Why Compare Outcomes Within Compliance Types?
An instrument can change treatment receipt for some people but not others. This means its effect on the outcome is not the same as a treatment effect for every person. To see exactly where the instrument-induced outcome contrast comes from, consider the contrast separately within each compliance type. Always-takers receive treatment at either instrument assignment, and never-takers receive no treatment at either assignment. Compliers change treatment receipt with the instrument.
The key distinction is between changing the instrument and changing treatment. Under exclusion, the instrument affects the outcome only through treatment. But changing the instrument does not change treatment for always-takers or never-takers. If instrument assignment is also independent of the relevant potential outcomes and treatment responses, those two types have no instrument-induced outcome contrast. Compliers, by contrast, switch treatment, so their contrast reflects their average treatment effect.
The four possible compliance types are defined by their potential treatment receipt: always-takers have \((D(0),D(1))=(1,1)\); never-takers have \((0,0)\); compliers have \((0,1)\); and defiers have \((1,0)\). The monotonicity assumption used for the usual LATE interpretation rules out defiers. The main result below focuses on always-takers, never-takers, and compliers, while noting what the type-specific contrast would be for defiers.
The notation \(Y(d,z)\) allows the instrument to affect the outcome directly. Exclusion rules out that direct effect: for each \(d\), \(Y(d,1)=Y(d,0)\). When exclusion holds, write this common potential outcome as \(Y(d)\). Independence requires that \(Z\) be independent of the joint collection \((Y(0),Y(1),D(0),D(1))\). In an experiment this can follow from randomized instrument assignment; in an observational setting it is a substantive as-if-random assignment assumption. Consistency connects potential outcomes to observed data: the observed treatment is \(D=D(Z)\), and the observed outcome is \(Y=Y(D,Z)\).
Deriving the Contrast for Each Type
The derivation has two parts. First, consistency and exclusion express the observed outcome, within a compliance type, as the potential outcome under the treatment that would be received at the assigned instrument value. Second, independence allows the comparison between \(Z=1\) and \(Z=0\) to be made within a type without changing the distribution of that type’s potential outcomes.
The first equality uses consistency and exclusion. The second uses independence: the instrument is independent of potential treatment receipt and potential outcomes, and hence of the compliance type and the potential outcome indexed by \(D(z)\). Subtract the \(z=0\) expectation from the \(z=1\) expectation. The resulting contrast depends on how \(D(z)\) changes for type \(k\).
They receive treatment under both instrument values, so the contrast is \(E[Y(1)\mid C=\text{always}]-E[Y(1)\mid C=\text{always}]=0\).
They receive no treatment under either instrument value, so the contrast is \(E[Y(0)\mid C=\text{never}]-E[Y(0)\mid C=\text{never}]=0\).
They receive treatment when \(Z=1\) and no treatment when \(Z=0\), so the contrast is \(E[Y(1)\mid C=\text{complier}]-E[Y(0)\mid C=\text{complier}]=E[Y(1)-Y(0)\mid C=\text{complier}]\).
The zero contrasts for always-takers and never-takers do not say that treatment has no effect for those people. Their treatment effects can be nonzero; they simply do not change treatment receipt when the instrument changes. For example, an always-taker receives treatment under both instrument values, so the comparison is between outcomes under treatment in both cases. The contrast is zero because independence makes the underlying potential-outcome distribution comparable across the two assignments, not because \(Y(1)=Y(0)\).
Worked Examples
Worked Example: Contrasts and Contributions Across Three Types
Hypothetical population. Suppose a population consists of always-takers with share \(0.25\), never-takers with share \(0.35\), and compliers with share \(0.40\). Assume exclusion and independence. The following table gives the mean potential outcomes within each type, in outcome units.
| Type | Population share | \(E[Y(0)\mid C]\) | \(E[Y(1)\mid C]\) | Instrument-induced contrast |
|---|---|---|---|---|
| Always-takers | 0.25 | 4 | 10 | 0 |
| Never-takers | 0.35 | 2 | 5 | 0 |
| Compliers | 0.40 | 3 | 8 | 5 |
For always-takers, \(D(1)=D(0)=1\), so their outcomes under either assignment have mean \(E[Y(1)\mid C=\text{always}]=10\). Their contrast is \(10-10=0\). For never-takers, \(D(1)=D(0)=0\), so their outcomes have mean \(E[Y(0)\mid C=\text{never}]=2\) under either assignment. Their contrast is \(2-2=0\).
Compliers receive treatment at \(Z=1\) and no treatment at \(Z=0\). Their contrast is \(8-3=5\) outcome units, which is also their mean treatment effect. Weighting these type-specific contrasts by their population shares gives the overall instrument-induced outcome contrast:
The same answer follows by computing each instrument-assignment mean. At \(Z=1\), the type-weighted mean is \(0.25(10)+0.35(2)+0.40(8)=6.40\). At \(Z=0\), it is \(0.25(10)+0.35(2)+0.40(3)=4.40\). The difference is \(6.40-4.40=2.00\) outcome units. The nonzero overall contrast comes entirely from the compliers’ treatment change.
