Why the Meaning of “Treatment” Matters
An instrumental-variable analysis may describe treatment with a binary indicator: a person either receives treatment or does not. That summary is useful only if it captures the intervention relevant to the outcome. In practice, an instrument may influence not just whether treatment is received, but also which version is received, when it begins, how intensely it is delivered, or which services accompany it. If those dimensions affect outcomes, a binary treatment indicator can hide a consequential change.
The exclusion restriction is stated relative to a specified treatment. If treatment is defined too broadly, a change in treatment version may look like an effect of the instrument that remains after “holding treatment fixed.” The issue is not merely wording: it determines which causal contrast the Wald ratio can represent.
This restriction is meaningful only after specifying what \(D=1\) represents. Suppose “received a program” includes a brief version and an intensive version. If the instrument makes intensive delivery more likely, then setting \(D=1\) does not necessarily set the same treatment in both instrument conditions. A more detailed description uses a version \(V\), with potential outcomes \(Y(z,d,v)\). Exclusion relative to a fully specified treatment can then be written \(Y(1,d,v)=Y(0,d,v)\). This allows the outcome to depend on the treatment version while ruling out a separate instrument effect at a fixed version.
From a Binary Indicator to a Well-Defined Treatment
A treatment is sufficiently specified for a causal contrast when its relevant features are defined well enough that “receiving treatment” refers to a coherent intervention. Depending on the setting, those features may include the active ingredient, dose, duration, timing, delivery method, provider, and accompanying services. Not every detail must be fixed in every study. The important question is whether an omitted feature varies in a way that matters for the outcome and is related to the instrument.
If versions differ in outcome consequences, there are two broad ways to clarify the causal question. One is to define the treatment as a particular version, such as a fixed dose delivered within a specified time window. The other is to define treatment as a package or strategy that includes the versions and services actually induced by the instrument. These are different estimands: an effect of the package is not automatically an effect of a single core component.
A binary instrument and binary \(D\) do not, by themselves, identify a causal effect for a multi-valued treatment. The familiar Wald ratio has a standard complier interpretation under the usual binary-treatment assumptions, including independence of the instrument, a nonzero first stage, exclusion for the specified treatment, and monotonicity. When the instrument changes versions at a fixed value of \(D\), the exclusion assumption for that binary treatment may fail, and the usual interpretation does not follow merely from calculating the ratio.
Worked Example: A Program With Brief and Intensive Versions
Consider a hypothetical randomized encouragement \(Z\) for a program. Among 100 people, 40 are compliers: they receive the program when encouraged and do not receive it otherwise. The other 60 never receive it. Thus, the treatment rate is 0.40 under \(Z=1\) and 0 under \(Z=0\). Assume the mean outcome without the program is 10 points. For these compliers, a brief version would raise the outcome by 2 points, while an intensive version would raise it by 6 points. The encouragement causes treated compliers to receive the intensive version; it does not change outcomes for the 60 never-treated people.
The observed mean outcome under encouragement is \(10+(40/100)(6)=12.4\) points. Under no encouragement, the mean is 10 points. The reduced-form assignment contrast is therefore \(12.4-10=2.4\) points, and the first-stage contrast is \(0.40-0=0.40\).
The value of 6 points is the outcome contrast between no program and the intensive version for the compliers in this construction. It is not the 2-point effect of the brief version. If the intended treatment had been defined simply as “any program,” the encouragement changes the version accompanying \(D=1\), so binary-treatment exclusion is not satisfied. If the intended intervention is instead “the intensive program as delivered under encouragement,” the ratio has a different, package-specific interpretation, subject to the stated assumptions.
How Version Changes Enter the Wald Ratio
The following short derivation isolates the problem. Suppose there are only compliers and never-takers, the instrument is independent of their potential outcomes and treatment behavior, and the instrument changes no outcomes among never-takers. Let the proportion of compliers be \(q>0\). For a complier, encouragement changes treatment from 0 to 1 and may also change the received version from \(V_0\) to \(V_1\). Write \(Y(d,v)\) for the outcome under treatment status \(d\) and version \(v\). Then the reduced form is the complier proportion times the average outcome change experienced by compliers:
Here \(C\) denotes the complier group. The first stage is \(E[D\mid Z=1]-E[D\mid Z=0]=q\). Dividing gives
This ratio is an average contrast between the outcome under the treatment-and-version combination induced by encouragement and the outcome under the untreated condition. It is not generally an effect of changing only \(D\) while holding version fixed. If the instrument also changes outcomes among people whose treatment status does not change, additional terms enter the reduced form, and this simple expression no longer applies.
