Why a Link Between Observed and Potential Outcomes Is Needed
Potential outcomes describe what an outcome would be under specified treatments. For a unit \(i\), let \(Y_i(1)\) be the outcome under treatment and \(Y_i(0)\) the outcome under control. The observed data, however, contain the unit’s actual treatment \(A_i\) and its observed outcome \(Y_i\). To connect these two descriptions, we need a rule: the observed outcome is the potential outcome corresponding to the treatment the unit actually received.
That rule is the consistency condition. It sounds simple, but it is essential. Without it, a treatment label and a set of potential outcomes would not tell us how to interpret the outcome recorded for a unit. Consistency is a link between variables and definitions; it is not a claim that treatment assignment is random or that the unobserved potential outcome can be recovered.
The notation \(Y_i(A_i)\) means “choose the potential outcome indexed by the treatment value \(A_i\).” If \(A_i=1\), the expression is \(Y_i(1)\); if \(A_i=0\), it is \(Y_i(0)\). Each unit has one observed outcome, but generally only one of its potential outcomes is observed. Consistency identifies which potential outcome corresponds to the observed one; it does not reveal the other.
Applying Consistency to Treatment-Specific Outcomes
Suppose a hypothetical study compares a defined program (\(A=1\)) with no program (\(A=0\)), and \(Y\) is a score measured one month after the treatment decision. If unit \(i\) receives the program, consistency says the recorded one-month score is \(Y_i(1)\). If the unit receives no program, the recorded score is \(Y_i(0)\). The outcome’s meaning and measurement time must be the same as those used to define the potential outcomes.
Consistency is stated unit by unit, not just as a statement about averages. In a group of treated units, it therefore implies that each observed outcome equals that unit’s treatment potential outcome. Averaging those equalities gives a group-level identity. This identity is useful for describing observed data, but it does not make the treated group’s average potential outcome equal to the average outcome that would have occurred under a different treatment.
Worked Example: Matching an Observed Outcome to Its Potential Outcome
Consider a hypothetical two-person study of a treatment for symptom severity. Scores range from 0 to 10, with lower scores indicating fewer symptoms. The potential outcomes below are shown for teaching purposes; in an actual study, both potential outcomes for the same person would not ordinarily be observed.
| Unit | \(Y_i(0)\): no treatment | \(Y_i(1)\): treatment | Actual treatment \(A_i\) | Observed outcome \(Y_i\) |
|---|---|---|---|---|
| 1 | 7 | 4 | 1 | 4 |
| 2 | 6 | 5 | 0 | 6 |
For unit 1, \(A_1=1\), so consistency requires \(Y_1=Y_1(1)\). Substitution gives \(4=4\), as required. The no-treatment potential outcome, \(Y_1(0)=7\), is not the observed outcome for this unit.
For unit 2, \(A_2=0\), so consistency requires \(Y_2=Y_2(0)\). Here \(6=6\). Its treatment potential outcome, \(Y_2(1)=5\), remains unobserved in the usual data structure. Thus the treatment label selects which potential outcome is linked to the observed score; it does not make both scores observable.
The same idea applies when treatment has more than two levels. If a treatment can be \(a=0\), \(a=1\), or \(a=2\), a unit receiving level 2 has \(Y_i=Y_i(2)\). The potential outcomes must be defined for the treatment levels being considered, and the observed treatment value must use those same labels.
Worked Example: Consistency With Three Treatment Levels
Suppose a hypothetical study records a blood-pressure change, in mmHg, after one of three defined dose levels: \(A=0\), \(A=1\), or \(A=2\). For one participant, let the potential outcomes be \(Y_i(0)=1\), \(Y_i(1)=-3\), and \(Y_i(2)=-5\). The participant receives level \(A_i=1\), and the observed change is \(Y_i=-3\).
Consistency selects the level-1 potential outcome: $$ Y_i=Y_i(A_i)=Y_i(1)=-3\text{ mmHg}. $$ The observed change is therefore consistent with the potential outcome under the received dose level. The values at levels 0 and 2 describe counterfactual outcomes for this participant, not additional observed changes. If the recorded change had instead been \(-4\) mmHg while these potential outcomes and the treatment definition were held fixed, the specified consistency relation would not hold for this record.
A Group-Level Consequence—and What It Does Not Establish
For binary treatment, let the treated group contain \(n_1\) units. If consistency holds for every treated unit, then \(Y_i=Y_i(1)\) for each unit with \(A_i=1\). Adding these equalities and dividing by \(n_1\) shows that the observed treated-group mean equals the mean of \(Y_i(1)\) among those same treated units. The corresponding identity holds for the control group.
