Two Sets of Potential Values
In an instrumental-variable setting, the instrument and the treatment are different variables with different roles. The instrument may shift whether a unit receives treatment; treatment may then affect the outcome. To represent those stages without confusing them, define one set of potential values for treatment receipt and another for outcomes.
Let \(i\) index units, such as people. The instrument \(Z_i\) is binary, with values 0 and 1. Treatment receipt \(D_i\) is also binary: \(D_i=1\) means the unit receives treatment, and \(D_i=0\) means it does not. The outcome is \(Y_i\), which may be binary or quantitative. A potential value describes what would happen to unit \(i\) under a specified intervention, whether or not that intervention is the one the unit actually experiences.
The instrument contrast \(D_i(1)-D_i(0)\) compares possible treatment receipt under the two instrument values. The treatment contrast \(Y_i(1)-Y_i(0)\) compares possible outcomes under the two treatment values. These are not the same contrast: one concerns how instrument assignment changes treatment receipt, while the other concerns how treatment changes the outcome.
Worked Example: Naming the Potential Values
Suppose a hypothetical clinic sends a reminder, coded \(Z_i=1\), or does not send one, coded \(Z_i=0\). The treatment is attending a follow-up appointment, and the outcome is a health score one month later. For one hypothetical patient, assume the potential values are \(D_i(0)=0\), \(D_i(1)=1\), \(Y_i(0)=62\), and \(Y_i(1)=70\).
The values \(D_i(0)=0\) and \(D_i(1)=1\) say that this patient would not attend without the reminder and would attend with it. Their difference is \(D_i(1)-D_i(0)=1-0=1\), a change in treatment receipt, not a change in the health score.
The values \(Y_i(0)=62\) and \(Y_i(1)=70\) are the potential health scores under no appointment and an appointment, respectively. The treatment contrast is \(Y_i(1)-Y_i(0)=70-62=8\) score points. It is distinct from the instrument contrast. These stipulated values describe a hypothetical unit; in an actual study, both instrument conditions and both treatment conditions cannot generally be observed for that same unit.
What the Notation Assumes—and What It Does Not
Potential values are defined relative to clearly specified interventions. “Set the instrument to 1” must refer to a meaningful version of the instrument, and “set treatment to 1” must refer to a sufficiently well-defined treatment. If different versions of treatment could produce different outcomes, the treatment needs a more detailed definition before \(Y_i(d)\) has an unambiguous meaning. The usual potential-outcomes setup also assumes that one unit’s assigned instrument or treatment does not alter another unit’s potential values; this is the no-interference component of the stable unit treatment value assumption.
These definitions alone do not assert that the instrument is randomly assigned, that it changes treatment receipt, that it has no direct effect on the outcome, or that units respond to it in a particular direction. Those are additional assumptions or empirical properties, not consequences of writing \(D_i(z)\) and \(Y_i(d)\). In particular, defining two potential outcomes does not make their difference observable or identify its average from observed data.
There is an important qualification to the short notation \(Y_i(d)\). It represents outcomes as depending on treatment but not separately on the instrument. That is an exclusion restriction. Without that restriction, the more general potential outcome is \(Y_i(z,d)\): the outcome unit \(i\) would have if both the instrument were set to \(z\) and treatment to \(d\). The short form is appropriate only when changing the instrument while holding treatment fixed does not change the outcome.
Worked Example: Separating Instrument and Treatment Contrasts
Consider two hypothetical people and suppose their potential values are as follows. Health scores are measured in points.
| Person | \(D_i(0)\) | \(D_i(1)\) | \(Y_i(0)\) | \(Y_i(1)\) |
|---|---|---|---|---|
| A | 0 | 1 | 50 | 58 |
| B | 1 | 1 | 65 | 70 |
For person A, the instrument contrast in treatment receipt is \(D_A(1)-D_A(0)=1-0=1\). Their treatment contrast in the outcome is \(Y_A(1)-Y_A(0)=58-50=8\) points. For person B, the corresponding contrasts are \(D_B(1)-D_B(0)=1-1=0\) and \(Y_B(1)-Y_B(0)=70-65=5\) points.
The average instrument contrast in treatment receipt across these two hypothetical people is \(\{1+0\}/2=0.5\). The average treatment contrast in outcomes is \(\{8+5\}/2=6.5\) points. These averages summarize different potential-value comparisons. Neither calculation, by itself, establishes what would be observed in a population or identifies a causal effect from an actual study.
From Potential Values to Observed Values
For a unit with observed instrument value \(Z_i=z\), consistency for treatment receipt states that the observed treatment is the potential treatment receipt under that value: \(D_i=D_i(z)\). If outcomes are represented by \(Y_i(d)\), outcome consistency says that the observed outcome is the potential outcome under the treatment actually received: \(Y_i=Y_i(D_i)\). Combining these statements gives the mapping below. Its final form uses the exclusion-based notation \(Y_i(d)\).
This mapping explains why observed data reveal only a limited part of a unit’s potential-value schedule. A unit is observed at its actual instrument value, so the other \(D_i(z)\) is not observed. The outcome is observed at the treatment actually received, so the outcome under the other treatment value is also not observed. Even with a large sample, the missing potential values are not supplied simply by collecting more observed outcomes; connecting observed comparisons to causal contrasts requires further assumptions.
Worked Example: Applying the Consistency Mapping
For a hypothetical participant, suppose \(Z_i=1\), \(D_i(0)=0\), \(D_i(1)=1\), \(Y_i(0)=40\), and \(Y_i(1)=47\). Assume the treatment and outcome are consistently defined, and use the exclusion-based outcome notation.