The derivation also explains why exclusion cannot be checked by looking only at the two treatment rates. A nonzero first stage establishes that the instrument changes the probability of the recorded binary treatment. It does not establish that all treated people receive the same intervention, or that version changes are irrelevant to outcomes.
Worked Example: An Intervention Bundled With Counseling
Suppose a hypothetical randomized instrument encourages a health intervention. Of 100 participants, 40 are compliers and 60 are never-takers. The mean outcome in the absence of the intervention is 20 units. When a complier receives the core intervention alone, the outcome rises by 3 units. Under encouragement, however, every treated complier also receives counseling that adds 2 units. Assume no effects of encouragement or counseling among never-takers.
Under \(Z=0\), no complier receives the intervention, so the mean outcome is 20 units. Under \(Z=1\), each of the 40 compliers receives the intervention-plus-counseling package, for an individual mean increase of \(3+2=5\) units. The overall mean is \(20+(40/100)(5)=22\) units. Thus the reduced form is \(22-20=2\) units, and the first stage is \(0.40-0=0.40\).
The ratio matches the 5-unit effect of the induced package in this hypothetical setup. It does not equal the 3-unit effect of the core intervention alone. Calling the binary variable simply “received the intervention” obscures the added counseling and risks presenting a package effect as a component effect.
Timing and Intensity Are Treatment Features Too
Version problems are not limited to visibly different products or services. Starting a treatment earlier, receiving more sessions, or taking a higher dose can define materially different interventions. If the instrument changes these features among people with \(D=1\), then “treatment received” may not describe a fixed intervention. A causal contrast should say whether it concerns treatment at a specified time and intensity, or the strategy of treatment as it is actually induced.
Worked Example: Early Versus Delayed Treatment
Imagine a hypothetical instrument that encourages a binary treatment. Among 50 compliers in a group of 100 people, the encouragement leads to treatment beginning immediately; without encouragement, these compliers remain untreated during the outcome window. The other 50 people are never-takers. The untreated mean outcome is 30 days, and immediate treatment reduces the outcome by 4 days. Assume the instrument has no effect on never-takers.
The treatment rate is 0 under \(Z=0\) and 0.50 under \(Z=1\). The mean outcome is 30 days under \(Z=0\), and \(30-(50/100)(4)=28\) days under \(Z=1\). The reduced form is \(28-30=-2\) days; the first stage is \(0.50\). Therefore,
Under the assumptions given, the result describes the effect for compliers of receiving immediate treatment rather than remaining untreated over the outcome window. It does not tell us the effect of starting later, or the effect of treatment at any unspecified time. If the instrument instead shifted people between early and delayed treatment while both groups had \(D=1\), that timing change would not appear in this binary first stage, even though it could affect the outcome.
Common Mistake: Treating a Coarse Indicator as the Intervention
A common error is to reason, “The instrument affects the outcome only by changing \(D\), because the outcome difference disappears conceptually once I condition on whether \(D=0\) or \(D=1\).” That argument assumes that everyone with \(D=1\) receives the same relevant intervention. If encouragement changes dose, timing, version, or accompanying services, then a comparison at \(D=1\) can still compare different treatments. The correction is to state the treatment precisely, assess whether its relevant features vary with the instrument, and interpret the estimand at the level of intervention actually defined.
A treatment version changing with the instrument does not automatically mean the instrument is unusable. If the target is the effect of a clearly defined package or delivery strategy, version differences may be part of that treatment rather than a violation of exclusion for the package. But a binary treatment indicator and the standard Wald ratio do not automatically identify the effect of each component. The treatment definition, assumptions, and causal claim must align.
Check Your Understanding
Use the treatment definition and instrument-induced changes to decide what causal contrast is supported.
- An instrument increases treatment uptake and also shifts treated participants from a low dose to a high dose. Why may exclusion fail for a binary indicator of “any treatment”?
- In the bundled-counseling example, what does the computed Wald ratio describe, and why is it not the effect of the core intervention alone?
- Suppose \(D\) records whether a person ever received a service, but the instrument changes when the service begins. Name a treatment definition that would make the timing dimension explicit.
- Under what condition can differences among treatment versions be irrelevant to the binary-treatment exclusion restriction?
- Why does a nonzero first stage not establish that the treatment is sufficiently well specified for the intended causal contrast?