# What is the relationship between LATE and TOT?

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

• @Parseltongue Is TOT the same thing as ATET? May 11, 2016 at 0:44
• if ATET is average treatment effect on the treated-- yes, it sounds the same. May 11, 2016 at 1:07
• I would agree with your first interpretation! ToT/ATET is a larger set! Which notes do you have? May 12, 2016 at 0:03

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 $$Z$$