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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.

Deterministic variables are variables that are fully explained by one or more parent variables. They commonly arise when a variable has been algebraically constructed from one or more parent variables, as with composite variables, and in compositional data, where the 'whole' variable is determined from its 'parts'. This article introduces how deterministic variables may be depicted within directed acyclic graphs (DAGs) to help with identifying and interpreting causal effects involving tautological associations, compositional data, and composite variables. We propose a two-step approach in which all variables are initially considered, and an explicit choice is then made whether to focus on the deterministic variable(s) or the determining parents. Depicting deterministic variables within DAGs bring several benefits. It is easier to identify and avoid misinterpreting tautological associations, i.e., self-fulfilling associations between variables with shared algebraic parent variables. In compositional data, it is easier to understand the consequences of conditioning on the 'whole' variable, and correctly identify total and relative causal effects. For composite variables, it encourages greater consideration of the target estimand and greater scrutiny of the consistency and exchangeability assumptions. DAGs with deterministic variables are a useful aid for planning and interpreting analyses involving tautological associations, compositional data, and/or composite variables., Comment: 18 pages, 5 figures