From Potential Receipt to Population Shares
An individual’s compliance type records how their treatment receipt would respond to the two possible values of a binary instrument. This is a property of the pair of potential treatment receipts, not simply a label for the treatment the person actually received. Knowing the population distribution of those pairs therefore gives the population shares of all four types directly.
Let \(Z\in\{0,1\}\) denote the instrument and \(D\in\{0,1\}\) the treatment received. For each individual, \(D(0)\) is the treatment they would receive if \(Z=0\), and \(D(1)\) is the treatment they would receive if \(Z=1\). Assume these potential receipts are well-defined and binary for the population under study. A person’s type is determined by the ordered pair \((D(0),D(1))\).
The order of the pair matters: its first entry is receipt under \(Z=0\), and its second is receipt under \(Z=1\). A complier switches from no treatment to treatment as the instrument changes from 0 to 1; a defier switches in the opposite direction. The other two types do not change receipt.
Reading the Joint Distribution
Write the joint distribution with \(D(0)\) as the row variable and \(D(1)\) as the column variable. Each cell is the probability of the pair named by that row and column. The type shares are the four cell probabilities:
Here \(AT\), \(NT\), \(C\), and \(F\) label always-takers, never-takers, compliers, and defiers. The joint distribution is a probability distribution: each cell must be between 0 and 1, and the four cells must sum to 1. No independence, exclusion, or monotonicity assumption is required just to define these shares. Monotonicity is an additional restriction on which cells may have positive probability.
Worked Example: Calculating All Four Shares
Suppose a hypothetical population has the following joint distribution. Each entry is the proportion of the population with the indicated potential-receipt pair.
| \(D(0)\) | \(D(1)=0\) | \(D(1)=1\) | Row total |
|---|---|---|---|
| 0 | 0.42 | 0.28 | 0.70 |
| 1 | 0.12 | 0.18 | 0.30 |
| Column total | 0.54 | 0.46 | 1.00 |
The \((0,0)\) cell gives the never-taker share: \(\pi_{NT}=0.42\). The \((1,1)\) cell gives the always-taker share: \(\pi_{AT}=0.18\). The \((0,1)\) cell gives the complier share: \(\pi_C=0.28\). The \((1,0)\) cell gives the defier share: \(\pi_F=0.12\).
As a check, \(0.42+0.18+0.28+0.12=1.00\). Thus the four type shares account for the entire population. In percentage terms, the shares are 42% never-takers, 18% always-takers, 28% compliers, and 12% defiers.
Why the Joint Distribution Matters
The joint distribution contains more information than the two marginal distributions considered separately. A marginal distribution records the distribution of \(D(0)\) without specifying how its values pair with \(D(1)\), or vice versa. Because type depends on the pair, marginals generally cannot tell us how the population is divided among the four types.
For example, the marginal probability \(P(D(0)=1)\) is the sum of the two cells in the \(D(0)=1\) row, \(\pi_{AT}+\pi_F\). Similarly, \(P(D(1)=1)=\pi_{AT}+\pi_C\). These are two sums of four unknown cell probabilities. Along with the requirement that the four cells sum to 1, the information does not generally determine every cell. Different joint distributions can share the same marginals while implying different compliance-type shares.
Worked Example: Same Marginals, Different Type Shares
Consider two hypothetical joint distributions, each with rows indexed by \(D(0)\) and columns by \(D(1)\). In both populations, \(P(D(0)=1)=0.40\) and \(P(D(1)=1)=0.60\).
| Distribution A: \(D(0)\) | \(D(1)=0\) | \(D(1)=1\) |
|---|---|---|
| 0 | 0.30 | 0.30 |
| 1 | 0.10 | 0.30 |
In Distribution A, the row total for \(D(0)=1\) is \(0.10+0.30=0.40\), and the column total for \(D(1)=1\) is \(0.30+0.30=0.60\). The type shares are \(\pi_{NT}=0.30\), \(\pi_C=0.30\), \(\pi_F=0.10\), and \(\pi_{AT}=0.30\).
| Distribution B: \(D(0)\) | \(D(1)=0\) | \(D(1)=1\) |
|---|---|---|
| 0 | 0.40 | 0.20 |
| 1 | 0.00 | 0.40 |
In Distribution B, the corresponding marginals are again \(0.00+0.40=0.40\) for \(D(0)=1\), and \(0.20+0.40=0.60\) for \(D(1)=1\). But its type shares are \(\pi_{NT}=0.40\), \(\pi_C=0.20\), \(\pi_F=0.00\), and \(\pi_{AT}=0.40\). Both tables are valid probability distributions and have the same marginals, yet their type shares differ. The pairing of the potential receipts—not just their separate distributions—is what determines type.
