  BACKGROUND: In 1967, Frederick Lord posed a conundrum that has confused
scientists for over 50-years. Subsequently named Lord's 'paradox', the puzzle
centres on the observation that two common approach to analyses of 'change'
between two time-points can produce radically different results. Approach 1
involves analysing the follow-up minus baseline (i.e., 'change score') and
Approach 2 involves analysing the follow-up conditional on baseline. METHODS:
At the heart of Lord's 'paradox' lies another puzzle concerning the use of
'change scores' in observational data. Using directed acyclic graphs and data
simulations, we introduce, explore, and explain the 'paradox', consider the
philosophy of change, and discuss the warnings and lessons of this 50-year
puzzle. RESULTS: Understanding Lord's 'paradox' starts with recognising that a
variable may change for three reasons: (A) 'endogenous change', which
represents simple changes in scale, (B) 'random change', which represents
change due to random processes, and (C) 'exogenous change', which represents
all non-endogenous, non-random change. Unfortunately, in observational data,
neither Approach 1 nor Approach 2 are able to reliably estimate the causes of
'exogenous change'. Approach 1 evaluates obscure estimands with little, if any,
real-world interpretation. Approach 2 is susceptible to mediator-outcome
confounding and cannot distinguish exogenous change from random change. Valid
and precise estimates of a useful causal estimand instead require appropriate
multivariate methods (such as g-methods) and more than two measures of the
outcome. CONCLUSION: Lord's 'paradox' reiterates the dangers of analysing
change scores in observational data and highlights the importance of
considering causal questions within a causal framework.
