Causal interpretation of the hazard ratio in randomized clinical trials

Abstract

Background: Although the hazard ratio has no straightforward causal interpretation, clinical trialists commonly use it as a measure of treatment effect.

Methods: We review the definition and examples of causal estimands. We discuss the causal interpretation of the hazard ratio from a two-arm randomized clinical trial, and the implications of proportional hazards assumptions in the context of potential outcomes. We illustrate the application of these concepts in a synthetic model and in a model of the time-varying effects of COVID-19 vaccination.

Results: We define causal estimands as having either an individual-level or population-level interpretation. Difference-in-expectation estimands are both individual-level and population-level estimands, whereas without strong untestable assumptions the causal rate ratio and hazard ratio have only population-level interpretations. We caution users against making an incorrect individual-level interpretation, emphasizing that in general a hazard ratio does not on average change each individual's hazard by a factor. We discuss a potentially valid interpretation of the constant hazard ratio as a population-level causal effect under the proportional hazards assumption.

Conclusion: We conclude that the population-level hazard ratio remains a useful estimand, but one must interpret it with appropriate attention to the underlying causal model. This is especially important for interpreting hazard ratios over time.

Keywords: Causal inference; clinical trial; estimand; hazard ratio; proportional hazards model; survival analysis.

Figure 1.

Example 1: Potential outcome bivariate distribution, $F$, plotted in the form of its bivariate probability mass function, $f$. Darker values have more mass.

Figure 2.

Example 1: Potential outcome survival curves.

Figure 3.

Example 1: Potential outcome probability mass functions.

Figure 4.

Example 1: Potential outcome hazard curves.

Figure 5.

Example 1: Hazard ratio estimands over time, except ${\theta }_{\text{CoxHB}}(t)$ is estimated from a simulated study with $n=10,000$ with the end of study censoring changing such that $\tau =t$.

Figure 6.

Example 2: Vaccine example based on a mixture of 4 exponential curves, with hazard rates from Table 1.