Why Put Potential Outcomes in a Table?
A causal study records which treatment each unit received and the outcome that followed. The potential-outcome framework adds a useful way to organize the causal question: for each unit, what outcome would occur under treatment, and what outcome would occur without treatment? A table makes those two possibilities visible side by side while also showing which one was observed.
The table is a bookkeeping device, not a way to recover information that the study did not observe. For any one unit, the observed treatment selects one potential outcome as observed; the other is that unit’s counterfactual. Keeping those entries separate prevents a common error: treating the unobserved outcome as if it were zero, or as if it could be read directly from the data.
A basic table has one row per unit and columns for \(T_i\), \(Y_i(1)\), \(Y_i(0)\), and \(Y_i\). The potential-outcome columns describe the two treatment states for the same unit. The treatment column records which state the unit actually received; the observed-outcome column records the outcome associated with that state. In real studies, only one of the two potential outcomes is observed for each unit.
For clarity, a table can mark an unobserved potential outcome with a question mark or “unobserved.” That mark means its value is not available in the observed data. It does not mean the value is zero, undefined, or random in the sense of being generated after treatment. The potential outcome is defined for the unit and treatment state, but only one state’s outcome is revealed.
Reading the Table
The selection rule can be written compactly as
When \(T_i=1\), the first coefficient is 1 and the second is 0, so \(Y_i=Y_i(1)\). When \(T_i=0\), the first coefficient is 0 and the second is 1, so \(Y_i=Y_i(0)\). This equation expresses consistency in the binary-treatment case. It does not say that the unobserved potential outcome equals the observed one.
In the table, the treatment value determines which potential-outcome entry must match the observed outcome. If the treated unit’s observed outcome does not match \(Y_i(1)\), or the untreated unit’s observed outcome does not match \(Y_i(0)\), then the table violates the stated consistency relationship. A mismatch could indicate a transcription error, a treatment definition that is not well specified, or that the table combines incompatible outcome definitions.
Worked Example: A Complete Hypothetical Table
Suppose a hypothetical teaching example stipulates both potential outcomes for four people, and records the treatment each person received. The outcome is a test score in points. The stipulated potential outcomes are shown to illustrate the full conceptual table; in an actual study, the unobserved entries would not be available.
| Unit | \(T_i\) | \(Y_i(1)\), treated | \(Y_i(0)\), untreated | Observed \(Y_i\) |
|---|---|---|---|---|
| A | 1 | 82 | 76 | 82 |
| B | 0 | 71 | 68 | 68 |
| C | 1 | 90 | 92 | 90 |
| D | 0 | 64 | 60 | 60 |
For A, \(T_A=1\), so the observed outcome must be \(Y_A(1)=82\); the score of 76 is the unobserved counterfactual for A. For B, \(T_B=0\), so \(Y_B=Y_B(0)=68\), and the score of 71 is unobserved. The same check gives \(Y_C=Y_C(1)=90\) and \(Y_D=Y_D(0)=60\).
The table also shows why potential outcomes are defined at the unit level: A’s two values belong to A, and B’s two values belong to B. It would be incorrect to treat A’s untreated outcome as B’s outcome merely because B was untreated. The entries describe different units.
Building a Table From Observed Records
Observed data usually provide treatment and outcome, not both potential outcomes. The appropriate table therefore has one known potential-outcome entry per row and one missing counterfactual entry. The question mark is a deliberate statement about what the data reveal, rather than a gap to fill using the observed outcome.
Worked Example: Marking the Unobserved Counterfactual
Consider a hypothetical program evaluation with an outcome measured as days absent from work during the following month. The observed records are: unit P received the program and had 2 days absent; unit Q did not receive it and had 5 days absent; unit R received it and had 1 day absent.
| Unit | \(T_i\) | \(Y_i(1)\) | \(Y_i(0)\) | Observed \(Y_i\) |
|---|---|---|---|---|
| P | 1 | 2 | Unobserved | 2 |
| Q | 0 | Unobserved | 5 | 5 |
| R | 1 | 1 | Unobserved | 1 |
For P, consistency puts the observed 2 days in the \(Y_P(1)\) column because \(T_P=1\). It gives no value for \(Y_P(0)\). For Q, \(T_Q=0\), so the observed 5 days go in \(Y_Q(0)\); \(Y_Q(1)\) remains unobserved. R is treated, so the observed 1 day fills \(Y_R(1)\), not \(Y_R(0)\).
The units on the outcome matter: the entries are days absent, not treatment effects or probabilities. If the outcome had instead been a binary indicator of any absence, the entries would be coded as 0 or 1, but the same placement rule would apply.
