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

  1. Transportability: effect in new population depends on modifier distribution.

  2. Targeting: identify sub-groups that truly benefit or are harmed (precision public health / medicine).

  3. 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)

  1. State primary effect measure & scale.

  2. Pre-specify modifiers or interacting exposures (biologic, social, mechanistic rationale).

  3. Provide stratum-specific estimates (RR, RD, OR, etc.) with confidence intervals and Ns.

  4. For interaction: present all 4 joint-exposure categories and additive/multiplicative metrics (e.g., RERI).

  5. Discuss potential confounding and whether adjusted.

  6. 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.