Why Treatment Versions Matter
An instrumental variable analysis often codes treatment as a binary indicator: a person either received treatment or did not. That coding is adequate only when the distinction captures the treatment features relevant to the outcome. If an instrument changes not just whether treatment occurs but also its timing, quality, dose, or mode, two people coded \(D=1\) may have received meaningfully different interventions.
The issue is not automatically a direct effect of the instrument. An instrument that changes treatment timing may affect the outcome through the treatment actually delivered. But if the analysis represents treatment only as a binary indicator, that timing change is hidden inside the same value \(D=1\). The exclusion restriction stated for that coarse binary treatment may then fail, even if the instrument has no effect on the outcome when the complete treatment state is held fixed.
Let \(Z\) denote a binary instrument, \(D\) a binary treatment indicator, and \(Y\) the outcome. Write \(V\) for the treatment version, with \(V=\varnothing\) when \(D=0\). The potential outcome \(Y(z,d,v)\) allows the outcome to depend on the instrument value \(z\), treatment status \(d\), and, when treated, version \(v\). Exclusion for the complete state is the condition
This condition rules out an instrument effect that bypasses the complete treatment state. But it does not say that treatment versions have identical effects. If \(Z\) changes which version is received, the outcome can change through \(V\), even though the displayed condition holds. A model that records only \(D\) cannot distinguish that version-mediated change from a change in the effect of the binary treatment.
Exclusion Depends on What Counts as Treatment
Suppose an encouragement increases treatment uptake and also leads treated people to start sooner. If early treatment has a different effect, then the encouragement changes outcomes partly by changing the treatment package. Calling the instrument “excluded” or “not excluded” without specifying the treatment definition is therefore incomplete. Relative to a binary indicator for any treatment, the timing difference is a hidden version shift. Relative to a richer treatment definition that includes timing, it is part of the treatment state.
This distinction separates two concerns. A direct effect occurs when \(Z\) affects \(Y\) while the full treatment state is held fixed. A version shift occurs when \(Z\) changes the treatment version, and that version affects \(Y\). Both can undermine the usual binary-treatment Wald interpretation, but they describe different causal pathways. The earlier material on exclusion in encouragement designs addresses the first concern; this tutorial focuses on the second and on the consequences of failing to represent versions.
Worked Example: Earlier Treatment as a Distinct Version
Consider a hypothetical encouragement \(Z\) to attend a clinic. Among people who receive a medication, the encouragement can also affect whether the first dose is taken within one day or after one week. Let \(D=1\) mean “received the medication,” without recording timing. For illustration, suppose that, for a particular person, the potential outcome under no medication is 50 points, under medication started within one day is 58 points, and under the same medication started after one week is 54 points.
Suppose this person receives the medication under either instrument value, but would start within one day if \(Z=1\) and after one week if \(Z=0\). The instrument contrast is \(58-54=4\) outcome points, despite \(D=1\) in both cases. Thus an outcome model using only \(D\) cannot represent the change: the person’s binary treatment status did not change, but the treatment version did.
This is not evidence of a direct effect while the complete treatment state is held fixed. It is a version-mediated effect. If timing is clinically relevant, the binary treatment definition is too coarse for the usual exclusion claim. If timing has no effect on the outcome in the setting under study, the version difference may be harmless for this outcome; that is a substantive claim to justify, not something implied by coding both regimens as \(D=1\).
How Version Shifts Enter the Reduced Form
The consequences can be made explicit using principal strata. Let \(D(z)\) be the treatment status that would occur under instrument value \(z\). Under monotonicity, \(D(1)\geq D(0)\), individuals belong to one of three strata: never-takers \(N\), with \(D(1)=D(0)=0\); always-takers \(A\), with \(D(1)=D(0)=1\); and compliers \(C\), with \(D(1)=1,D(0)=0\). Assume instrument independence, \(Z\perp\{D(0),D(1),\text{relevant potential outcomes}\}\), as in a randomized instrument or an appropriate conditional analysis. The following decomposition also assumes well-defined potential outcomes for the relevant treatment states.
