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A genetic risk score combines allele counts from several genetic variants, usually using specified weights. In Mendelian randomization it can act as a combined instrumental variable for an exposure; interpretation still requires instrument relevance, instrumental-variable independence, and the exclusion restriction.

A genetic variant is a heritable difference in a DNA sequence relative to a chosen reference. Genetic variants can be used as instrumental variables in Mendelian randomization when the required causal assumptions are credible.

Horizontal pleiotropy occurs when a genetic instrument affects an outcome through a pathway other than the exposure under study. It violates the exclusion restriction in Mendelian randomization.

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 $P=4.75\times 10^{-13}$.

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.

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.

Three useful extensions are:

The overall odds ratio is $0.99$ with 95% interval $(0.95,1.02)$, so there is little evidence for an average effect. The estimate changes sharply across residual-vitamin-D strata: it is $0.69$ $(0.59,0.80)$ in the deficient group, $0.94$ $(0.89,0.99)$ in the insufficient group, and close to one in the two higher groups. The pattern is consistent with effect modification: raising vitamin D may reduce mortality among deficient people but offer little benefit once concentration is adequate. The claim still depends on the Mendelian randomization assumptions and should account for the fact that several subgroup estimates were examined.

Population stratification is systematic variation in allele frequencies between subpopulations. If those subpopulations also differ in outcome causes, genetic variant associations can exhibit confounding, violating instrumental-variable independence in Mendelian randomization.

Serum calcium is the concentration of calcium in blood serum. It can be an exposure in an observational study or Mendelian randomization, but genetically changing it need not reproduce the effect of changing dietary calcium intake.

Two-sample Mendelian randomization combines genetic variant associations with an exposure estimated in one sample and their associations with an outcome estimated in another. It requires the instrumental variable assumptions and compatible associations across the populations; sample overlap can change the behaviour of weak instruments.

Within-family Mendelian randomization uses genetic contrasts within families, such as differences between siblings. Conditioning on shared parental background can reduce population stratification and family-level sources of confounding, although the exclusion restriction and sufficient instrument relevance remain necessary.