  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
built-in selection bias as, over time, the individuals at risk in the exposed
and unexposed are no longer exchangeable. In this work, we formalize how the
observed hazard ratio evolves and deviates from the causal hazard ratio of
interest in the presence of heterogeneity of the hazard 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 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) 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 can not
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.