The placement rule can be checked systematically. First write down the unit identifier and the treatment coding. Next, enter the observed outcome in the potential-outcome column corresponding to the received treatment. Finally, leave the other potential-outcome entry marked as unobserved unless independent information genuinely supplies it. This order reduces the chance of reversing the treatment labels.
State which intervention is coded 1 and which comparison condition is coded 0.
Use the same unit, outcome definition, and measurement period across the columns.
If \(T_i=1\), enter \(Y_i\) as \(Y_i(1)\); if \(T_i=0\), enter it as \(Y_i(0)\).
Do not infer it merely by copying the observed outcome or entering zero.
Checking the Coding and the Assumptions
The table relies on a coherent binary treatment: every unit is assigned to one of two clearly described states, and \(Y_i(1)\) and \(Y_i(0)\) refer to the same outcome for the same unit under those states. The consistency relationship also presumes that the treatment and outcome are defined in a way that gives the potential outcomes their intended meaning. The earlier material on SUTVA and treatment versions gives the relevant framework: this table uses a well-defined treatment and the no-interference setup, so a unit’s potential outcome is indexed by its own treatment state.
These conditions are not verified simply because a table can be drawn. If “treatment” combines meaningfully different versions, or if one unit’s outcome depends on other units’ treatment, the two-column notation may fail to represent the causal setup adequately. The table should not conceal such design problems.
Worked Example: Detecting a Coding Error
In a hypothetical study, a researcher codes \(T_i=1\) for untreated participants and \(T_i=0\) for treated participants, but labels the potential-outcome columns \(Y_i(1)\) “treated” and \(Y_i(0)\) “untreated.” One participant is recorded as treated with an observed outcome of 7 units. The draft table places 7 in \(Y_i(0)\).
The entries conflict: according to the treatment column and stated labels, this participant received treatment, so consistency requires \(Y_i=Y_i(1)\). The value 7 belongs in the treated potential-outcome column, not \(Y_i(0)\). The table should be corrected by making the coding convention consistent throughout: either recode treatment so 1 means treated, or retain the reversed coding and relabel the potential outcomes and selection rule to match.
This check is useful because an internally inconsistent table can make later calculations meaningless. Always verify the mapping between treatment codes, column labels, and observed outcomes before interpreting the entries.
What the Table Does—and Does Not—Show
A complete conceptual table displays both potential outcomes for every unit. An observed-data table displays only the one potential outcome revealed by treatment for each unit. This distinction is central: the complete table is a way to define causal comparisons, while the observed records alone generally leave some entries unknown. As introduced in the earlier material on the missing counterfactual problem, each unit’s two potential outcomes cannot both be observed under the same realized treatment assignment.
The table is not a method for determining the unobserved entries. Nor does a table containing treated and untreated units, by itself, show that their observed outcomes can be interpreted causally. It organizes the information and makes the missing comparisons explicit; additional assumptions and study-design information are needed for causal identification. Those questions are distinct from correctly constructing this table.
A useful alternative is an observed-data table containing only unit, treatment, and outcome. It may be simpler for reporting raw records, but it does not display the counterfactual question as clearly. The potential-outcome table adds the two treatment-state columns and makes clear which entry is observed and which is not. Neither format can recover the missing outcome without further information or assumptions.
Common diagnostic signs of a table problem include an observed outcome placed under the wrong treatment state, treatment labels that switch meaning between rows, units that do not share the same outcome definition, and an unobserved entry silently replaced by zero. Check the unit identifiers as well: one row should represent one unit, and both potential outcomes in that row must refer to that same unit.
Check Your Understanding
Answer each question using the treatment coding in the definition: 1 means treated and 0 means untreated.
- A unit has \(T_i=0\) and observed outcome \(Y_i=12\). Which potential-outcome entry equals 12, and what is the status of the other entry?
- A table shows \(T_i=1\), \(Y_i(1)=4\), \(Y_i(0)=9\), and observed \(Y_i=9\). Is the table consistent with the stated coding?
- Why should an unobserved counterfactual not be entered as zero?
- What does a complete conceptual potential-outcome table contain that a typical observed-data table does not?
Solutions. (1) \(Y_i(0)=12\); \(Y_i(1)\) is unobserved for this unit. (2) No. Since \(T_i=1\), consistency requires \(Y_i=Y_i(1)=4\), not 9. (3) Zero is a particular outcome value, whereas “unobserved” means the data do not reveal that potential outcome; replacing one with the other changes the information represented. (4) It contains both \(Y_i(1)\) and \(Y_i(0)\) for every unit, including the potential outcome not observed under the unit’s actual treatment.