It is known that the hazard ratio lacks a useful causal interpretation. Even for data from a randomized controlled trial, the hazard ratio suffers from so-called built-in selection bias as, over time, the individuals at risk among the exposed and unexposed are no longer exchangeable. In this paper, we formalize how the expectation of the observed hazard ratio evolves and deviates from the causal effect of interest in the presence of heterogeneity of the hazard rate of unexposed individuals (frailty) and heterogeneity in effect (individual modification). For the case of effect heterogeneity, we define the causal hazard ratio. We show that the expected observed hazard ratio equals the ratio of expectations of the latent variables (frailty and modifier) conditionally on survival in the world with and without exposure, respectively. Examples with gamma, inverse Gaussian and compound Poisson distributed frailty and categorical (harming, beneficial or neutral) distributed effect modifiers are presented for illustration. This set of examples shows that an observed hazard ratio with a particular value can arise for all values of the causal hazard ratio. Therefore, the hazard ratio cannot be used as a measure of the causal effect without making untestable assumptions, stressing the importance of using more appropriate estimands, such as contrasts of the survival probabilities.

众所周知，风险比缺乏有用的因果解释。即使对于来自随机对照试验的数据，风险比也会受到所谓的内置选择偏差的影响，因为随着时间的推移，暴露者和未暴露者中面临风险的个体不再可以互换。在本文中，我们形式化了在未暴露个体的风险率异质性（虚弱）和效应异质性（个体修改）存在的情况下，观察到的风险比的期望如何演变并偏离感兴趣的因果效应。对于效应异质性的情况，我们定义因果风险比。我们表明，预期观察到的风险比等于分别在有暴露和无暴露的世界中以生存为条件的潜在变量（脆弱性和调节剂）的预期比率。给出了伽玛分布、逆高斯分布和复合泊松分布脆弱性和分类（有害、有益或中性）分布效应修饰符的示例以供说明。这组示例表明，对于因果风险比的所有值，都可能出现具有特定值的观察到的风险比。因此，如果不做出不可检验的假设，就不能将风险比用作因果效应的衡量标准，强调使用更合适的估计值的重要性，例如生存概率的对比。