Past exam of the mathematics course of the University of Cambridge 2019 iii Paper 207 2 2 4 Solution 2026-10-03
The mediation estimate relies on several strong assumptions. Possible failures include mediator-outcome confounding, residual exposure-outcome or exposure-mediator confounding, and an exposure-induced mediator-outcome confounder. Smoking duration may itself be part of the causal pathway, making adjustment inappropriate. Self-reported cigarettes per day has measurement error and does not fully measure tobacco exposure. The case-control sampling can create selection bias; population stratification can confound the genotype relations; cancer may alter reported smoking; and the product-of-coefficients calculation can be inappropriate for a binary outcome because odds ratio effects are nonlinear and noncollapsible. Any of these can attenuate or distort the indirect effect.
Past exam of the mathematics course of the University of Cambridge 2022 iii Paper 207 2 2 3 Solution 2026-09-28
A complete-case analysis estimates each success probability among clinic attenders. If attendance depends on the unobserved outcome even after conditioning on treatment, the observed success proportions differ systematically from those in the randomized groups. This outcome-dependent dropout causes selection bias and can bias their difference.
Past exam of the mathematics course of the University of Cambridge 2022 iii Paper 221 4 i Solution 2026-09-28
The administrative data include only encounters with . Conditioning on the stop indicator selects on the colliderwhich opens the noncausal path and creates collider bias. The equal entries estimate only the selected risks . They ignore racial differences in the probability of being stopped and therefore do not identify the total causal effect of race on violence. This is also selection bias, because encounters with never enter the dataset.
Past exam of the mathematics course of the University of Cambridge 2025 iii Paper 207 1 b Solution Created 2026-09-24 Updated 2026-09-25
The birth-period comparison requires consistency, no interference between individuals, and a real discontinuity in sugar exposure at the end of rationing. Near the cutoff, potential diabetes outcomes must otherwise vary continuously with birth date: no simultaneous policy, nutritional, diagnostic, seasonal, or demographic discontinuity may affect them. Birth dates must not be manipulable around the cutoff, and survival and UK Biobank participation must not create differential selection bias. These assumptions make cutoff assignment locally exchangeable and give a regression discontinuity design an exclusion restriction in which birth period affects diabetes through sugar exposure.
By 1867 the central change was from confidence in bloodletting as a general remedy to a disputed, increasingly restricted practice tested against clinical outcomes. Bloodletting includes opening a vein, local wet cupping, and the application of medicinal leeches. The history must explain both the theories that supported it and why clinical comparisons weakened that support.
Ancient explanations and qualifications. In the Hippocratic Corpus, disease could be explained through disturbances of bodily fluids. The four-fluid account underlying humorism named blood, humoral phlegm, yellow bile and black bile; restoring their proper mixture offered a rationale for evacuation. This was a historical explanatory system, not a modern description of blood composition. Nor was ancient practice uniformly indiscriminate. Aulus Cornelius Celsus describes bloodletting as a powerful intervention whose suitability depends on bodily strength and whether the condition represents excess rather than exhaustion. Galen likewise distinguishes whether evacuation is needed, whether it should be by venesection, and whether the patient can tolerate it; he discusses the site, quantity and repetition. Thus the ancient debate already included selection and restraint. [ref]
On the Nature of Man in the Hippocratic Corpus. Selection 1, The Elements of Life, paragraphs 1–2: four distinct fluids; translated primary excerpt
[ref] [ref]On Treatment by Bloodletting by Galen. §1, successive questions governing evacuation, tolerance, site and amount
Medieval continuity and the instruments. The Canon of Medicine by Ibn Sina preserves a carefully qualified doctrine of venesection, including cases where removing good blood would leave harmful fluids behind. Its reasoning belongs to humorism; its distinctions should not be mistaken for verified modern indications. In late medieval Europe, bloodletting was also organized by calendars: a surviving Augsburg calendar for 1472 specifies supposedly favourable times according to the patient and the Moon. This documents an additional astrological framework, not evidence of therapeutic efficacy. [ref]
Canon of Medicine by Ibn Sina. Book I, translation §§1019–1022, contraindications, timing and choice of vessels
[ref]Bloodletting calendar for Augsburg (1472). Princeton copy and catalogue description: days selected by lunar phases, patient age, affected part and temperament
The material distinction between general and local removal mattered. Venesection opened a vein with a sharp instrument. Wet cupping combined superficial cuts in the skin with suction under a cup; Celsus describes bronze cups in which burning material reduced the enclosed air's pressure, and horn cups from which air was sucked through a small opening. Dry cupping did not include the cuts and therefore did not itself remove blood. Medicinal leeches provided another means of local removal. These instruments made depletion visibly demonstrable, but the visible removal of blood did not establish that a disease had been removed. [ref]
