Stanghellini and Doretti (2019) studied the exact formulae relating parameters in conditional and marginalized logistic models when the mediator is binary. Those formulae generally do not hold for the reduced model as the reduced model is generally not the same as the marginalized model. For a conditional model that allows for treatmentmediator interaction, I present 1) alternative exact formulae relating parameters in the conditional model to those in the marginalized logistic model. They are equivalent to but simpler and easier to interpret than those given in Stanghellini and Doretti (2019); 2) a decomposition of the total treatment effect into the natural direct effect and the natural indirect effect without assuming the outcome is rare; 3) exact formulae relating parameters in the conditional model to those in the reduced logistic model by using likelihood equations; 4) a bound on the size of the natural direct effect regardless of whether the treatment is numeric or discrete; and 5) a numerical assessment of the bias of the approximate formulae reported in Valeri and VanderWeele (2013). The relative bias can be greater than 15% even when the prevalence is less than 10%.

中文翻译：

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当调解员为二进制时，在条件，边际和边际逻辑模型中关联参数

Stanghellini和Doretti（2019）研究了调解员为二元时条件和边际逻辑模型中与参数有关的精确公式。这些公式通常不适用于简化模型，因为简化模型通常与边际化模型不同。对于允许治疗介质交互作用的条件模型，我提出1）将条件模型中的参数与边际逻辑模型中的参数相关的替代精确公式。它们等效于Stanghellini和Doretti（2019）中给出的内容，但更易于解释。2）在不假设结果很少的情况下将总治疗效果分解为自然直接效果和自然间接效果；3）使用似然方程，将条件模型中的参数与简化逻辑模型中的参数相关的精确公式；4）限制自然直接效应的大小，无论是数字处理还是离散处理；5）Valeri和VanderWeele（2013）中报告的近似公式偏差的数值评估。即使患病率小于10％，相对偏差也可以大于15％。