Here \(n_1\) and \(n_0\) are the numbers of treated and control units, respectively. Each equality follows by substituting the unit-level consistency equality into the sum. The two means refer to different groups, and each involves the potential outcome corresponding to that group’s observed treatment.
Worked Example: Reading Observed Group Means Correctly
Consider four hypothetical units with the following complete potential-outcome values for illustration. In real data, only the outcome matching each unit’s actual treatment would be observed.
| Unit | \(Y_i(0)\) | \(Y_i(1)\) | \(A_i\) | Observed \(Y_i\) |
|---|---|---|---|---|
| 1 | 4 | 8 | 1 | 8 |
| 2 | 5 | 7 | 0 | 5 |
| 3 | 3 | 6 | 1 | 6 |
| 4 | 7 | 9 | 0 | 7 |
For the treated units, the observed mean is $$ \frac{8+6}{2}=7. $$ Their treatment-potential-outcome mean is also $$ \frac{Y_1(1)+Y_3(1)}{2}=\frac{8+6}{2}=7. $$ For the control units, the observed mean is $$ \frac{5+7}{2}=6, $$ which equals their mean \(Y_i(0)\), \((5+7)/2=6\). These are direct consequences of consistency.
The observed difference is \(7-6=1\) outcome unit. Consistency alone does not show that this difference is the average causal effect for all four units. That causal contrast would require comparing treatment and control potential outcomes over the same target group. The group means above instead concern the treatment potential outcome among treated units and the control potential outcome among control units.
Checking the Link and Recognizing Its Limits
Consistency is not usually testable in full from ordinary individual-level data: for each unit, one potential outcome is missing, so the equality with that unobserved value cannot generally be checked directly. Still, analysts can look for operational mismatches that make the intended equality implausible or the variables unclear. Before interpreting the condition, check that:
- The treatment variable represents the treatment in the potential-outcome definition. If \(A_i\) records assignment but the potential outcomes are defined by treatment received, the two variables do not automatically match.
- The outcome is defined consistently. A score at one follow-up time is not interchangeable with a score at another time, or with a change score, unless that is how the potential outcomes were defined.
- The data coding preserves the intended categories. A miscoded treatment value can select the wrong potential outcome in the notation.
These checks do not prove consistency. They help identify situations in which a treatment or outcome label is being used inconsistently. If the treatment is not clearly specified, the expression \(Y_i(a)\) itself may not refer to a sufficiently clear outcome. Detailed questions about treatment definitions and versions require additional assumptions beyond the basic link stated here.
The central limitation is that consistency addresses only the connection between an actual treatment and its corresponding outcome. It does not explain why a unit received that treatment, establish comparability between treatment groups, or reveal the missing potential outcome. Consequently, consistency by itself does not identify an average causal effect from observed data. Other assumptions and design information are needed for causal identification.
Common Mistake: Treating Consistency as a Causal Comparison
A common error is to reason, “The observed outcome for every treated unit equals that unit’s treated potential outcome, so the treated-group mean is the causal effect of treatment.” The first part is the consistency statement; the conclusion does not follow. A potential outcome mean is not itself an effect. An effect compares potential outcomes under different treatment levels for a specified group, while consistency links the observed outcome to only one of those levels for each unit.
A related mistake is to say that consistency means the unobserved outcome “would have been the same” as the observed outcome. It means the observed outcome equals the potential outcome under the treatment actually received. It makes no claim about the potential outcome under another treatment. Keeping these statements separate prevents an assumption about data interpretation from being mistaken for a solution to the missing-counterfactual problem.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A unit receives \(A_i=0\), with \(Y_i(0)=12\) and \(Y_i(1)=9\). What observed outcome does consistency specify, and which potential outcome remains unobserved?
- In a hypothetical sample, three treated units have observed outcomes 2, 4, and 6. Under consistency, what is the mean of \(Y_i(1)\) for these treated units?
- A study defines \(Y_i(1)\) as a score at one month, but records \(Y_i\) at three months. What should an analyst check before applying the consistency equality?
- Does consistency alone show that the observed mean difference between treated and control units equals the average causal effect? Explain briefly.
Solutions. (1) Since \(A_i=0\), consistency specifies \(Y_i=Y_i(0)=12\); \(Y_i(1)=9\) is the unobserved counterfactual. (2) The treated units’ observed outcomes equal their \(Y_i(1)\) values, so the mean is \((2+4+6)/3=4\) outcome units. (3) Check that the observed outcome and the potential outcome refer to the same measurement time and definition; as stated, they do not, so the link has not been specified consistently. (4) No. Consistency links each observed outcome to the potential outcome under the treatment received, but the treated and control means concern different groups. It does not alone supply the missing potential outcomes or establish the required causal comparison.