Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 207 / 2

Causal mediation analysis decomposes the total causal effect of an exposure into a pathway operating through a mediator and a pathway not operating through that mediator. Let $M(x)$ be the potential mediator under exposure $x$, and let $Y(x,m)$ be the potential outcome under exposure $x$ with mediator set to $m$. One conventional decomposition usesfor the natural direct effect, andfor the natural indirect effect. In words, the direct effect changes exposure while holding the mediator at the value it would naturally have under no exposure; the indirect effect changes only that natural mediator value while holding exposure fixed at one.

Taken at face value, the mediation analysis says that the rs8034191 allele raises lung-cancer risk mainly through a pathway not captured by reported cigarettes per day: the direct-effect odds ratio is $1.35$, whereas the estimated indirect-effect odds ratio through smoking intensity is $1.01$ and is compatible with no mediation. It does not show that smoking intensity has no causal effect on lung cancer. It concerns mediation of this genetic variant's effect through this measured mediator, and a weak variant-to-intensity association, measurement error, or mediation through smoking duration, inhalation, or nicotine exposure could make that indirect pathway appear small.

The significant additive-scale interaction and the association of the variant with cancer among smokers but not nonsmokers indicate effect modification: the joint effect of genotype and smoking exceeds additivity on the risk scale. Under adequate control of confounding and selection, that pattern supports a causal role for smoking in activating or amplifying the genetic pathway. Interaction alone is not proof that smoking is causal, because smoking was not randomized and the stratum-specific estimates can be affected by confounding, selection, and low power among nonsmokers.

The mediation estimate relies on several strong assumptions. Possible failures include mediator-outcome confounding, residual exposure-outcome or exposure-mediator confounding, and an exposure-induced mediator-outcome confounder. Smoking duration may itself be part of the causal pathway, making adjustment inappropriate. Self-reported cigarettes per day has measurement error and does not fully measure tobacco exposure. The case-control sampling can create selection bias; population stratification can confound the genotype relations; cancer may alter reported smoking; and the product-of-coefficients calculation can be inappropriate for a binary outcome because odds ratio effects are nonlinear and noncollapsible. Any of these can attenuate or distort the indirect effect.

The evidence would be stronger with prospectively measured smoking before diagnosis, repeated measures of intensity and duration, objective biomarkers such as cotinine, and explicit modeling of cumulative exposure. The authors could use modern counterfactual mediation estimators suited to case-control data and binary outcomes, allow exposure-mediator interaction, report effects on interpretable risk scales, and perform sensitivity analyses for unmeasured mediator-outcome confounding and measurement error. Replication, ancestry adjustment, negative controls, and a Mendelian randomization analysis using additional smoking instruments with credible exclusion restrictions would help distinguish smoking-mediated effects from direct pleiotropic effects of this locus.