Why Compliance Types Matter
In an instrumental variable study, an instrument can change whether a person receives treatment without determining treatment receipt for everyone. For example, an encouragement may increase the likelihood that a person takes up a service, while some people would take it up either way and others would decline either way. To describe these different response patterns precisely, we compare each person’s potential treatment receipt under both possible instrument values.
The key distinction is between a person’s compliance type and the treatment receipt that is actually observed. The type is defined by two potential decisions, one for each instrument setting. In a study, only the decision under the assigned instrument value is observed for each person. The other decision is counterfactual, so observing actual receipt does not usually reveal the full type.
Here, \(Z_i=1\) means the instrument takes its designated “on” value, such as receiving an encouragement, and \(Z_i=0\) means its designated “off” value. \(D_i=1\) means the person receives the treatment; \(D_i=0\) means the person does not. These labels depend on the coding and definition of the instrument and treatment, so those must be stated before assigning types.
The Four Potential-Receipt Patterns
| Potential receipt under \(Z=0\) | Potential receipt under \(Z=1\) | Type | Interpretation |
|---|---|---|---|
| \(D_i(0)=1\) | \(D_i(1)=1\) | Always-taker | Receives treatment under either instrument value |
| \(D_i(0)=0\) | \(D_i(1)=0\) | Never-taker | Does not receive treatment under either instrument value |
| \(D_i(0)=0\) | \(D_i(1)=1\) | Complier | Receives treatment when the instrument is 1, but not when it is 0 |
| \(D_i(0)=1\) | \(D_i(1)=0\) | Defier | Receives treatment when the instrument is 0, but not when it is 1 |
These four possibilities are exhaustive. Each of \(D_i(0)\) and \(D_i(1)\) is binary, so the first entry has two possible values and the second entry has two possible values. Their Cartesian product contains \(2\times2=4\) ordered pairs:
The names assign one label to each pair: \((0,0)\) is never-taker, \((0,1)\) is complier, \((1,0)\) is defier, and \((1,1)\) is always-taker. The labels describe how treatment receipt would respond to the instrument; they do not describe a person’s outcome or the size of that person’s treatment effect.
Worked Example: Classifying Complete Potential-Receipt Profiles
Suppose a hypothetical encouragement program is coded \(Z=1\) when a person receives an invitation and \(Z=0\) otherwise. A researcher has, for illustration, a complete potential-receipt profile for four hypothetical people.
| Person | \(D_i(0)\) | \(D_i(1)\) |
|---|---|---|
| A | 0 | 1 |
| B | 1 | 1 |
| C | 0 | 0 |
| D | 1 | 0 |
For A, the pair is \((0,1)\): A would not receive the treatment without an invitation and would receive it with one. A is a complier. For B, the pair is \((1,1)\), so B is an always-taker. For C, \((0,0)\) makes C a never-taker. For D, \((1,0)\) makes D a defier.
This example classifies types directly because both potential receipt values are supplied. In an ordinary study, the researcher does not observe both values for the same person. The complete profiles here are hypothetical information used to demonstrate the definition, not a claim that a study can normally record both counterfactual decisions.
What One Observed Decision Can Tell You
For a given person, only the potential receipt corresponding to the observed instrument value is observed. If \(Z_i=0\), then \(D_i=D_i(0)\); the value \(D_i(1)\) remains unobserved. If \(Z_i=1\), then \(D_i=D_i(1)\); \(D_i(0)\) remains unobserved. Consequently, observed assignment and receipt can narrow the set of compatible types, but ordinarily cannot select one type from that set.
| Observed instrument and receipt | Compatible types | Reason |
|---|---|---|
| \(Z_i=0,\ D_i=0\) | Never-taker or complier | Both types have \(D_i(0)=0\) |
| \(Z_i=0,\ D_i=1\) | Always-taker or defier | Both types have \(D_i(0)=1\) |
| \(Z_i=1,\ D_i=0\) | Never-taker or defier | Both types have \(D_i(1)=0\) |
| \(Z_i=1,\ D_i=1\) | Always-taker or complier | Both types have \(D_i(1)=1\) |
Worked Example: Finding the Types Compatible With an Observation
A hypothetical participant receives an invitation, so \(Z_i=1\), and does not receive the treatment, so \(D_i=0\). Which compliance types are compatible with this record?
Because \(Z_i=1\), consistency tells us that the observed receipt is \(D_i(1)\). Thus \(D_i(1)=0\), and the person’s type must be one of the pairs whose second entry is zero: \((0,0)\) or \((1,0)\).
The pair \((0,0)\) is the never-taker type, while \((1,0)\) is the defier type. The observation does not distinguish between them because it provides no value for \(D_i(0)\). Calling this person a never-taker simply because they did not take treatment after an invitation would therefore assert more than the data show.
