The monotonicity assumption

In the previous lesson, we saw that estimating the ITT (intention-to-treat) causal effect is easy because the intention to treat is randomized. In contrast, estimating the effect of the treatment $D$ is tricky because it may not be randomized!

Because of non-compliance and the existence of confounders in determining who actually receives treatment, estimating the treatment effect $D$ is similar to estimating the treatment effect in an observational study.

Each subject has two potential values of treatment:

• $D^{Z=1}$ or simply $D^1$ which is the value of treatment received if assigned to treatment or if $Z=1$. $Z=1$ also means that the subject is encouraged,
• $D^{Z=0}$ or simply $D^0$ which is the value of treatment received if not assigned to treatment or if $Z=0$. $Z=0$ also means that the subject is not encouraged.

Remember, these are not potential outcomes but rather potential treatments (received). For each person, we only observe one of the potential treatments $D^0$ or $D^1$. If a subject’s treatment assignment is 0, $Z=0$, then we only observe $D^0$. If a subject’s treatment assignment is 1, $Z=1$, then we only observe $D^1$. It’s important not to get confused here. $D^0$ is not the same as $D=0$ nor is $D^1$ the same thing as $D=1$. This should become clear in a minute.

Now, imagine for a subject $D^1=1$ and $D^0=0$. This means if the subject is in the encouraged group (i.e., assigned to receive the treatment), he’ll actually receive the treatment. If he is not in the encouraged group, he won’t end up receiving the treatment. This person is our favorite type of subject because even when treatment can’t be enforced, he does what the treatment assignment dictates. Subjects such as these are usually referred to as compliers.

Practice

For a given subject, $D^1=0$ and $D^0=0$. Which of the following statements are correct. Choose all that apply.

If the subject is assigned to treatment, they will receive treatment

If the subject is assigned to the control they will not receive treatment

If the subject is assigned to the control, they will receive treatment.

For our next subject, $D^0=1$ and $D^1=0$. This person clearly does the opposite of what they’re told. If assigned (or encouraged) to the treatment group, they choose not to receive treatment and if assigned (or encouraged) to the control group, they choose to receive the treatment. These types of subjects are called defiers.

There are two more types of subjects. You’ll recognize never-takers from the quiz question above. For never-takers, $D^0=0$ and $D^1=0$. These subjects always choose not to receive the treatment regardless of the treatment assignment.

Finally, always-takers are those for whom $D^0=1$ and $D^1=1$. These subjects always choose to receive the treatment regardless of their assignment.

The table below summarizes the four types of subjects:

Clearly, if everybody in the sample is a complier, treatment assignment $Z$ will be equal to receiving the treatment $D$, and you won’t have any non-compliance issues. When this is the case, treatment received is randomized because treatment assigned is randomized. Estimating the causal effect of treatment $D$ will be a breeze!

If you only have defiers in your sample, treatment assigned $Z$ and treatment received $D$ are exactly the opposite of each other. This is actually good in the sense that treatment received will still be randomized because treatment assigned is randomized.

Because there are variations in taking the treatment among complier and defiers, a causal effect can be estimated for them.

The disappointing fact about never-takers and always-takers is that we can not learn anything about the treatment’s causal effect because, for them, there are no variations in treatment status $D$. Regardless of $Z$, always-takers will always receive the treatment ($D=1$). The opposite is true of the never-takers.

Now, what happens when we have a mixed bag of subjects within our study? Sadly, compliance types are always not directly observed 🙁

However, one assumptions comes to help us in big ways. Typically, we expect there to be very few defiers in an experimental study. Unless you’ve done a really poor job designing your experiment, you wouldn’t expect there to be a large group of subjects doing the exact opposite of what the experiment is intended to have them do. For this reason, researchers typically assume away the defiers and focus their attention on the three remaining groups: compliers, always-takers, and never-takers. This assumption, although minor, will help us in big ways.

All of the compliance types we’ve seen are based on the idea of potential treatments, i.e., what the treatment received would be given the treatment assignment. Potential treatments similar to potential outcomes are not always observed. We observe the treatment assigned and treatment received for each subject. But let’s see what we don’t observe…

Imagine a person who is assigned to treatment, $Z=1$, and receives the treatment, $D=1$. For this person, we only observe potential treatment $D^1$ which is equal to 1; we do not observe $D^0$. We will never know what the subject would have done had they been assigned to the control group. For this subject, $D^0$ could be 1 or 0. In other words, they may be a complier, but they could just as easily be an always-taker.

Imagine another person. This person is not assigned to the treatment, $Z=0$, but they receive the treatment anyways, $D^0=1$. For this person, we only observe potential treatment $D^0$; we do not observe $D^1$. We don’t know what would have happened if the subject had been assigned to the treatment group. $D^1$ could either 1 or 0. The subject could be an always-taker or a defier.

Practice

For a subject in the control group, $D^0=1$. Which of the following do we know for sure?

The subject is a complier.

The subject is a defier.

The subject is not a never-taker.

The subject is an always-taker.

There are four different combinations we may observe. The following table summarizes each of them, and what we are able to discern from each:

Looking at this table, let’s revisit our assumption that there is a negligible amount of defiers in the sample. This assumption is called the monotonicity assumption or no-defier assumption. Does it help us at all in dealing with non-compliance and identifying unobserved potential outcomes?

If there are no defiers, the table above would change to this.

The no-defier assumption helps us identify those in row two as always-takers and those in row three as never-takers. In other words, if $Z=0$ and $D=1$, we know the subject is an always-taker. If $Z=1$ and $D=0$, we know the subject is a never-taker.

As a result, we are able to identify the share of always-takers and the share of never-takers in our sample. We know the share of subjects in the group $Z=0, D=1$ and the share of subjects in the group $Z=1, D=0$ and as a result, we know the share of always-takers and never-takers. Finally, if we know the share of always-takers and the share of never-takers, we know the share of compliers. Again, all because of our convenient assumption that there are no defiers.

For instance, row four in the table above signifies those who are assigned to treatment, $Z=1$, and end up receiving it, $D=1$. These subjects can either be always-takers or compliers. If we subtract the share of always-takers from this total share, we can easily find the share of compliers:

The share of subjects that are compliers is also called the compliance rate.

All the hard work we did above, wasn’t for nothing. In the next lesson, we will see how they all come together.