New physiology did not immediately settle treatment. William Harvey's 1628 account of blood circulation undermined the older picture of blood simply being supplied and consumed in the parts. His observations of venous valves and experiments with arm ligatures showed directional movement: filling below a ligature and the behaviour of compressed veins helped demonstrate return towards the heart. Nevertheless, a circulating blood supply could still be invoked to explain relief through depletion or redistribution. Establishing blood circulation and establishing the benefit of bloodletting were different questions. [ref]
On the Motion of the Heart and Blood in Animals by William Harvey. chapters 8–9, quantitative argument for circulation; chapter 13, ligatures and venous valves
In the early nineteenth century, François-Joseph-Victor Broussais's emphasis on local irritation and inflammation encouraged extensive use of medicinal leeches. This illustrates why changing theory could increase bloodletting: a local inflammatory explanation supplied a fresh reason to remove blood near the supposed seat of disease. The accompanying claims of efficacy still needed comparison with what happened under other treatments. [ref]
Re-imagining Bleeders by Robert G. W. Kirk and Neil Pemberton. section discussing Broussais, local irritation, leeching, and the claim that leeches offered a relatively gentle intervention
Comparative evidence weakened general claims. In 1835 Pierre Charles Alexandre Louis asked whether comparable patients with pneumonia did better when first subjected to bloodletting during days 1–4 than when first bled later. He explicitly considered age and other differences. This numerical method in medicine shifted attention from selected recoveries to outcomes in defined groups. His work restricted confidence in a large, general benefit; it should not be presented as a randomized controlled trial proving that every use was ineffective. In modern terms, these were observational studies: confounding by indication could make the earlier-treated group differ in severity, and surviving long enough to enter the later-treated group could introduce selection bias. Counting cases was an advance, but did not automatically remove these problems. [ref]
Research on the Effects of Bloodletting by Pierre Charles Alexandre Louis. 1835, pp. 70–76; primary passages reproduced by the James Lind Library on timing, age and comparisons between groups
The dispute still alive in 1867. William Pulteney Alison defended earlier experience of benefit, while John Hughes Bennett opposed routine depletion and developed a restorative approach. One defence of the altered practice was the change-of-type theory of disease: formerly vigorous or “sthenic” inflammatory illness was said to have become weak or “asthenic”, so that measures once borne well were now inappropriate. The rival explanation was that understanding of pathology and the evidence about treatment had improved. Alison's 1856 argument and the contemporary 1858 review show that this was an active disagreement, not a completed unanimous rejection. [ref]
Reflections on the Results of Experience as to the Symptoms of Internal Inflammations by William Pulteney Alison. 1856, pp. 769–788, opening response to Bennett and defence of previous clinical experience
[ref]The Bloodletting Controversy (1858 review). opening pp. 1–3, contrasting altered disease type with altered pathological understanding
Bennett's 1864 discussion made the theoretical contrast explicit: an account centred on altered blood and vessels favoured depletion, whereas his account of tissue irritation and inflammatory exudation led him towards the opposite practice. That historical account of inflammation should itself be evaluated historically, rather than adopted as current pathology. The durable methodological lesson is that recovery after treatment is insufficient evidence of recovery because of it: acute disease may resolve naturally, as even the Latin extract acknowledges. Changes in theory, recognition of the patient's limited reserve, and comparative clinical observation together explain the retreat from routine bloodletting. [ref]
Treatment of Inflammation by John Hughes Bennett. 1864, p. 142, opening comparison of vascular and extravascular accounts of inflammation and their therapeutic consequences
Present-day distinction. Contemporary venesection has specific indications: in haemochromatosis, removal of red blood cells helps reduce excess iron; in polycythaemia vera, it can reduce an excessive red blood cell burden. Those uses concern defined abnormalities, rather than correction of historical humorism or a general remedy for fever and inflammation. They are a modern explanatory coda, not knowledge attributed to the 1867 candidate. [ref]
Haemochromatosis treatment by the NHS. Venesection: removal of iron-containing red blood cells and use of iron to replace them
[ref]Erythrocytosis by the NHS. Treatment, Venesection: use in polycythaemia vera and selected other circumstances
Inherited variation contributes to psychiatric vulnerability and to some measured differences in adaptation after psychological stress, but its importance is phenotype-, population- and environment-dependent. Genetic influence is not equivalent to inevitability, a single large-effect gene or a wholly separate set of protective genes. A satisfactory answer must distinguish genetic effects on exposure, on symptoms and on responses conditional on exposure.
Family resemblance provides an initial clue, but families transmit environments and social circumstances as well as DNA. Twin comparisons exploit differences in relatedness: monozygotic pairs share essentially all inherited sequence variation, whereas dizygotic pairs share about half their segregating additive variation on average. Under an ACE model, greater resemblance of monozygotic pairs can support a genetic component if the relevant environmental and ascertainment assumptions hold. Equal environments means comparable effects of relevant environmental similarity, not literally identical treatment of every twin.
