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Calcium intake is the amount of calcium consumed, including food and supplements. It is an exposure that can be changed by an intervention, and should not be identified automatically with serum calcium in a causal inference analysis.

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.

InSIDE stands for instrument strength independent of direct effect. It requires the genetic associations with the exposure to be independent of the direct, pleiotropic effects of those genetic variants on the outcome. It is a key identifying assumption for MR-Egger regression.

An instrumental variable changes an exposure or treatment but affects the outcome only through that exposure. Together, instrument relevance, instrumental-variable independence, and the exclusion restriction identify causal effects under suitable structural assumptions.

Mendelian randomization uses inherited genetic variants associated with an exposure as instrumental variables. Its causal interpretation requires instrument relevance, instrumental-variable independence, and the exclusion restriction.

Migraine is a neurological disorder that can be an outcome in a clinical trial or an observational study. Its statistical association with an exposure does not alone establish a causal effect.

MR-Egger regression fits genetic outcome associations to genetic exposure associations with a free intercept. Its slope estimates a causal effect under the InSIDE assumption and appropriate instrument-strength assumptions; its intercept assesses average directional horizontal pleiotropy. A nonsignificant intercept, especially with few genetic variants, does not prove the exclusion restriction for every instrument.

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, isThe average is over the target population. This compares hypothetical outcomes under two interventions, rather than the observed difference $\mathbb{E}[Y\mid X=1]-\mathbb{E}[Y\mid X=0]$, 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.

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:

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.

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:

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.

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.

Two useful ways to strengthen the causal inference are:

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

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.