For each stratum \(s\in\{N,A,C\}\), define \(\Delta_s\) as the mean outcome contrast between instrument values within that stratum, allowing the treatment version to change as the instrument changes. For example, among always-takers, \(\Delta_A\) can reflect an instrument-induced change in timing or mode. By partitioning the population into these mutually exclusive strata, the reduced form is exactly
This identity does not assume that the instrument affects outcomes only among compliers. It simply writes the overall mean contrast as a weighted average of the within-stratum contrasts. Under monotonicity, never-takers have \(D=0\) at both instrument values, so \(\Delta_N=0\) if there is no instrument effect on their outcomes while untreated. Always-takers have \(D=1\) at both values, but \(\Delta_A\) need not be zero: their treatment version may change. A direct effect among never-takers, or a version shift among always-takers, must be included rather than silently omitted.
The first stage under independence and monotonicity is
Never-takers and always-takers have no treatment-status change; each complier changes from \(D=0\) to \(D=1\). Therefore the Wald ratio is
The familiar complier-effect interpretation follows only under further restrictions that remove the extra terms and give \(\Delta_C\) the intended treatment-effect meaning. In particular, it is not enough to assume independence and monotonicity. If the instrument affects never-takers directly, or changes versions for always-takers, those effects enter the numerator. If compliers receive different versions because of \(Z\), their contrast is an effect of the instrument-induced treatment package, not automatically an effect of a single, uniform binary treatment.
Worked Example: A Version Shift Among Always-Takers
In a hypothetical population, suppose \(P(N)=0.50\), \(P(A)=0.20\), and \(P(C)=0.30\). The instrument is independent of stratum membership and potential outcomes, and there are no defiers. Suppose the mean outcome contrast is 0 among never-takers, \(+1\) point among always-takers because the instrument changes their treatment version, and \(+4\) points among compliers because the instrument induces treatment. Thus \(\Delta_N=0\), \(\Delta_A=1\), and \(\Delta_C=4\).
The reduced form is
The first stage is \(P(C)=0.30\), so the Wald ratio is
It is not the complier contrast of 4 points: the extra \(0.20\) points in the reduced form comes from always-takers’ version shift. If, in the same hypothetical setting, the instrument also raised never-takers’ mean outcome by 1 point, then \(\Delta_N=1\). The reduced form would instead be \(0.50(1)+0.20(1)+0.30(4)=1.90\) points, and the Wald ratio would be \(1.90/0.30=6.333\). This additional term shows why a decomposition that omits \(\Delta_N\) requires the explicit assumption \(\Delta_N=0\), as well as any other restrictions needed for its remaining terms.
Assessing the Assumption in Practice
Exclusion is a substantive design claim; observed data alone generally cannot establish that no relevant direct effect or version shift exists. A useful assessment asks what the instrument changes in the treatment process, not just whether it changes the binary uptake indicator.
State whether treatment means any receipt, or whether it includes timing, dose, quality, provider, or delivery mode. Explain why the chosen features are or are not relevant to the outcome.
Ask whether the instrument changes access, timing, follow-up, provider contact, or other care even for people whose binary treatment status does not change. Such changes may be direct effects or shifts in treatment versions, depending on the defined treatment state.
Compare protocols, treatment logs, dose and timing records, and implementation procedures across instrument groups. These can reveal plausible version shifts, though they cannot prove that outcome-relevant differences are absent.
If versions differ and may matter, narrow the causal interpretation, redefine treatment to include the relevant versions, or use a design that standardizes delivery. Do not report a conventional binary-treatment LATE as if those differences had been ruled out.
A tempting but unreliable diagnostic is to compare outcomes between instrument groups only among observed recipients. Receipt is affected by the instrument, so selecting on \(D=1\) can change which people are being compared. Such a comparison does not, by itself, identify the version effect among always-takers or establish exclusion. Design knowledge and treatment-process measurements are useful for diagnosing plausibility, but they do not replace the assumptions required for causal interpretation.
