What is the difference between ITT and ATE? I am having trouble understanding the different estimators that can be used in an impact evaluation. I know that the intention-to-treat (ITT) estimator compares differences between eligible individuals without the program, and eligible individuals with the program, regardless of compliance. However, I thought the average treatment effect (ATE) also measured the same thing. However, it seems that the ATE takes into consideration the compliance. Therefore, it compares outcomes between those eligible and taking up treatment with those who are not eligible. Is this correct? 
 A: Suppose that I am a doctor and I tell everyone in a treatment group to go home and exercise for an hour per day and tell the control group nothing. After a month, I evaluate the difference in their blood pressure. If I just compare the difference in mean blood pressures between the two groups, I get the intention to treat estimator. This doesn't tell me the causal effect of exercise on blood pressure, but the causal effect of telling people to exercise on blood pressure. We would presume that this estimate would be smaller than the treatment effect of exercise per se, as only a (small!) fraction of people in the treatment group would follow my advise. You need to take this difference into account.
A prime example is instrumental variables. This procedure aims to recover the ATE from the ITT. See, for example, 
Joshua D. Angrist; Guido W. Imbens; Donald B. Rubin. 1996. "Identification of Causal Effects Using Instrumental Variables." JASA 91(434): 444--455.
A: For pedagogical purposes, it's actually much better to think about three quantities:
ITT: Intent to Treat Effect -- effect of treatment ASSIGNMENT on outcome (for everybody)
LATE: Local Average Treatment Effect -- effect of treatment no outcome FOR COMPLIERS
ATE: Average Treatment Effect -- effect of treatment on outcome FOR EVERYBODY
The ITT is the most straightforward. If we randomize some individuals into treatment and some into control, we can certainly recover the causal effect of assignment into treatment. That's the ITT.
The LATE is a bit more complicated, but the measure most often gleaned via instrumental variables / two stage least squares, etc. Assuming we're not in a lab setting, even if we assign some folks to treatment (T = 1) and assign some to control (T = 0), people will do what they will do! Some will takeup treatment (D = 1) and some will not takeup treatment (D = 0). We can imagine that some people are just willing to comply with our assignments. We may want to know what kind of person everybody in our data is -- are they the type of person who will do what we say, who will rebel, who will always takeup, who will never takeup? To know this without making any assumptions, we would actually need to know, for each person, what they would do if assigned to treatment and what they would do if assigned to control. Let's imagine Fred, for example. In one universe, we assign Fred treatment. He takes it up! In an alternate universe, we assign Fred control. He doesn't take up treatment! Fred has complied! Thus:


*

*compliers are those that would takeup treatment only if assigned to treatment, and would not takeup treatment only if assigned to control. They would comply with our assignment. 

*Always takers would takeup treatment whether assigned to it or not. 

*Never takers would not takeup treatment whether assigned to it or not. And 

*defiers would do the opposite of what we assign them to do (i.e. would not takeup treatment if assigned treatment, would takeup treatment if assigned control). 


We can't actually glean the type of person each of the folks in our data is, unfortunately. We live in one universe... but if we make an assumption (monotonicity) we can use folks ACTUAL behavior to glean their "type." Once we've done that, we can make a few more assumptions (exclusion restriction, valid randomization, no SUTVA violations on D or Y, relevancy) to calculate the average effect of treatment FOR COMPLIERS. This is the LATE. It's called a "local" average treatment effect b/c it doesn't calculate the treatment effect "globally" (i.e. for all) but instead calculates the effect of treatment "locally" (i.e. for some, specifically, for compliers). It's also sometimes called the CATE or Complier Average Treatment Effect for that reason.
Now we get to the mythical ATE! The ATE is the Average Treatment Effect -- the average effect of treatment for everybody, regardless of the type of person they are. Alas! Our assumptions will not allow us to recover the ATE! Even with them, we can only recover the treatment effect for compliers, or the LATE!  The most straightforward way to recover the ATE is to ensure there is no non-compliance. Then your complier average treatment effect IS the Average Treatment Effect becuase everybody is a complier!
So there you have it! 


*

*ITT -- effect of ASSIGNMENT on outcome. 

*LATE -- effect of treatment on outcome FOR COMPLIERS.

*ATE -- effect of treatment on outcome for EVERYBODY.

A: I think your problem is an ambiguity in language. I have always seen "Average Treatment Effect" with Intent-to-Treat as a subset of ATE.
For example:
ITT analysis is estimating the ATE among those in the treatment arm of a trial. "Treatment of treated" is estimating the ATE among those actually treated.
