Why “Received Treatment” May Be Too Simple
In an encouragement study, the instrument may change whether a person receives treatment. But it may also change what treatment the person receives, when it is received, or at what intensity. A reminder might increase medication uptake while also shifting patients toward a different drug or a more regular dosing schedule. If the analysis records only “treated” versus “not treated,” these distinct treatment states are collapsed into one category.
That collapse matters for the exclusion restriction. The restriction rules out an effect of encouragement on the outcome except through treatment. If “treatment” means only a binary indicator, a change in drug, timing, or intensity can affect the outcome while the indicator remains one. The encouragement then has an outcome pathway through a treatment feature that the binary variable does not represent.
The appropriate definition depends on the causal question. If the target is the effect of receiving any member of a clinically interchangeable class, a binary indicator may be adequate, provided the versions are genuinely interchangeable for the outcome question. If the target concerns a particular regimen, or if encouragement can shift people between meaningfully different regimens, the treatment definition should preserve those distinctions. Merely changing the variable name does not make versions equivalent.
Exclusion Depends on the Treatment State
For a complete treatment state \(T\), write \(T_i(z)\) for the state person \(i\) would receive under encouragement level \(z\), and \(Y_i(t)\) for the outcome under treatment state \(t\). The exclusion restriction for this representation says that encouragement affects the outcome only through the state actually received:
Here \(Y_i(z)\) is the outcome under encouragement \(z\). This restriction still permits \(Z\) to change treatment: the treatment state may differ between \(T_i(0)\) and \(T_i(1)\). It rules out other routes from encouragement to outcome, such as reassurance, extra clinical attention, or a change in follow-up that affects the outcome apart from the treatment state represented by \(T\).
The usual binary-treatment formulation instead writes \(Y_i(d)\), with exclusion requiring that the outcome under encouragement \(z\) be \(Y_i(D_i(z))\). That formulation treats every state coded \(D=1\) as the same treatment state for the relevant causal question. If encouragement changes the version among people with \(D=1\), the same binary value can correspond to different outcome-relevant states. In that case, the binary exclusion restriction is not justified just because encouragement has no other direct pathway.
Worked Example: A Reminder Changes the Medication Version
Suppose a hypothetical clinic randomizes a reminder to encourage medication use. The data record \(D=1\) whenever a patient takes any prescribed medication. Under no reminder, a patient who would take medication uses Drug A; under a reminder, that same patient would still take medication but use Drug B. Assume, for illustration, that the drug choice can affect the outcome.
With \(T=(D,V)\), the patient’s states are \(T_i(0)=(1,\text{A})\) and \(T_i(1)=(1,\text{B})\). Under the complete-state exclusion restriction, an outcome difference is allowed if \(Y_i(1,\text{A})\ne Y_i(1,\text{B})\): the reminder changed the treatment state. But under a binary definition alone, \(D_i(0)=D_i(1)=1\). A difference in outcomes between those states would look like an effect of encouragement that occurred without any change in \(D\), contradicting binary-treatment exclusion.
The distinction is about the model, not about whether the reminder was randomized. Randomization can support a causal comparison of encouragement levels, but it does not make the binary treatment variable a complete description of what changed.
When the Usual Wald Interpretation Applies
A Wald ratio divides the outcome contrast by the treatment-receipt contrast. To interpret it as a complier average causal effect for a binary treatment, the binary representation must be adequate. The following standard conditions state the relevant setup. Encouragement is randomized (or independent of the potential treatment and outcome states); the observed outcome and receipt equal the potential values under the encouragement actually assigned; there is no interference between individuals; the first stage is nonzero; treatment receipt is monotone, \(D_i(1)\ge D_i(0)\); and exclusion holds for the binary treatment state. In addition, the versions must not create a change in outcome-relevant treatment state among people whose binary receipt is unchanged from \(Z=0\) to \(Z=1\). A sufficient condition is that the version is stable for those treated under both encouragement levels, with the binary treatment outcome well-defined for each person.
Under these assumptions, people are either compliers, who take treatment only when encouraged; always-takers, who take it either way; or never-takers, who do not take it either way. Monotonicity rules out people who would take treatment only without encouragement. This tutorial uses that condition only to establish the familiar Wald interpretation; it does not address departures from monotonicity.
The justification can be seen by splitting the outcome contrast across these groups. Random assignment and consistency let the observed encouragement contrast identify the mean contrast in potential outcomes. Under binary exclusion, always-takers have the same treatment state under both assignments, and never-takers have the same untreated state. Their outcome contrasts are therefore zero. Only compliers contribute to the outcome contrast. In the first stage, only compliers change \(D\), so the treatment-receipt contrast equals their population share.
Here \(C\) denotes the complier group, and the denominator must be nonzero. The final quantity is the average effect of the binary treatment for compliers, not necessarily an average effect for everyone in the study. The argument fails if encouragement changes versions in a way that affects outcomes among people whose binary receipt remains unchanged: those people can then contribute to the outcome contrast even though they contribute nothing to the first stage.
Worked Example: Checking Whether a Binary Treatment Definition Is Defensible
Consider a hypothetical study in which \(D\) records whether a participant receives a course of therapy. The encouragement offers help arranging an appointment. The therapy is always the same procedure, at the same dose, and within the same time window; the encouragement changes access, not the procedure or its delivery. Assume the encouragement is randomized, has no direct effect on the outcome, and affects treatment receipt for some participants.