Disorder-level evidence is substantial. Patrick F. Sullivan and colleagues' ascertainment-aware synthesis of schizophrenia twin studies estimated liability heritability at roughly , with uncertainty. Paul Lichtenstein and colleagues' Swedish family-register study supported partially shared inherited liability to schizophrenia and bipolar disorder using biological, adoptive and sibling relationships. The convergence supports inherited vulnerability while showing that diagnostic categories do not necessarily partition genetic causes into entirely independent sets.[ref]
Schizophrenia as a complex trait by Patrick F. Sullivan and others. 2003, twin-liability synthesis
[ref]Common genetic determinants of schizophrenia and bipolar disorder in Swedish families by Paul Lichtenstein and others. 2009, family/adoption comparisons and cross-disorder liability
Molecular evidence provides a different test. The Psychiatric Genomics Consortium's 2014 schizophrenia study identified associations spanning 108 loci in a large multi-stage genome-wide analysis. Many common variants contribute small effects; an associated locus is not automatically one causal gene or a known mechanism. Population stratification, linkage disequilibrium, case ascertainment and sample overlap still require control. A genetic risk score can predict risk differences in independent data while remaining far from a deterministic diagnosis or a transportable measure of one person's psychological resilience.[ref]
Biological insights from 108 schizophrenia-associated genetic loci by Psychiatric Genomics Consortium. 2014, multi-stage associations and polygenic interpretation
Psychological resilience needs additional definition because favorable outcomes without adversity do not establish psychological stress resistance. Ananda B. Amstadter and colleagues studied 7,500 adult twins at two assessments, modeling internalizing symptoms relative to cumulative stressful events. Their regression residual-based phenotype had estimated heritability around at each wave; longitudinal modeling that accounted for measurement error and occasion-specific influences gave a larger latent estimate, around . This supports both inherited and environmental contributions to that operationalization, not a universal percentage for every meaning of psychological resilience.[ref]
Psychiatric resilience: longitudinal twin study by Ananda B. Amstadter and others. 2014, residual phenotype, wave-specific and longitudinal estimates
The shift in estimates is conceptually informative. Independent measurement error adds to observed variance without adding the corresponding genetic signal. In a simple measurement model, if is genetic variance, true-score variance and independent error variance, thenRemoving noise can therefore raise estimated heritability without changing anyone's genes. This illustration is not a reconstruction of the study's full model: occasion-specific effects and longitudinal assumptions also matter, so should not be reported as an independently established measurement reliability coefficient.
A regression residual phenotype also inherits assumptions from its construction. If adversity is undermeasured or a prediction model is misspecified, the regression residual includes more than protective capacity. Genetic effects on depression or anxiety can appear as genetic effects on low symptom burden after psychological stress, even without a separate buffering mechanism. Demonstrating a psychological resilience-specific effect would require stronger evidence that inherited variation changes the response to comparable exposure, with adequate exposure measurement and explicit competing models.
Gene–environment processes further complicate simple partitioning. Gene-environment correlation occurs when inherited characteristics influence environments encountered or elicited, or when parental characteristics affect both inherited liability and the rearing environment. Genotype-environment interaction instead means that inherited differences have different consequences across environments. Neither can be inferred merely because both genes and psychological stress predict the outcome. Restricting analysis to people with a particular exposure can also create selection bias if exposure is influenced by several causes relevant to symptoms.
Historical candidate-gene findings illustrate the need for stringent evidence. Richard Border and colleagues' large, preregistered analyses did not support widely proposed depression candidate-gene or candidate-gene-by-environment effects. This challenges those specific claims and the power of small discovery studies; it does not demonstrate that all interactions are absent. Replication, correction for multiple testing, ancestry-aware analyses and precise phenotype definitions are essential before a plausible biological story becomes a reliable finding.[ref]
No support for historic candidate gene or candidate gene-by-interaction hypotheses for major depression across multiple large samples by Richard Border and others. 2019, candidate-variant, gene and interaction tests
Finally, high heritability does not mean an outcome cannot be changed. It describes sources of variation under a particular range of environments, not the response to an environment absent from the data. Genetic effects can act through modifiable developmental systems, and external support can benefit people across inherited-risk levels. Predictive accuracy and equity can also deteriorate when scores are transferred to populations unlike the discovery sample.
Genes contribute meaningfully to liability and to particular measured adaptation phenotypes, while evidence for specific protective mechanisms is narrower than evidence for heritability. The strongest account combines family and molecular results with explicit exposure and developmental models, rather than assigning a fixed genetic percentage to an individual or replacing environmental prevention with genetic classification.
The panel supports feasibility in these samples, but not the full strength or generality of the claim. The method produces modification estimates for three patient samples, and the displayed selected genes generally have higher estimates than the knockout controls. That is evidence that an analysis can be performed and yields biological-looking contrasts.
The comparison simultaneously changes cell background, disease status and potentially sequencing conditions. The displayed genes were selected for high estimated modification, creating selection bias; the same estimates do not independently validate their accuracy. The panel also does not itself show input-RNA amounts, uncertainty, or performance across controlled input levels. It therefore cannot alone establish reliable low-input performance across genetically diverse clinical samples or disease-specific modification differences.
Selection bias 2026-09-24
Selection bias occurs when inclusion in the analyzed sample depends on variables related to the exposure and outcome, making the observed association differ from its target-population counterpart.