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Incoming links: Survivorship bias
Past exam of the mathematics course of the University of Cambridge 2006 iii Paper 44 1 c Solution Created 2026-10-03 Updated 2026-10-14
A cross-sectional age association need not equal within-person growth. The following alternatives each suggest a concrete extension of the study.
- Birth-cohort differences. Older participants were born in a different period. Persistent differences established early in life could produce a positive age slope even if every person's adult ear length were constant. Repeat measurements in a longitudinal study of several birth cohorts, separating within-person changes from differences in their initial measurements. Calendar age, birth year and observation year are algebraically related, so all three unrestricted linear effects cannot be identified without additional assumptions.
- Confounding by sex, body size or ancestry. Suppose a characteristic is associated both with ear length and with age among the sampled patients. Omitting it then changes the apparent age slope. Measure these characteristics, examine their distributions across ages, and fit a multiple linear regression or compare age slopes within appropriately defined groups. A guessed overall sex proportion supplies none of the required age-by-sex information.
- Selection bias from attendance or consent. Consulting patients and people willing to be measured may differ from the general population. For example, attendance could select different combinations of age and body characteristics. Sample from a population register, recruit people regardless of consultation, record refusals and compare age-specific recruitment patterns; compare the resulting slope with the practice-based slope.
- Survivorship bias. Ear length or a correlated characteristic might be associated with survival. Then people with larger ears could be overrepresented among older survivors even without growth. Follow a population cohort, measure ears before subsequent deaths or dropout, and investigate whether survival or retention depends on initial ear length after accounting for age and other covariates. This tests the proposed selection mechanism rather than assuming it.
- Age-related measurement error. Different practitioners may see different age mixes and have different measurement offsets; observers might also measure older ears differently. Use one protocol, calibrate instruments, randomly allocate observers where possible, conceal previous readings, obtain duplicate independent measurements and record observer identity. Analyze observer effects and the repeat-measurement differences before interpreting an age slope biologically.
These explanations are hypotheses to be checked, not established facts about the participants. Neither a small p-value nor an approximately linear scatter plot excludes them.
Past exam of the mathematics course of the University of Cambridge 2019 iii Paper 207 4 a Solution 2026-10-03
Place discrete hazards at the ordered distinct event times . If events occur among individuals at risk immediately before , the nonparametric empirical likelihood isEquivalently, an observed event at contributes the probability mass at , while a right-censored observation contributes the survivor probability beyond its censoring time. Maximization gives and the Kaplan–Meier estimator
Left truncation means an individual is observed only conditional on surviving beyond an entry time . Ignoring it overrepresents long survivors and creates survivorship bias. A subject with event or censoring time contributes its usual likelihood divided by ; in risk-set form, that individual enters each only for event times satisfying .