Because the observed instrument is \(Z_i=1\), treatment consistency gives \(D_i=D_i(1)=1\). The observed treatment is therefore receipt. Outcome consistency then gives \(Y_i=Y_i(1)=47\). The values \(D_i(0)=0\) and \(Y_i(0)=40\) remain counterfactual for this participant in this observation; they are not additional measurements.
If instead the same participant had been observed under \(Z_i=0\), consistency would give \(D_i=D_i(0)=0\) and, under the same exclusion-based representation, \(Y_i=Y_i(0)=40\). The arithmetic illustrates the mapping, not a claim that both scenarios can be observed for one person.
When the Instrument May Affect the Outcome Directly
If the instrument can affect the outcome through a route other than treatment, potential outcomes indexed only by \(d\) omit a relevant cause. For example, a reminder might also provide health information that changes a score even for someone who does not attend the appointment. In that case, write \(Y_i(z,d)\), and the exclusion restriction is the equality \(Y_i(1,d)=Y_i(0,d)\) for each treatment value \(d\). The notation \(Y_i(d)\) is then justified by setting \(Y_i(d)=Y_i(0,d)=Y_i(1,d)\).
Worked Example: Checking the Outcome Index
Suppose a hypothetical participant has the following potential scores under each combination of instrument and treatment:
| \(d=0\) | \(d=1\) | |
|---|---|---|
| \(z=0\) | \(Y_i(0,0)=50\) | \(Y_i(0,1)=60\) |
| \(z=1\) | \(Y_i(1,0)=53\) | \(Y_i(1,1)=63\) |
Holding treatment at \(d=0\), the instrument contrast is \(Y_i(1,0)-Y_i(0,0)=53-50=3\) points. Holding treatment at \(d=1\), it is \(Y_i(1,1)-Y_i(0,1)=63-60=3\) points. Since these contrasts are nonzero, the table does not satisfy exclusion: the instrument changes the outcome by 3 points at either treatment value.
The treatment contrast is also distinct. At \(z=0\), it is \(Y_i(0,1)-Y_i(0,0)=60-50=10\) points; at \(z=1\), it is \(Y_i(1,1)-Y_i(1,0)=63-53=10\) points. Treating these four values as only two values \(Y_i(0)\) and \(Y_i(1)\) would erase the instrument’s direct contribution. Thus the reduced notation would be inappropriate for this example.
Diagnostics, Alternatives, and Limitations
A useful first diagnostic is an index check: every potential value should answer one clear “what if?” question. \(D_i(z)\) asks what treatment receipt would be under an instrument value; \(Y_i(d)\) asks what outcome would be under a treatment value. If a proposed quantity uses an instrument value to describe the outcome, or a treatment value to describe instrument assignment, check whether the intended object is instead \(Y_i(z,d)\) or \(D_i(z)\).
A second check is to distinguish definitional consistency from empirical evidence. A table that lists every potential value for a few hypothetical units is useful for reasoning, but real data do not ordinarily reveal those full tables. Observed treatment rates across instrument groups can describe an instrument-treatment association; they do not reveal each person’s pair \(D_i(0),D_i(1)\). Likewise, observed outcomes do not reveal each person’s pair \(Y_i(0),Y_i(1)\). Conclusions about causal effects require assumptions beyond these definitions.
Potential outcomes are a bookkeeping framework, not a method that automatically solves confounding or identifies effects. Other approaches may target different quantities or use different assumptions; for example, randomized treatment assignment and propensity score methods address treatment comparisons in their own settings. An instrumental-variable analysis instead uses variation in the instrument, and its interpretation depends on additional IV assumptions. Those approaches are not interchangeable merely because each can be written using potential outcomes.
A common mistake is to say, “\(D_i(1)-D_i(0)\) is the treatment effect, because changing the instrument changed treatment.” This is wrong: that difference measures the effect of the instrument on treatment receipt. The outcome treatment effect is \(Y_i(1)-Y_i(0)\). A related mistake is to write \(Y_i(d)\) while allowing a direct effect of \(Z_i\); in that case use \(Y_i(z,d)\), or state and defend exclusion before using the shorter notation.
Check Your Understanding
Use the definitions above. The solutions are included after the questions so you can check each distinction.
- In a hypothetical study, \(D_i(0)=0\) and \(D_i(1)=1\). What does their difference describe?
- If \(Y_i(0)=12\) and \(Y_i(1)=17\), calculate the treatment contrast and state its units if the outcome is measured in points.
- Under exclusion-based notation, a unit has \(Z_i=0\), \(D_i(0)=1\), and outcomes \(Y_i(0)=8\), \(Y_i(1)=11\). What treatment and outcome are observed under consistency?
- Why is \(Y_i(d)\) insufficient if the instrument can directly change the outcome while treatment is held fixed?
- Does defining \(D_i(0)\) and \(D_i(1)\) imply that the instrument is randomized? Explain.
Solutions. (1) \(D_i(1)-D_i(0)=1-0=1\), the instrument contrast in treatment receipt for this unit. (2) \(Y_i(1)-Y_i(0)=17-12=5\) points, the treatment contrast in the potential outcome. (3) Treatment consistency gives \(D_i=D_i(0)=1\); outcome consistency gives \(Y_i=Y_i(1)=11\). (4) The outcome can depend on both assignments, so use \(Y_i(z,d)\); \(Y_i(d)\) suppresses the instrument index and corresponds to exclusion. (5) No. Potential treatment values can be defined regardless of how the instrument is assigned; randomization is an additional design feature or assumption.