Monotonicity as a Restriction on the Shares
Monotonicity says that the instrument does not make anyone less likely to receive treatment in the individual-level potential-receipt comparison: \(D(1)\geq D(0)\) for every individual. With binary potential receipts, this rules out the pair \((1,0)\). In terms of the joint distribution, it requires \(\pi_F=0\). The other three cells remain possible, and their probabilities still have to sum to 1.
This restriction changes the set of distributions under consideration; it does not change the definition of the four types. It also does not mean that the defier share is known to be zero merely because no defiers appear in an observed sample. Whether monotonicity is credible depends on the substantive setting and the meaning of the instrument and treatment.
Worked Example: Checking a Monotone Joint Distribution
Suppose a hypothetical population has \(\pi_{NT}=0.35\), \(\pi_C=0.45\), \(\pi_F=0\), and \(\pi_{AT}=0.20\). First check that these entries form a probability distribution:
The defier cell is zero, so every pair with positive probability satisfies \(D(1)\geq D(0)\): the possible pairs are \((0,0)\), \((0,1)\), and \((1,1)\). This joint distribution is consistent with monotonicity. If instead the \((1,0)\) cell were 0.05, monotonicity would fail for that distribution, because a positive fraction would have \(D(1)=0<D(0)=1\). The existence of that cell is a direct diagnostic of a contradiction between the proposed joint distribution and the monotonicity restriction.
Derivation, Interpretation, and Limits
The cell formulas follow by defining an indicator for membership in each type. For example, the indicator for being a complier is 1 exactly when \(D(0)=0\) and \(D(1)=1\), so it equals \((1-D(0))D(1)\). The other type indicators are obtained in the same way. Since \(D(0)\) and \(D(1)\) are binary, exactly one of the four pairings occurs for each individual. Therefore the four indicators sum to 1 for every individual. Taking population expectations gives the four cell probabilities and shows that their sum is 1.
Thus, if the joint distribution is specified, calculating population shares is direct: map each cell to its type and report its probability. But the joint distribution of an individual’s two potential receipts is not ordinarily observed directly. For each individual, only the receipt under the instrument value they actually encounter is observed. Consequently, a table of joint potential receipts should be understood as known only when the problem explicitly supplies it or when additional design or assumptions justify the information used to obtain it. The arithmetic of the shares does not, by itself, establish that the shares can be recovered from observed data.
A useful audit is to check the table’s orientation, cell validity, total, and any imposed restriction. Reversing the row and column meanings swaps the interpretation of the switching cells: \((0,1)\) is a complier only when the pair is ordered as \((D(0),D(1))\). A total other than 1 signals a missing population category or an invalid table. A negative cell or a cell greater than 1 cannot be a probability. A positive \((1,0)\) cell conflicts with monotonicity.
The main alternative to using the full joint distribution is to summarize each potential receipt separately. Such marginal summaries are often simpler, but they lose the pairing needed to assign all four shares. Imposing monotonicity is another approach: it rules out defiers, but it is an assumption about individual responses, not a consequence of the arithmetic or a guarantee that the remaining shares are known. These shares describe how a population is divided by potential receipt; they do not alone describe outcomes or establish a treatment effect.
Common Pitfalls
- Reversing the pair. Calling \((1,0)\) a complier confuses the response direction. With the convention \((D(0),D(1))\), compliers are \((0,1)\) and defiers are \((1,0)\).
- Using marginals as if they were the joint distribution. Marginals give sums of cells, not each cell separately. Different pairings can produce the same marginal probabilities.
- Treating monotonicity as observed fact. Monotonicity sets the defier share to zero only when it is assumed to hold for every individual in the target population.
- Calling a population share an observed individual label. The two potential receipts for one person are not both observed together. A population type is defined by those potential receipts, not by the single receipt recorded for that person.
Check Your Understanding
Answer each question, then compare with the solutions.
- A joint distribution has cell probabilities \(P(0,0)=0.25\), \(P(0,1)=0.50\), \(P(1,0)=0.05\), and \(P(1,1)=0.20\), where pairs are ordered as \((D(0),D(1))\). What are the four type shares?
- Does the distribution in question 1 satisfy monotonicity? Explain using the relevant cell.
- Why do the two marginal distributions of \(D(0)\) and \(D(1)\) generally fail to determine all four type shares?
- A proposed joint table has cell probabilities 0.30, 0.20, 0.40, and 0.15. Is it a valid population distribution? Explain.
Solutions. (1) The shares are \(\pi_{NT}=0.25\), \(\pi_C=0.50\), \(\pi_F=0.05\), and \(\pi_{AT}=0.20\); they sum to 1.00. (2) No. The defier share is \(\pi_F=0.05>0\), corresponding to \(D(0)=1,D(1)=0\), which violates \(D(1)\geq D(0)\). (3) Marginals report row and column sums, not how every \(D(0)\) value is paired with \(D(1)\); distinct joint tables can have identical marginals. (4) No. The cells sum to \(0.30+0.20+0.40+0.15=1.05\), so they cannot be probabilities for a complete population partition.