative: run separate models within strata (simple for EMM, not for interaction).

PAGE 13 – Graphical Illustration of Product Term

• Without product term: two parallel lines (same slope, different intercept).
• With product term: slopes diverge – visual evidence of EMM/interaction.

PAGE 14 – Reporting Recommendations (STROBE-Inspired)

State primary effect measure & scale.
Pre-specify modifiers or interacting exposures (biologic, social, mechanistic rationale).
Provide stratum-specific estimates (RR, RD, OR, etc.) with confidence intervals and Ns.
For interaction: present all 4 joint-exposure categories and additive/multiplicative metrics (e.g., RERI).
Discuss potential confounding and whether adjusted.
Clearly distinguish EMM from interaction in text.

PAGE 15 – Practical Pitfalls & Sample-Size Considerations

• Small cell counts inflate variance → wide CIs; may need pooling or larger study.
• Multiple testing: avoid data-dredging; limit to biologically/plausibly motivated modifiers.
• Remember: statistical non-significance ≠ no EMM; examine magnitude & CI overlap.
• Precision Medicine hype: truly individual-level effects are unidentifiable—need groupings.

PAGE 16 – Key Exam Take-Aways

• Define & distinguish:
– Effect Heterogeneity / Effect-Measure Modification.
– Interaction / Joint Effects.
• Know scales (additive vs multiplicative) and metrics (RD, RR, RERI, etc.).
• Understand confounding vs EMM.
• Recognise product term in regression and interpret accordingly.

PAGE 17 – Ethical & Policy Implications

• Ignoring EMM can hide harm in vulnerable sub-groups.
• Public-health resources can be wasted if interventions are targeted to low-benefit groups.
• Reporting absolute and relative results guides fair risk communication.
• Equity: social modifiers (SES, housing, region) highlight structural determinants needing policy change.

PAGE 18 – Connections to Prior Material

• Builds on causal inference principles: counterfactuals, DAGs, confounding control.
• Links to sufficient-component-cause model (interaction relates to causal pies).
• Regression fundamentals re-appear (interpretation of coefficients, product terms).
• Sets stage for later sessions on mediation analysis & precision public health.

PAGE 19 – Formulae Cheat-Sheet

• Risk Difference: $RD = R1 - R0$
• Risk Ratio: $RR = R1 / R0$
• RERI (additive interaction): $RERI = RR{11} - RR{10} - RR{01} + 1$ • Attributable Proportion due to Interaction (AP): $AP = RERI / RR{11}$
• Synergy Index: $S = (RR{11} -1) / [(RR{10}-1) + (RR{01}-1)]$ • Product-term regression coefficient (log-scale): $\beta3 = \log RR{11} - \log RR{10} - \log RR{01} + \log RR{00}$

PAGE 20 – Further Reading

• Hernán & Robins – What If (Ch-18, Effect Heterogeneity).
• Knol & VanderWeele (2012) – "Recommendations for Presenting Effect Modification and Interaction".
• VanderWeele – Explanation in Causal Inference (Ch-4).
• STROBE guidelines – pages on interaction & subgroup analyses.