The Instrument as an Assignment That Shifts Treatment

Assignment Is Not the Same as Treatment

In an instrumental-variable design, the instrument is often an assignment, offer, or encouragement that changes who receives a treatment without determining receipt for everyone. This distinction matters because people may decline an offer, fail to access the treatment, or obtain it through another route. The assigned condition and the treatment actually received are therefore different variables.

Let \(Z\) denote an assignment or encouragement, \(D\) indicate whether a person receives the treatment, and \(Y\) denote an outcome. In a common binary setup, \(Z=1\) means assigned or offered the encouragement and \(Z=0\) means not assigned or offered it; \(D=1\) means treatment was received and \(D=0\) means it was not. These labels describe separate stages: \(Z\) is the assignment, \(D\) is treatment receipt, and \(Y\) is the outcome of interest.

For example, randomizing an invitation to a counseling program does not randomize counseling attendance. Some invited people may not attend, and some people not invited may attend through another route. The invitation is \(Z\); attendance is \(D\). Treating these as interchangeable would erase the very distinction that makes an encouragement design useful.

A direct randomized treatment assignment can look different. If every person assigned to treatment receives it, then assignment and receipt coincide in the data. With nonadherence, however, the assigned treatment group and the treatment-receipt groups diverge. An instrumental-variable analysis uses the assignment variation, not a claim that every person followed the assignment.

What an Uptake Shift Shows

For a binary assignment, a basic descriptive check compares the treatment-receipt rates in the two assignment groups. If \(D\) is binary, the rate in each group is the proportion with \(D=1\). The difference tells us whether the encouragement is associated with a change in treatment uptake in the observed data.

When \(Z\) is randomized, properly implemented, and analyzed according to assignment, this contrast estimates the effect of assignment on treatment receipt in the study population, subject to the usual design and measurement conditions. It is an effect of the offer on uptake—not the effect of the treatment on the outcome \(Y\). Even a large uptake difference does not show that the assignment affects \(Y\) only through treatment, or that the assignment is independent of all relevant potential outcomes. Those are separate requirements for instrumental-variable interpretation.

The arithmetic is straightforward. If \(n_1\) people are assigned the encouragement and \(r_1\) receive treatment, then the observed uptake rate in that group is \(r_1/n_1\). Define \(n_0\) and \(r_0\) analogously for the comparison group. Subtracting the two rates gives the observed uptake contrast. A positive value indicates higher uptake among those assigned the encouragement; a negative value indicates lower uptake. A value near zero may indicate little practical shift, but sampling uncertainty should be considered before concluding that the assignment has no effect on uptake.

Encouragement Designs in Practice

An encouragement can be an invitation, reminder, referral, scheduling offer, or other assignment intended to make treatment receipt more likely. It need not force treatment, and it should not be confused with a treatment itself. Its defining role is to create a difference in the opportunity or motivation to receive treatment.

What Must Be Checked Before Calling It an Instrument

“Instrument” is sometimes used informally for any encouragement. For causal analysis, it is safer to distinguish an instrument candidate from an instrument whose assumptions are credible for the question at hand. An observed uptake shift is necessary for the usual binary-instrument strategy to provide useful treatment variation, but it is not sufficient for a causal treatment interpretation.

The design and analysis should establish, or justify, several points. First, the assignment must be defined clearly and recorded correctly. Second, assignment should be implemented as intended; a randomized allocation that is overridden or misrecorded may not provide the planned comparison. Third, treatment receipt should be measured consistently in both groups, including treatment received outside the study. Finally, the assignment must satisfy the instrumental-variable assumptions relevant to the chosen analysis. The standard names include independence, exclusion, and—when targeting a local average treatment effect—monotonicity. These are distinct from the question of whether uptake differs.

For a randomized encouragement, randomization supports independence of assignment from pre-assignment characteristics in expectation. It does not guarantee that assignment has no direct effect on \(Y\), and it does not guarantee that every person’s receipt moves in the same direction. A reminder might change attendance for some people and have no effect for others; another assignment might discourage some people. The assignment procedure and the real-world pathways it creates both matter.

A simple uptake contrast is also a sample estimate. Random assignment can produce different groups by chance, so a small observed difference does not prove the population uptake shift is exactly zero, and a positive difference does not prove it is large or precisely estimated. Consider uncertainty, the size of the shift, and whether the comparison was pre-specified. Do not rely on a single noisy percentage difference as a complete diagnostic.

Derivation: Why the Difference in Rates Is the Assignment Contrast

For binary treatment receipt, the sample mean of \(D\) in an assignment group equals the proportion receiving treatment, because each person’s \(D\) is either zero or one. If the group contains \(n_z\) people and \(r_z\) of them receive treatment, the sum of their \(D\) values is \(r_z\). Therefore, the group mean is \(r_z/n_z\), which is also the group’s treatment-receipt rate. Subtracting the two group means gives the rate difference above.

Thus, \(\bar{D}_{Z=1}-\bar{D}_{Z=0}\) is not a special adjustment: it is exactly the difference in observed treatment-receipt proportions. Randomization gives this comparison a causal interpretation as the assignment’s effect on receipt under appropriate implementation and measurement conditions. It does not transform the comparison into the causal effect of receipt on the outcome.

Diagnostics, Alternatives, and Limitations

A useful review starts with a flow of people from assignment to receipt. Report the number assigned to each group, the number with observed treatment status, and the number receiving treatment in each group. Check missing treatment records and treatment access outside the study. If missingness differs by assignment, the observed rates may not represent the actual uptake rates. Also verify that the group labels correspond to the original assignment rather than a post-assignment status.

Common warning signs include an almost identical uptake rate across groups, substantial missing receipt information, inconsistent definitions of treatment, or an encouragement that changes other services at the same time. A weak observed shift can make an IV estimate unstable; a strong shift cannot repair an exclusion or independence problem. These are different failure modes and should not be conflated.

The main alternatives answer different questions. A direct randomized treatment assignment with nearly complete adherence can support a comparison of assigned treatment groups, while an encouragement design retains the assignment comparison when uptake is incomplete. A per-protocol or as-treated comparison instead groups people by treatment received and can lose the protection of random assignment, because uptake may be related to prognosis or motivation. An observational instrument may offer variation in treatment receipt, but its credibility depends on a defensible source of assignment variation rather than randomization.

An encouragement design is limited by the population and setting in which the offer operates. Its uptake shift may be modest, its effect on receipt may differ across groups, or the offer may create pathways to the outcome besides treatment. The uptake contrast is therefore an essential design diagnostic, not a complete analysis. Interpret it narrowly, then evaluate the remaining assumptions and the target causal question.

• Mistake: “The invitation raised attendance by 30 percentage points, so treatment reduced the outcome by 30%.”
• Correction: The 30-percentage-point contrast is in treatment receipt across assignment groups. It is not an outcome effect, and it is not a treatment effect.
• Mistake: “Assignment was randomized, so it must be a valid instrument.”
• Correction: Randomization supports independence, but instrument validity also requires other conditions, including a credible exclusion restriction; randomization alone does not establish them.