Worked Example: Classifying a Hypothetical Roster With Partial Information
Consider four hypothetical participants in an encouragement study. For each one, the roster records the assigned encouragement and actual treatment receipt, but not the unobserved potential receipt.
| Person | Observed \(Z_i\) | Observed \(D_i\) |
|---|---|---|
| E | 0 | 1 |
| F | 1 | 1 |
| G | 0 | 0 |
| H | 1 | 0 |
For E, \(Z_i=0\) means the observed value is \(D_i(0)=1\). The first entry of the type pair is therefore 1, leaving \((1,1)\) and \((1,0)\): E could be an always-taker or a defier.
For F, \(Z_i=1\) reveals \(D_i(1)=1\). The second entry is 1, leaving \((1,1)\) and \((0,1)\): F could be an always-taker or a complier.
For G, \(Z_i=0\) reveals \(D_i(0)=0\). The compatible pairs are \((0,0)\) and \((0,1)\): G could be a never-taker or a complier. For H, \(Z_i=1\) reveals \(D_i(1)=0\), so H could be a never-taker or a defier. In all four cases, the table supports a set of possible types, not a unique type.
Definitions Do Not Require a Type Assumption
The four labels are definitions based on the potential-receipt pair. Defining all four does not assume that every type occurs in a particular population, that the instrument changes treatment receipt for any particular person, or that the instrument is assigned independently of other variables. Those are separate questions about the study and its assumptions.
In particular, one should not remove a type from the list merely because its pattern seems unlikely in a given setting. For example, a defier’s potential receipt goes from 1 under \(Z=0\) to 0 under \(Z=1\). Whether that response pattern is ruled out requires an additional substantive assumption; the definition alone does not rule it out. Here, it is enough to recognize the pattern and keep the labels distinct.
Common Misinterpretations and Diagnostic Checks
- Equating assignment with receipt. \(Z_i\) is the instrument value and \(D_i\) is treatment receipt. A person can have \(Z_i=1\) but \(D_i=0\), or \(Z_i=0\) but \(D_i=1\). The type definitions concern potential values of \(D_i\), not the assigned value of \(Z_i\).
- Assigning a type from observed receipt alone. Seeing \(D_i=1\) does not show that a person is an always-taker. The relevant interpretation also depends on the instrument value, and even then the unobserved potential receipt generally leaves multiple types compatible.
- Calling only positive responders compliers. A complier has the specific pair \((0,1)\), not merely a positive observed treatment value. The pair includes both the response under \(Z=0\) and the response under \(Z=1\).
- Confusing compliance type with treatment effect. Compliance type concerns treatment receipt under the instrument. It does not by itself say whether treatment improves or worsens an outcome.
- Using labels without specifying coding. If the meanings of \(Z=0\), \(Z=1\), \(D=0\), and \(D=1\) are unclear, the type labels can be reversed or misapplied. State the instrument and treatment coding first.
A practical diagnostic is to write the two-entry pair before writing the type name. If a proposed classification cannot be tied to both \(D_i(0)\) and \(D_i(1)\), it may be relying on an unobserved counterfactual as if it were known. Also check that each potential receipt is coded as 0 or 1 and that the instrument’s two settings are defined consistently for the entire study.
How This Classification Differs From Other Descriptions
Compliance types are not the same as groups formed by observed treatment status. “Treated” and “untreated” describe the realized value \(D_i\); a compliance type describes the pair \((D_i(0),D_i(1))\). Two people who both received treatment may have different types, and two people with the same type can have different observed receipts if they were assigned different instrument values.
The classification is also different from comparing outcomes by instrument assignment or by treatment received. Such comparisons summarize observed data; they do not, by themselves, reveal each person’s two potential receipt decisions. The main advantage of the type framework is precise language for describing how receipt could respond to the instrument. Its principal limitation at the individual level is that one of those two decisions is counterfactual and unobserved.
Check Your Understanding
For each question, use the ordered-pair definition. The solutions follow the questions.
- A hypothetical person would receive treatment under either instrument value. Write the potential-receipt pair and name the type.
- A participant has \(Z_i=0\) and \(D_i=0\). Which two types are compatible with this observation, and what potential receipt remains unknown?
- A participant has \(Z_i=1\) and \(D_i=1\). Is the participant necessarily a complier? Explain.
- What type corresponds to \((D_i(0),D_i(1))=(1,0)\)? State the person’s receipt under each instrument value.
- Why does a complete list of four compliance types not, by itself, imply that all four types occur in a study population?
Solutions. (1) The pair is \((1,1)\), so the person is an always-taker. (2) The observed value gives \(D_i(0)=0\); \(D_i(1)\) is unknown. The compatible types are never-taker \((0,0)\) and complier \((0,1)\). (3) No. The observation reveals only \(D_i(1)=1\), which is compatible with an always-taker \((1,1)\) or a complier \((0,1)\); \(D_i(0)\) is not observed. (4) This is a defier: the person receives treatment when \(Z_i=0\) and does not receive it when \(Z_i=1\). (5) The four labels enumerate the logically possible binary pairs. Whether any particular pair occurs in the population is an empirical or substantive matter, not a consequence of the definition.