However, sometimes in a randomized experiment, some members selected for treatment may not actually receive treatment (if participation is voluntary, for example the Medicaid expansion in Oregon). In this case, sometimes researchers will compare differences in outcome between those selected for treatment vs those assigned to control groups. This analysis, as assigned or as randomized, is referred to as an intent-to-treat analysis (ITT). With perfect compliance, ITT = ATE.
As discussed previously, using treatment assignment as an instrumental variable (IV) is another approach to estimating treatment effects. This is referred to as a local average treatment effect (LATE).
What is LATE and how does it give us an unbiased estimate of causal effects?
In simplest terms, LATE is the ATE for the sub-population of compliers in an RCT (or other natural experiment where an instrument is used).
In a randomized controlled trial you can characterize participants as follows: (see this reference from egap.org for a really great primer on this)
Never Takers: those that refuse treatment regardless of treatment/control assignment.
Always Takers: those that get the treatment even if they are assigned to the control group.
Defiers: Those that get the treatment when assigned to the control group and do not receive treatment when assigned to the treatment group. (these people violate an IV assumption referred to monotonicity)
Compliers: those that comply or receive treatment if assigned to a treatment group but do not recieve treatment when assigned to control group.
The outcome for never takers is the same regardless of treatment assignment and in effect cancel out in an IV analysis. As discussed by Angrist and Pishke in Mastering Metrics, the always takers are prime suspects for creating bias in non-compliance scenarios. These folks are typically the more motivated participants and likely would have higher potential outcomes or potentially have a greater benefit from treatment than other participants. The compliers are characterized as participants that receive treatment only as a result of random assignment. The estimated treatment effect for these folks is often very desirable and in an IV framework can give us an unbiased causal estimate of the treatment effect. This is what is referred to as a local average treatment effect or LATE.
How do we estimate LATE with IVs?
One way LATE estimates are often described is as dividing the ITT effect by the share of compliers. This can also be done in a regression context. Let D be an indicator equal to 1 if treatment is received vs. 0, and let Z be our indicator (0,1) for the original randomization i.e. our instrumental variable. We first regress:
D = β0 + β1 Z + e
This captures all of the variation in our treatment that is related to our instrument Z, or random assignment. This is 'quasi-experimental' variation. It is also an estimate of the rate of compliance. β1 only picks up the variation in treatment D that is related to Z and leaves all of the variation and unobservable factors related to self selection (i.e. bias) in the residual term. You can think of this as the filtering process. We can represent this as: COV(D,Z)/V(Z).
Then, to relate changes in Z to changes in our target Y we estimate β2 or COV(Y,Z)/V(Z).
Y = β0 +β2 Z + e
Our instrumental variable estimator then becomes:
βIV = β2 / β1 or (Z’Z)-1Z’Y / (Z’Z)-1Z’D or COV(Y,Z)/COV(D,Z)
The last term gives us the total proportion of ‘quasi-experimental variation’ in D related to Y. We can also view this through a 2SLS modeling strategy:
Stage 1: Regress D on Z to get D* or D = β0 + β1 Z + e
Stage 2: Regress Y on D* or Y = β0 +βIV D* + e
As described in Mostly Harmless Econometrics, "Intuitively, conditional on covariates, 2SLS retains only the variation in s [D in our example above] that is generated by quasi-experimental variation- that is generated by the instrument z"
Regardless of how you want to interpret βIV, we can see that it teases out only that variation in our treatment D that is unrelated to selection bias and relates it to Y giving us an estimate for the treatment effect of D that is less biased.
The causal path can be represented as:
There are lots of other ways to think about how to interpret IVs. Ultimately they provide us with an estiamate of the LATE which can be interpreted as an average causal effect of treatment for those participants in a study whose enrollment status is determined completely by Z (the treatment assignment) i.e. the compliers and this is often a very relevant effect of interest.
Marc Bellemare has some really good posts related to this see here, here, and here.