A generalized definition of the average causal effect for both binary and continuous treatments

One of the main tasks of causal inference is estimating well-defined causal parameters. One of the main causal parameters is the average causal effect (ACE) - the expected value of the individual level causal effects in the target population. For binary treatments, the individual level causal effect is defined as contrast between potential outcomes. For continuous outcomes, however, there are many such contrasts in finite samples, thus hampering their use as a useful summary of the causal relationship. Here, we proposed a generalized version of the ACE, where individual level causal effects are defined as the derivative (with respect to the treatment) of the individual level causal dose-response function evaluated at treatment value that the individual has. This definition is equivalent to the conventional definition for binary treatments, but also incorporates continuous treatments. We demonstrate that this quantity can be estimated under conventional causal assumptions and illustrate the theoretical ideas with a simulation study.