Why Compliance Types Are Defined Counterfactually
A binary instrument has two levels, conventionally coded \(Z=0\) and \(Z=1\). For example, \(Z=1\) might denote receiving an offer of an appointment, while \(Z=0\) denotes not receiving that offer. The treatment \(D\), also coded 0 or 1, is what the unit actually receives. An offer can affect treatment uptake without determining it: some units may receive treatment regardless of the offer, while others may respond to the offer by changing their treatment status.
To describe that response, we compare each unit’s treatment status under both possible instrument levels. This is a comparison of potential statuses, not merely a label based on the treatment the unit happened to receive. The four resulting categories are called compliance types.
The ordering matters. A pair \((0,1)\) means the unit would not receive treatment at \(Z=0\) but would receive it at \(Z=1\). Reversing the entries would describe a different response. The instrument’s levels must also be defined in the study context; the labels \(0\) and \(1\) alone do not say what an offer or assignment represents.
The Four Possible Types
Because each potential treatment status is binary, there are exactly four possible ordered pairs. The standard names describe how treatment status changes, or does not change, between the two instrument levels.
- Always-taker: \((D_i(0),D_i(1))=(1,1)\). The unit receives treatment at either instrument level.
- Never-taker: \((D_i(0),D_i(1))=(0,0)\). The unit does not receive treatment at either instrument level.
- Complier: \((D_i(0),D_i(1))=(0,1)\). The unit receives treatment at \(Z=1\) but not at \(Z=0\).
- Defier: \((D_i(0),D_i(1))=(1,0)\). The unit receives treatment at \(Z=0\) but not at \(Z=1\).
The complete set of possible ordered pairs is
This is the set of possibilities for an individual unit, not a claim that every type must occur in a particular population. A population may contain all four types, only some of them, or just one type.
| Potential status at \(Z=0\) | Potential status at \(Z=1\) | Ordered pair | Type |
|---|---|---|---|
| 0 | 0 | (0, 0) | Never-taker |
| 0 | 1 | (0, 1) | Complier |
| 1 | 0 | (1, 0) | Defier |
| 1 | 1 | (1, 1) | Always-taker |
A complier responds to the instrument by switching from no treatment to treatment as \(Z\) changes from 0 to 1. A defier switches in the opposite direction. Always-takers and never-takers have treatment status that does not change across the two levels. These names describe potential treatment behavior; they do not assert that a unit’s behavior is known from observing its assigned instrument level alone.
Worked Classifications
Worked Example: Classifying Individual Potential-Status Pairs
Suppose a hypothetical clinic records the potential treatment statuses of four units for a binary appointment-offer instrument. For this example, \(Z=1\) means an offer is made, \(Z=0\) means no offer is made, and \(D=1\) means the unit receives the treatment. The following pairs are specified hypothetically; they are not presented as observed data from a real study.
| Unit | \(D_i(0)\) | \(D_i(1)\) | Classification |
|---|---|---|---|
| A | 0 | 1 | Complier |
| B | 1 | 1 | Always-taker |
| C | 0 | 0 | Never-taker |
| D | 1 | 0 | Defier |
For unit A, the ordered pair is \((0,1)\): no treatment without the offer and treatment with it, so A is a complier. For unit B, \((1,1)\) means treatment under either condition, so B is an always-taker. Unit C has pair \((0,0)\) and is a never-taker. Unit D has pair \((1,0)\), so D would receive treatment without the offer but not with it; D is a defier.
The classification follows from comparing both potential statuses for each unit. It does not follow from whether the unit received an offer in one particular instance.
Worked Example: Counting Types in a Hypothetical Roster
Consider a hypothetical finite roster of \(n=20\) units whose two potential statuses are fully specified for instructional purposes. Suppose 6 units have pair \((1,1)\), 5 have pair \((0,0)\), 7 have pair \((0,1)\), and 2 have pair \((1,0)\). Classify the roster and calculate each type’s proportion.