The relevant treatment state can reasonably be represented by \(D\) if those features truly do not vary in outcome-relevant ways. The proposed restriction is then \(Y_i(z)=Y_i(D_i(z))\): the encouragement affects the outcome only by changing receipt of this well-defined therapy. Randomization supports independence, the stated absence of other pathways supports exclusion, and a nonzero change in receipt is required for a Wald ratio. To interpret that ratio as the complier effect, monotonicity is also needed.
If, instead, the encouragement also leads treated participants to receive the therapy earlier, the binary definition needs further justification. If earlier treatment can affect outcomes, then a binary receipt indicator no longer captures the full treatment state. The procedure may still be studied, but timing must either be included in the treatment definition or defended as irrelevant to the particular outcome and target.
What a Version-Induced Wald Ratio Combines
When versions change, the observed outcome contrast can combine several contributions: changes from no treatment to treatment among people whose receipt changes, changes in version among people already treated, and possibly changes in timing or intensity. The first stage, however, measures only the change in the proportion with \(D=1\). Dividing by that first stage cannot generally separate the different outcome contributions.
Worked Example: The Ratio Is Not the Complier Effect
Suppose a hypothetical randomized study has 30% always-takers, 20% compliers, and 50% never-takers. Receipt rises from 30% under \(Z=0\) to 50% under \(Z=1\), so the first stage is \(0.50-0.30=0.20\). Assume the encouragement also changes the treatment version for the always-takers. On the outcome scale, that version change raises their mean outcome by 2 units. Among compliers, the effect of moving from no treatment to the encouraged version is 12 units. Never-takers have no treatment change, and there are no other encouragement pathways.
The population outcome contrast has a contribution of \(0.30(2)=0.60\) units from always-takers and \(0.20(12)=2.40\) units from compliers. Thus:
The Wald ratio is \(3.00/0.20=15.0\) outcome units per unit of treatment receipt. But the complier effect specified in this example is 12 units. The extra \(0.60\) in the outcome contrast comes from a version change among people whose receipt remains \(D=1\); dividing by the complier share adds \(0.60/0.20=3\) units to the ratio. Thus the ratio is not the complier effect for the binary receipt indicator, even though the encouragement is randomized and the treatment first stage is positive.
A complete treatment-state representation can describe the version change as part of the pathway from encouragement to outcome. That does not by itself make this single Wald ratio an effect of one component of the state. The ratio’s interpretation depends on the treatment definition and additional assumptions about which components vary and whose outcomes they affect.
Choosing and Reporting the Treatment Definition
Before using a binary treatment indicator in an encouragement analysis, specify what counts as the same treatment. A practical check is to compare the treatment states that could occur under each encouragement level, including among people whose recorded receipt does not change. Ask whether version, timing, intensity, duration, and delivery are fixed, irrelevant to the outcome question, or potentially changed by encouragement. If a changed feature can affect outcomes, the binary indicator may conceal a consequential pathway.
This is not solved by simply adjusting for the received version in an outcome regression. Version is itself potentially affected by encouragement; treating it as an ordinary baseline covariate does not restore the binary exclusion restriction or automatically identify a version-specific causal effect. The causal question and the treatment state must be specified first.
| Situation | What the treatment variable represents | Implication for exclusion |
|---|---|---|
| Encouragement changes receipt only; the treatment received is well-defined | Binary receipt may represent the relevant state | Binary exclusion may be plausible if other pathways are ruled out |
| Encouragement changes drug, timing, dose, or delivery in outcome-relevant ways | Binary receipt merges distinct treatment states | Exclusion for receipt alone is not justified by exclusion for a complete state |
| Versions vary, but are irrelevant for the stated outcome contrast | Binary receipt may be an adequate simplification for that question | State the version-irrelevance assumption; do not treat it as automatic |
Common Mistake
Wrong reasoning: “The encouragement is randomized, and it changes whether participants are treated. Therefore the Wald ratio is the causal effect of treatment for compliers.” Randomization helps identify the effect of encouragement, but it does not ensure that the encouragement changes only the binary receipt indicator. A version shift among people who remain treated can affect outcomes without affecting the first stage.
Correction: Define the treatment state finely enough for the causal question, then assess exclusion for that state. Use the standard binary Wald interpretation only when the binary treatment is a defensible representation and the other instrumental-variable conditions hold. Otherwise, describe the ratio as an outcome contrast scaled by the change in binary receipt, not automatically as a complier effect.
Check Your Understanding
For each question, focus on how the treatment state is defined and what the encouragement changes.
- An encouragement increases whether patients receive a procedure and also shifts treated patients to a higher dose. What additional question must be answered before using binary receipt in the exclusion restriction?
- In potential-outcome notation, how does the exclusion restriction for a complete treatment state differ from exclusion written only in terms of \(D\)?
- Under the standard binary-treatment conditions, why do always-takers and never-takers contribute zero to the outcome contrast?
- In the numerical example, calculate the Wald ratio and identify the contribution that makes it differ from the complier effect.
- Why does randomization of encouragement not, by itself, justify interpreting the Wald ratio as a complier effect?