Let

Y (1)

and

Y (0)

be the potential outcomes for the same individual under well-defined interventions setting the binary exposure to 1 and 0. The average treatment effect, also called the average causal effect on the additive scale, is

ACE = E [Y (1) - Y (0)] = E [Y (1)] - E [Y (0)] .

(1)

The average is over the target population. This compares hypothetical outcomes under two interventions, rather than the observed difference

E [Y ∣ X = 1] - E [Y ∣ X = 0]

, which can be distorted by confounding. For binary

Y

, the displayed contrast is a causal risk difference. Consistency in causal inference and a well-defined intervention, including an appropriate no-interference in causal inference assumption, give these potential outcomes their meaning.

b

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An instrumental variable

G

supplies variation in an exposure

B

that can be used for causal inference about its effect on

Y

. Its three core requirements are:

Instrument relevance: $G$ changes the distribution of $B$ ; in a simple linear model this requires a nonzero first-stage association.
Instrumental-variable independence: $G$ is independent of unmeasured common causes of $B$ and $Y$ , or independent of the relevant potential outcomes conditional on specified measured covariates.
Exclusion restriction: interventions on $G$ affect $Y$ only through $B$ , with no other causal pathway. In potential outcomes notation, $Y (b, g) = Y (b)$ .

Relevance, independence, and exclusion are the core instrumental-variable assumptions. They do not by themselves identify an arbitrary population-average effect. For a binary instrument and treatment, instrumental-variable monotonicity and the usual consistency in causal inference assumptions can identify a local average treatment effect; stronger effect-homogeneity or structural-model assumptions are needed for other effect interpretations.

c

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Under the displayed causal directed acyclic graphs, with nonzero effects along the relevant arrows and no exact cancellation of the effects on

B

, the valid instrumental variables are scenarios (i), (ii), and (iv). The path arguments are:

In (i), $G \to B$ supplies instrument relevance, and every directed path from $G$ to $Y$ goes through $B$ . The unmeasured $U$ does not cause $G$ , so instrumental-variable independence holds.
In (ii), $G \to A \to B$ supplies instrument relevance. Although $A$ is unmeasured, every directed path from $G$ to $Y$ still passes through $B$ . The path $G \to A \leftarrow U \to Y$ is blocked at its collider $A$ without conditioning, so it does not violate instrumental-variable independence.
In (iii), $G \to C \to Y$ bypasses $B$ , violating the exclusion restriction. There is also no directed effect of $G$ on $B$ , so instrument relevance is absent in this graph.
In (iv), both $G \to B$ and $G \to A \to B$ affect $B$ , and all directed paths to $Y$ still pass through $B$ . The unconditioned collider $G \to A \leftarrow U$ blocks the route to $U$ , so both instrumental-variable independence and the exclusion restriction hold. Conditioning on $A$ could introduce collider bias, but $A$ is not being conditioned on here.
In (v), $G \to C \to Y$ bypasses $B$ , so the exclusion restriction fails despite the additional $G \to B$ arrow.
In (vi), $G \to U \to Y$ bypasses $B$ , and $G$ affects the unmeasured common cause $U$ . Thus the exclusion restriction and the stated instrumental-variable independence requirement fail.

An upstream intermediate variable need not invalidate an instrumental variable if its routes to the outcome all pass through the target exposure; an outcome intermediate reached without that exposure does invalidate it.

d

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The supplementary excerpt uses an adjusted observational study and a two-sample Mendelian randomization analysis.

The observational study examines co-occurrence of hypercalcaemia and migraine diagnoses in electronic health records, adjusting for age, sex, and ancestry. Its adjusted odds ratio is 1.58, with

P = 4.75 \times 1 0^{- 13}

The two-sample Mendelian randomization analysis combines the associations of eight genetic variants with serum calcium and with migraine, using a weighted genetic risk score. It estimates an odds ratio of 1.80 per genetically predicted 1 mg/dL higher serum calcium, with 95% confidence interval

(1.31, 2.46)

. The weighted-median Mendelian randomization estimator and MR-Egger regression provide sensitivity analyses.

The measured and instrumented exposure is serum calcium, rather than dietary calcium intake itself. Calcium intake and serum calcium are not interchangeable interventions, and an odds ratio of 1.80 is an odds contrast rather than necessarily an 80% increase in probability. The shortened excerpt supplies two approaches; a further analysis in the full research paper is outside this excerpt.

e

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For the electronic-record observational study, three strengths are:

The large dataset improves precision of the estimated statistical association and allows detection of associations too small for a modest study.
Adjusting for age, sex, and ancestry addresses specified measured sources of confounding.
Routinely collected hypercalcaemia and migraine diagnoses give evidence from clinical practice, rather than depending solely on recalled calcium intake.

Three weaknesses are:

Residual confounding remains possible: for example, other diseases, medicines, or health behaviours may affect both serum calcium and migraine.
Co-occurrence does not establish temporal order, leaving reverse causality possible. Migraine, its management, or the decision to investigate it may alter recorded serum calcium.
Selection bias and diagnostic misclassification can arise because patients enter health records and receive testing nonrandomly. Moreover, hypercalcaemia is not a direct measure of dietary calcium intake, limiting the claimed intervention's interpretation.

For two-sample Mendelian randomization, three strengths are:

Genetic variants are inherited before adult migraine develops, making ordinary disease-to-genotype reverse causality implausible.
Under instrumental-variable independence, genetic assignment reduces the environmental confounding that undermines a conventional observational study.
The large outcome sample and combined genetic risk score improve precision, while concordant weighted-median Mendelian randomization estimator and MR-Egger regression results offer checks under different assumptions about horizontal pleiotropy.

Three weaknesses are:

Horizontal pleiotropy can violate the exclusion restriction, while population stratification or inherited family effects can violate instrumental-variable independence. In the supplied figure, one variant has a much larger serum calcium association than the others, motivating a check for dependence on that single variant.
The score explains only 1.25% of serum calcium variance, so instrument strength must be assessed rather than assumed. Different populations in the two samples can also undermine transport of genetic associations. A small variance fraction is not itself proof of a weak instrument in a large sample.
A lifelong genetically influenced shift in serum calcium is not equivalent to changing dietary calcium intake. The robustness methods also need assumptions: the weighted-median Mendelian randomization estimator needs more than half of the weight from valid instruments, and MR-Egger regression needs the InSIDE assumption. The nonsignificant MR-Egger regression intercept is compatible with zero average directional horizontal pleiotropy, but with eight variants it cannot prove all instruments valid.

Thus the agreement supports the hypothesis, but does not establish the effect of a dietary intervention.

f

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Two useful ways to strengthen the causal inference are:

Run a randomized controlled trial of the proposed dietary intervention. Assign a well-defined change in calcium intake, measure its achieved effect on serum calcium, and prospectively ascertain migraine outcomes. Randomization addresses baseline confounding, and measuring the intermediate exposure tests whether the intervention actually changes the quantity implicated by the genetic analysis. This directly evaluates the proposed dietary intervention rather than assuming equivalence with a lifelong genetic shift.
Replicate and challenge the genetic result in independent data. Use a prespecified two-sample Mendelian randomization analysis with additional well-supported genetic variants, examine associations with possible confounders, and perform leave-one-variant-out and horizontal pleiotropy sensitivity analyses. In particular, removing the strongest serum calcium variant tests whether the result depends on one possibly invalid instrument. Replication checks reproducibility, while within-family Mendelian randomization can reduce population stratification and family-level genetic/environmental biases.

These are complementary: the first changes the proposed exposure experimentally, and the second probes the validity and reproducibility of its genetic evidence.

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 3

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 3

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