The counts correspond respectively to always-takers, never-takers, compliers, and defiers. First check that the counts account for every unit:
For each type, divide its count by the roster size. The always-taker proportion is \(6/20=0.30\), or 30%. The never-taker proportion is \(5/20=0.25\), or 25%. The complier proportion is \(7/20=0.35\), or 35%. The defier proportion is \(2/20=0.10\), or 10%.
As a check, the proportions sum to one:
These proportions summarize the stipulated potential-status pairs in this hypothetical roster. They do not imply that all four types must be present in any particular population, and the complete pairs in this exercise should not be mistaken for information generally available from observing each unit only once.
What a Single Observation Can Tell You
In a study, a unit ordinarily experiences only one instrument level. Under consistency, its observed treatment status is the potential treatment status corresponding to its realized instrument value: \(D_i=D_i(Z_i)\). Thus, if \(Z_i=1\), the observed treatment reveals \(D_i(1)\), but not also \(D_i(0)\). If \(Z_i=0\), it reveals \(D_i(0)\), but not also \(D_i(1)\). The missing potential status is needed to determine the unit’s type.
This gives a practical way to see why an observed treatment status alone does not identify compliance type. For instance, observing treatment when \(Z=1\) is compatible with both an always-taker, whose pair is \((1,1)\), and a complier, whose pair is \((0,1)\). It does not distinguish those two possibilities.
Worked Example: Narrowing the Possible Types From an Observed Status
Suppose a hypothetical unit is assigned \(Z=0\), and the observed treatment status is \(D=0\). Use the observed information to determine which compliance types remain possible.
By consistency, \(D=D_i(0)\) for this unit. The observation therefore fixes the first entry of its pair at 0. The possible binary values for the unobserved second entry are 0 and 1, giving the candidate pairs
The pair \((0,0)\) is a never-taker type, and \((0,1)\) is a complier type. Therefore the observation rules out an always-taker and a defier, but it does not distinguish a never-taker from a complier. No additional classification is justified from this unit’s single observed instrument level and treatment status alone.
For comparison, if the same unit had \(Z=1\) and observed \(D=1\), then \(D_i(1)=1\); the compatible pairs would be \((0,1)\) and \((1,1)\), corresponding to a complier or an always-taker. The observed status still would not reveal the other potential status.
A Short Justification and a Common Pitfall
The four-type classification is exhaustive because each coordinate of the potential-status pair has two possibilities. There are two choices for \(D_i(0)\), and for each choice there are two choices for \(D_i(1)\). Hence there are \(2\times2=4\) ordered pairs. Assigning the standard name to each pair gives exactly the four categories listed above. Since the pairs differ in at least one coordinate, no unit can belong to two of these categories at once.
A common mistake is to classify units using only their observed behavior. For example, one might reason, “This person received treatment after an offer, so they are a complier.” That conclusion is not warranted: an always-taker would also receive treatment at \(Z=1\). The correction is to reserve the type label for the joint potential pair. An observed treatment status supplies only the coordinate corresponding to the instrument level that occurred; it does not, by itself, supply the counterfactual coordinate.
Another error is to assume from the names that an offer must increase treatment for every unit. The definition of a complier describes one type’s response, not a fact about all units. A defier is one of the four logically possible types unless an additional assumption rules that type out. Whether such an assumption is appropriate is a separate question; the classification itself does not impose it.
Check Your Understanding
For each question, keep the order of the pair fixed as \((D_i(0),D_i(1))\).
- A unit has potential statuses \(D_i(0)=1\) and \(D_i(1)=1\). What is its compliance type, and what does the pair mean in context?
- Why are there exactly four possible compliance types when both the instrument and treatment are binary?
- A unit is observed at \(Z=1\) with \(D=0\). Which compliance types are compatible with this observation under consistency?
- In a hypothetical roster of 40 units, 12 are compliers. What proportion of the roster are compliers?
- Why does observing \(D=1\) for a unit assigned \(Z=1\) not establish that the unit is a complier?