Alternatives and Limitations
| Approach | What it clarifies | Main limitation |
|---|---|---|
| Keep a binary treatment and defend version irrelevance | Preserves a simple uptake definition when versions do not differ or do not affect the outcome. | Requires a strong substantive argument; the binary code alone does not establish it. |
| Define treatment as a richer package | Makes timing, dose, or mode part of the treatment state and distinguishes instrument effects on delivery from effects holding that state fixed. | May create multiple treatment categories and require enough data and a suitable identification strategy for them. |
| Standardize treatment delivery | Reduces the opportunity for the instrument to change versions among treated people. | May be infeasible, and standardization does not rule out other direct effects of the instrument. |
| Report a qualified package-effect interpretation | Describes the treatment change actually induced by the instrument rather than claiming an effect of a uniform treatment. | Does not isolate the effect of each version without additional assumptions or design information. |
No statistical adjustment can make an ill-defined treatment state well-defined by itself. A richer definition improves conceptual clarity, but the resulting causal effect may concern a particular policy of versions—such as the timing pattern induced by the instrument—rather than a universal effect of “treatment.” The key limitation is interpretive: evidence that \(Z\) changes uptake does not reveal whether it also changes outcome-relevant treatment versions or affects outcomes through other routes.
Common Mistakes
- Calling every version pathway a direct effect. If the instrument changes timing and timing is part of the treatment state, the outcome pathway runs through treatment. It may still violate exclusion for a coarse binary treatment.
- Assuming stable uptake means stable treatment. Always-takers can remain treated under either instrument value while receiving different versions, so their reduced-form contrast need not be zero.
- Using only the complier term in the reduced form. The decomposition includes contributions from all principal strata. Terms for noncompliers can be dropped only when their relevant outcome contrasts are assumed to be zero.
- Treating a larger treatment definition as a free solution. Including version in the treatment state improves the definition, but does not automatically identify the effects of multiple versions.
Check Your Understanding
For each question, identify the assumption or interpretation at issue before checking the solution.
- An instrument changes only whether a person receives a therapy, and all treated people receive the same protocol. What version-related concern remains, and what must still be assumed about exclusion?
- Why can an instrument-induced timing change among always-takers affect the reduced form even when their binary treatment status is unchanged?
- Suppose \(P(N)=0.40\), \(P(A)=0.10\), and \(P(C)=0.50\), with \(\Delta_N=0\), \(\Delta_A=2\) points, and \(\Delta_C=3\) points. Calculate the reduced form and Wald ratio.
- In the same setting, suppose instead \(\Delta_N=0.5\) points. Recalculate the reduced form and Wald ratio. What does this show?
- Why does comparing outcomes by instrument among observed treated individuals not, by itself, establish that treatment versions violate exclusion?
Solutions. (1) No version-related concern is apparent if the protocol truly is the same, but exclusion still requires no instrument effect on outcomes while that complete treatment state is fixed. (2) Their version can change despite \(D=1\) under both instrument values; if the version affects outcomes, their within-stratum contrast \(\Delta_A\) contributes \(P(A)\Delta_A\) to the reduced form. (3) The reduced form is \(0.40(0)+0.10(2)+0.50(3)=1.70\) points. The first stage is \(P(C)=0.50\), so the Wald ratio is \(1.70/0.50=3.40\) outcome points per unit increase in treatment probability. (4) The reduced form becomes \(0.40(0.5)+0.10(2)+0.50(3)=1.90\) points, and the Wald ratio is \(1.90/0.50=3.80\). The change shows that a nonzero contrast among never-takers adds to the numerator; monotonicity and instrument independence do not remove it. (5) Treatment receipt is affected by the instrument, so conditioning on observed receipt can change the composition of the compared groups. The comparison is therefore not, by itself, a causal estimate of a version effect or proof of an exclusion violation.