 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-207/3/e/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

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.

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.

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