Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 44 / 1 / a / Solution

Use a longitudinal study with repeated measurements of the same individuals. Recruit people across several starting ages and measure the same ear, with a standardized anatomical definition and instrument, at baseline and subsequent visits over several years. Repeat measurements at each visit to estimate measurement error; where practical, conceal previous measurements from the observer. Record sex, body size, birth cohort and observer, and document reasons for loss to follow-up.

A simple random-intercept linear mixed model for ear length $Y_{ij}$ at elapsed follow-up time $t_{ij}$ isThe individual intercept ${\alpha }_i$ absorbs persistent differences between people. Alternatively, differences $Y_{ij}-Y_{i0}$ remove that intercept directly. Estimate the within-person regression coefficient $\beta$ and its confidence interval, accounting for the correlation of repeated measurements. Compare it with the cross-sectional slope rather than treating the two slopes as automatically identical.

This longitudinal study directly checks whether an individual's ears tend to lengthen as time passes. Several entry-age cohorts help assess whether the change varies with starting age. Calendar-period changes and selective dropout can still affect interpretation, so repeated observation alone does not eliminate every possible source of confounding or selection bias.