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.

翻译：背景：1967年，弗雷德里克·洛德提出一个困扰科学界逾50年的谜题，后被称为“洛德悖论”。该谜题的核心在于：两种常用方法分析两个时间点间的“变化”时，可能得出截然不同的结果。方法一分析随访值减去基线值（即“变化分数”），方法二则在基线值条件下分析随访值。方法：洛德“悖论”的核心还涉及另一个关于在观测数据中使用“变化分数”的谜题。我们通过有向无环图和数据模拟，引入、探讨并解释该“悖论”，思考变化的哲学内涵，并讨论这一历时50年谜题的警示与教训。结果：理解洛德“悖论”需首先认识到变量可能因三种原因发生变化：（A）“内源变化”，代表单纯尺度缩放的变化；（B）“随机变化”，代表随机过程引起的变化；（C）“外源变化”，代表所有非内源、非随机的变化。遗憾的是，在观测数据中，方法一与方法二均无法可靠估计“外源变化”的成因。方法一评估的模糊估计量几乎不具备现实解释意义；方法二则易受中介-结局混杂影响，且无法区分外源变化与随机变化。要获得有用因果估计量的有效精确估计，需采用合适的多元方法（如g方法）并收集超过两期的结局变量测量。结论：洛德“悖论”重申了在观测数据中分析变化分数的危险，并凸显了在因果框架内思考因果问题的重要性。