# The difference between average and marginal treatment effect

I have been reading some papers, and I am unclear about the specific definitions of Average Treatment Effect (ATE), and Marginal Treatment Effect (MTE). Are they the same?

According to Austin...

A conditional effect is the average effect, at the subject level, of moving a subject from untreated to treated. The regression coefficient for a treatment assignment indicator variable from a multivariable regression model is an estimate of a conditional or adjusted effect. In contrast, a marginal effect is the average effect, at the population level, of moving an entire population from untreated to treated [10]. Linear treatment effects (differences in means and differences in proportions) are collapsible: the conditional and marginal treatment effects will coincide. However, when outcomes are binary or time to event in nature, the odds ratio and the hazard ratio are not collapsible [11]. Rosenbaum has noted that propensity score methods allow one to estimate marginal, rather than conditional, effects [12]. There is a paucity of research into the performance of different propensity score methods to estimate marginal treatment effects.

But in another Austin paper, he says

For each subject, the effect of treatment is defined to be $Y_i(1)- Y_i(0)$. The average treatment effect (ATE) is defined to be $E[Y_i(1)- Y_i(0)]$. (Imbens, 2004). The ATE is the average effect, at the population level, of moving an entire population from untreated to treated.

So the question that I have is...What is the difference between the average treatment effect and the marginal treatment effect?

As well, how should I classify my estimand? I have a propensity score weighted (IPTW) Cox model. My only covariate is the treatment indicator. Should the resulting hazard ratio be considered the ATE or the MTE?

Edit: To add to the confusion, Guo, in his book propensity score analysis claims that the marginal treatment effect is

...special case of the treatment effect for the people at the margin of indifference (EOTM). In some policy and practice situations, it is important to distinguish between the marginal and average returns. For instance, the average student going to college may do better (i.e. have higher grades) than the marginal student who is indifferent about going to school or not.

I feel like this should be taken with a grain of salt, because this is directed for social sciences (where I believe marginal has a different definition), but I thought I would include it here to display why I am confused.

• You're right that "marginal" means something different in economics. There, it can mean the effect of adding 1 more, which is not the same as unconditional. Eg, if we sorted college applicants descending by IQ, & admitted 1:i & didn't admit (i+1):N, we could ask if the benefit of college for the (i+1)th applicant is the same as the average of the benefit accrued to the top i applicants. In that case the benefit would probably be lesser, but in cases w/ increasing returns to scale, the benefit could be greater. Oct 23, 2016 at 22:06