Intention to treat estimator

Before we see what instrumental variables are, we need to understand non-compliance and randomized encouragement designs.

Non-compliance

In a randomized experiment, non-compliance arises when people don’t stick to the treatment. Let’s see an example!

Imagine an experiment designed to estimate the effect of regular visits to health clinics on individuals’ health outcomes. The treatment is whether the individual regularly visits health clinics and has regular checkups (a simple yes/no treatment for the sake of argument), and the outcome is the probability of having diabetes.

In this randomized experiment, we can imagine randomly assigning individuals to a treatment group where they are told to have two or more annual checkups or a control group where they are told to only have one or no checkup each year. The assignment to treatment is fully randomized.

Can you guess where the non-compliance issues may arise?

Individuals have the freedom of doing what they want. For ethical reasons, we can’t stop them from visiting clinics more or less than we want them to. Treatment assignment alone won’t guarantee that individuals will go by our treatment assignment. They might forget to go to the clinics. Some in the control group might want to have more frequent checkups.

What this example demonstrates is that while treatment may be assigned randomly, actually receiving the treatment may not be.

If within the group that is assigned to treatment, receiving the treatment is non-random, you have a non-compliance problem. For example, if within the treatment group, individuals who are more health-conscious are more likely to go to the clinics. Likewise, within the control group, a systematic (non-random) deviation from the control assignment is also considered a non-compliance issue.

Randomized encouragement designs

Now let’s look at randomized encouragement design with an example looking at a different causal question. What is the effect of a mother’s smoking habit on their child’s birth-weight?

This has always been a question of interest in medical sciences, but conducting a randomized experiment to help answer this question would be far from ethical; you can’t force mothers in the treatment group to smoke, especially when your hypothesis is that smoking will negatively affect their children’s health. Researchers may implement a randomized encouragement design (RED) rather than falling back on observational data in situations like these.

A randomized encouragement design (RED) is an experimental design in which participants are randomized to receive encouragement for treatment rather than the treatment itself. The encouragement can take different forms, but it usually takes the form of additional incentives (financial or non-financial benefits) or information.

In our example, encouraging mothers to smoke is just as unethical as making them smoke, but there may be a way to design the experiment so that the encouragement is positive. We start by randomly assigning mothers into two groups. One group receives an information packet about the negative health effects of smoking during pregnancy, while the other group (the control group) doesn’t receive any information. Note that there may still be some ethical concerns here, but the design should not leave individuals in the treatment and the control group worse off than they would be having not participated in the experiment.

Of course, not everyone who receives encouragement is going to stop smoking. However, those who receive no encouragement are likely to continue smoking.

Similar to randomized trials with non-compliance, treatment assignment (encouragement) is randomized, but receiving the treatment isn’t. Treatment received is likely associated with individual characteristics.

Practice

Imagine we’re interested in understanding the effect of an already-existing government program designed to provide financial assistance to single parents. Specifically, we want to understand how this program affects the academic performance of children. Which of the following best describes the reason why we can’t use a randomized experiment and why instead we should use a RED?

It is unethical to study the academic performance of children.

To conduct a randomized experiment, we would have to deny some single parents access to the government program in order to have a control group, and doing so would be unethical.

We would need to force parents to separate, which is unethical.

Conducting a randomized experiment would be too expensive.

Intention-to-treat estimator

The interesting thing about randomized experiments with non-compliance issues and REDs is that BOTH can suffer from non-compliance. In REDs, non-compliance occurs when those who are encouraged to take the treatment don’t necessarily take it and those who are encouraged not to take the treatment may take it.

Let’s think of a variable $Z$ as the variable capturing treatment assignment and $D$ as the variable capturing who, in fact, receives the treatment and who doesn’t. Therefore, in both randomized experiments with non-compliance and REDs, the DAG would look like something like this:

Note that $X$ is a confounder. Let’s assume $X$ is not observed.

$Z$ affects the treatment $D$ but does not directly affect the outcome (it only affects the outcome through the actual treatment status, $D$). $Z$ is randomized by design (we flip a coin to assign subjects to either the treatment or control group).

Unlike $Z$, $D$ isn’t randomized because it is associated with some individual characteristic. People with this characteristic are more likely to receive the treatment once it’s assigned to them.

The causal effect of $Z$, being assigned to treatment, is called the intention-to-treat (ITT) effect. The ITT effect is all about the treatment assignment and not receiving the treatment, and it is simple and easy to obtain because of random assignment. To estimate the ITT causal effect, we do not have to worry about the confounder.

The ITT estimator is simply:

Estimating the true treatment effect $D$ is much trickier, and something we’ll save for the next lesson!