I was wondering how does one study the average treatment affect in scenarios suchs as mortality rates.
For example: suppose we want to study the effect that a certain medicine has on the mortality rates os the patients. How can we do a study such as Difference-In-Differences or Propensity Scores if the differences before the treatment are zero? (for a patient to receive or not the treatment he/she has to not have died before being given the treatment, so the mortality rates of the control and treatment group are zero)
Can someone help me understand causal inference in this situations where there's no difference before the treatment is implemented?