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Joined 2026-09-21 Message
Incoming links: Reverse causality
Past exam of the mathematics course of the University of Cambridge 2018 iii Paper 207 3 e Solution 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.
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.
Past exam of the mathematics course of the University of Cambridge 2019 ii Paper 2 5J b Solution Created 2026-09-24 Updated 2026-10-03
The first model alone would suggest that more cycling is beneficial, but cycling was not randomized in that model: its coefficient can be distorted by confounding or reverse causality. The randomized incentive both substantially increased cycling and substantially increased asthma episodes.
If the incentive affects asthma only through cycling, it is an instrumental variable, and the corresponding Wald estimator of the effect per additional weekly mile isadditional episodes. Under that exclusion assumption the evidence therefore advises against cycling to work more often for this population. Without the exclusion assumption, the experiment establishes the harmful effect of the incentive itself but cannot by itself distinguish an effect of cycling from a direct effect of the monetary incentive.
Past exam of the mathematics course of the University of Cambridge 2021 iii Paper 207 2 a Solution 2026-09-28
Education is a causal risk factor if an intervention that changes educational exposure changes the distribution of the corresponding potential outcome for myopia. An observed statistical association need not have that interpretation. Shared causes such as socioeconomic circumstances, parental behavior, genetic traits, or preference for indoor near work can confound education and myopia. Reverse causality is also possible: children with impaired distance vision may select more reading and indoor study, which can increase educational attainment without education causing the impairment.
Past exam of the mathematics course of the University of Cambridge 2022 iii Paper 207 1 1 2 Solution 2026-09-28
First, an observational study can suffer from confounding or reverse causality. Ill health may both lower circulating vitamin D and raise mortality, while lifestyle, socioeconomic status, and comorbidity may affect both variables. Randomization breaks these baseline associations in expectation.
Second, the interventions answer different questions. Supplementation may be too small, too late, too short, or poorly adhered to, and an average effect can be nearly zero when benefit is confined to people with severe deficiency. Such heterogeneous treatment effects can coexist with a strong observational gradient.