What is the relationship between LATE and TOT?

Question

My understanding of LATE was that it was the effect of a treatment on individuals who were induced to be treated by the experiment. That is, the effect on compliers. My understanding of Treatment-on-the-treated (TOT) is that it is the effect of individuals who are in the treatment group (which can include both individuals who were induced to be treated, and always-takers who would have accepted the treatment regardless)

However, my notes say the opposite -- that TOT is a subset of LATE, and that LATE allows for the existence of treated individuals who would have been treated even in the absence of experiment. But I don't really understand why one of them is capturing always-takers and the other isn't

What is the right way to think about this?

Answer 1

Writing out the estimands may help. With $Y$ as the observed outcome, $A$ is the treatment of interest, $Z$ is the instrument, and $Y^a$ is the potential outcome under treatment plan $A=a$.

For the TOT (or average treatment effect in the treated, ATT), the estimand is $$E[Y^{a=1} - Y^{a=0} |A=1]$$ For LATE $$E[Y^{a=1} - Y^{a=0} | A^{z=1} = 1, A^{z=0}=0]$$ From this, we can see that the TOT is estimated in the population all treated ($A=1$) individuals, whereas LATE applies to a subset of the treated individuals in the population. More specifically, LATE applies only to compliers as defined by $Z$

Answer 2

LATE is a subset of TOT....I quote from Mastering 'Metrics by Angrist and Pischke:

"Researchers and policy makers are sometimes interested in average causal effects for the entire treated population, as well as in LATE. This average causal effect is called the treatment effect on the treated (TOT for short)" (Chapter 3 Section 1)