Revisiting Hazard Ratios: Can We Define Causal Estimands for Time‐Dependent Treatment Effects?

In this paper, some aspects concerning the causal interpretation of hazard contrasts are revisited. It is first investigated, in which sense the hazard ratio constitutes a causal effect. It is demonstrated that the hazard ratio at a timepoint <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>t</mml:mi> <mml:annotation>$t$</mml:annotation></mml:semantics> </mml:math> represents a causal effect for the population at baseline, but in general not for any population at risk at time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>t</mml:mi> <mml:annotation>$t$</mml:annotation></mml:semantics> </mml:math> . Moreover, the scenario is studied, in which the survival curves coincide up to some timepoint <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>t</mml:mi> <mml:annotation>$t$</mml:annotation></mml:semantics> </mml:math> and then separate. This investigation provides valuable insight both on the causal interpretation of the conventional hazard ratio and on properties of the recently proposed causal hazard ratio. The findings suggest that, without making further assumptions, there is in general no meaningful estimand for a treatment effect at time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn></mml:mrow> <mml:annotation>$t > 0$</mml:annotation></mml:semantics> </mml:math> . It is therefore advocated to develop alternative estimands grounded in medically plausible assumptions about the joint distribution of counterfactual survival times.