Worked Example: Why Exclusion Alone Is Not Enough
Hypothetical illustration. Consider two equally weighted always-takers. Both receive treatment regardless of \(Z\), and exclusion holds. Their potential outcomes under treatment are \(Y(1)=10\) and \(Y(1)=20\), respectively. Suppose instrument assignment is associated with which person is observed: the person with outcome 20 is assigned \(Z=1\), and the person with outcome 10 is assigned \(Z=0\).
The observed contrast within this type is \(20-10=10\) outcome units. Yet neither person changes treatment receipt: both are treated for either value of \(Z\). The nonzero contrast arises because instrument assignment is not independent of potential outcomes within the type. Thus exclusion by itself does not establish a zero contrast for always-takers or never-takers. Independence is also required.
If assignment were independent, the two always-takers would have the same expected distribution of \(Y(1)\) in the \(Z=1\) and \(Z=0\) groups. In this two-person example, that means the expected mean under either assignment is \((10+20)/2=15\), giving an expected contrast of \(15-15=0\). The example identifies the assumption at issue; it does not treat a particular finite random assignment as guaranteed to produce exactly equal observed means.
Worked Example: Heterogeneous Effects Among Compliers
Hypothetical population. Suppose two equally weighted complier subgroups have potential outcome pairs \((Y(0),Y(1))=(10,12)\) and \((20,28)\), measured in outcome units. Assume exclusion and independence. The first subgroup’s treatment effect is \(12-10=2\); the second’s is \(28-20=8\).
For the first subgroup, the instrument-induced outcome contrast is \(12-10=2\). For the second, it is \(28-20=8\). Averaging over the two equally weighted subgroups gives a complier contrast of \((2+8)/2=5\) outcome units. Equivalently, the mean outcome under \(Z=1\) among compliers is \((12+28)/2=20\), while under \(Z=0\) it is \((10+20)/2=15\), and \(20-15=5\).
The result is a mean treatment effect among compliers. It does not imply that every complier has an effect of 5, or that the effect is the same for always-takers and never-takers.
What the Decomposition Does—and Does Not—Establish
Because each unit belongs to one compliance type, the population outcome contrast can be written as the share-weighted sum of type-specific contrasts. Under monotonicity, there are no defiers, so the always-taker and never-taker terms are zero and the complier term is their population share multiplied by their mean treatment effect. This explains why the overall outcome contrast can be nonzero even though two of the three types contribute zero.
This decomposition is a statement about potential outcomes and latent types. It does not mean an analyst can usually calculate the contrast separately for each type from a standard observed data set. A person’s compliance type depends on both \(D(0)\) and \(D(1)\), but only one of those potential treatment receipts is observed for that person. The earlier material on the LATE identification result uses the decomposition to connect population contrasts to a complier effect; that identification argument does not make individual compliance types directly observed.
A type-specific contrast also differs from comparing observed treated and untreated people. Treatment receipt is affected by the instrument and may reflect underlying characteristics. The contrast here compares instrument assignments within the same latent type, under independence and exclusion. It should not be replaced by \(E[Y\mid D=1]-E[Y\mid D=0]\), which generally mixes types and need not have a causal interpretation.
The main checks are therefore conceptual. Ask whether exclusion is plausible: could the instrument affect the outcome through a pathway other than treatment? Ask whether independence is plausible: could assignment be associated with potential outcomes or potential treatment receipt? Ask whether monotonicity is appropriate for the three-type decomposition. If exclusion fails, a direct instrument effect can create a contrast even when treatment receipt is unchanged. If independence fails, different instrument groups can have different potential-outcome distributions within a type. If defiers exist, their term has the opposite treatment direction rather than being zero.
These assumptions are not verified merely because estimated contrasts look plausible. In particular, a zero estimated contrast among a group does not prove that exclusion and independence hold, and a nonzero contrast does not by itself identify which assumption failed. The value of the type-specific decomposition is that it makes the assumptions’ roles explicit: exclusion removes direct instrument effects, while independence makes the instrument groups comparable in their potential outcomes.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions below.
- Under exclusion and independence, an always-taker’s mean \(Y(1)\) is 14 outcome units. What is the type-specific instrument-induced contrast, and why?
- A never-taker’s mean \(Y(0)\) is 6 outcome units, and its mean treatment effect is 4 units. What is its instrument-induced contrast? Does the treatment effect change that answer?
- Among compliers, \(E[Y(1)]=11\) and \(E[Y(0)]=7\). State the type-specific contrast and its interpretation.
- In a hypothetical population, compliers make up 30% and have a mean treatment effect of 4 outcome units. Always-takers and never-takers have zero instrument-induced contrasts. What is the overall outcome contrast under monotonicity?
- An always-taker group has a contrast of 3 outcome units, although treatment receipt is 1 under both instrument values. Which assumption could fail, and why does exclusion alone not rule out the contrast?
Solutions. (1) The contrast is \(14-14=0\), because always-takers receive treatment under both assignments and the potential-outcome distribution is comparable by independence. (2) It is \(6-6=0\). The treatment effect can be 4 units without changing the instrument-induced contrast, since the instrument does not change this type’s treatment receipt. (3) It is \(11-7=4\) outcome units, the mean treatment effect among compliers. (4) The overall contrast is \(0.30(4)=1.20\) outcome units; the other types contribute zero. (5) Independence could fail: the instrument groups may differ in potential outcomes even within the always-taker type. Exclusion rules out a direct effect of \(Z\), but does not make instrument assignment independent of those outcomes.