From a Medical Offer to Potential Treatment Receipt
Suppose a clinic offers patients a new treatment program. Some patients offered the program take it up; some do not. Some patients not offered the program may obtain the treatment another way. The offer and the treatment are therefore different variables: the offer is the instrument, while actual receipt is the treatment. A patient’s response to the offer is described by asking what that patient would do under either offer condition.
Let \(Z_i=1\) mean patient \(i\) is offered the program and \(Z_i=0\) mean the patient is not offered it. Let \(D_i=1\) mean the patient receives the treatment and \(D_i=0\) mean the patient does not. Define \(D_i(1)\) as the treatment receipt that would occur for patient \(i\) if offered, and \(D_i(0)\) as the treatment receipt that would occur if not offered. Each patient has one pair of potential treatment receipts, \((D_i(0),D_i(1))\), although only one member of the pair is observed.
These four labels exhaust the possibilities because each of the two potential receipts is binary. A complier takes up treatment when offered but would not receive it without the offer. A defier does the reverse: they would receive treatment without the offer but not if offered. “Complier” is not a synonym for someone who accepted an offer in the observed data; it describes how the same patient’s receipt would respond across both offer conditions.
The basic interpretation presumes a clearly defined offer, a clearly defined treatment, and consistent assignment and receipt: the observed receipt is \(D_i=D_i(Z_i)\). It also presumes that one patient’s offer does not alter another patient’s treatment receipt. These conditions make the potential receipts meaningful for the study being described. They do not make both potential receipts observable for any one patient.
Mapping Types to What Is Observed
The observed data contain \(Z_i\) and \(D_i(Z_i)\), not both \(D_i(0)\) and \(D_i(1)\). The mapping below shows why an observed uptake pattern does not always determine a patient’s type.
| Observed offer and receipt | Potential receipt revealed | Compatible types without monotonicity |
|---|---|---|
| No offer, receives treatment: \(Z_i=0,D_i=1\) | \(D_i(0)=1\) | Always-taker or defier |
| No offer, does not receive treatment: \(Z_i=0,D_i=0\) | \(D_i(0)=0\) | Never-taker or complier |
| Offered, receives treatment: \(Z_i=1,D_i=1\) | \(D_i(1)=1\) | Always-taker or complier |
| Offered, does not receive treatment: \(Z_i=1,D_i=0\) | \(D_i(1)=0\) | Never-taker or defier |
For example, seeing a patient offered the program who did not receive it tells us \(D_i(1)=0\). It does not reveal whether the patient would also have declined without the offer, as a never-taker would, or would have received treatment without the offer, as a defier would. The missing potential receipt is precisely what prevents direct individual classification in this case.
Worked Example: Classifying Four Hypothetical Patients From Their Potential Receipts
For illustration, suppose a study could somehow list both potential receipts for four hypothetical patients. These pairs are stipulated for the example; a real study cannot observe both potential receipts for the same patient.
| Patient | \(D_i(0)\) | \(D_i(1)\) | Classification |
|---|---|---|---|
| A | 0 | 1 | Complier |
| B | 1 | 1 | Always-taker |
| C | 0 | 0 | Never-taker |
| D | 1 | 0 | Defier |
For patient A, the pair is \((0,1)\): no treatment without the offer, treatment with it. That is the definition of a complier. Patient B’s pair is \((1,1)\), so receipt does not change with the offer and the patient is an always-taker. Patient C’s \((0,0)\) pair identifies a never-taker. Patient D’s \((1,0)\) pair identifies a defier because receipt changes in the direction opposite to the offer.
The exercise illustrates the definition, not a practical way to label each enrolled patient. In an actual study, each patient reveals only the potential receipt corresponding to their assigned offer condition.
Monotonicity and the Types It Rules Out
A common assumption in an instrumental-variable analysis is monotonicity: no patient would receive treatment if not offered it and then refuse treatment if offered. In this notation, the condition is \(D_i(1)\geq D_i(0)\) for every patient. Because receipt is binary, this rules out the \((1,0)\) pair and hence rules out defiers. It does not rule out always-takers, never-takers, or compliers.
Under monotonicity, two observed patterns identify a type for the particular observed patient. If \(Z_i=0\) and \(D_i=1\), then \(D_i(0)=1\); monotonicity requires \(D_i(1)\geq1\), so \(D_i(1)=1\), making the patient an always-taker. If \(Z_i=1\) and \(D_i=0\), then \(D_i(1)=0\); monotonicity requires \(0\geq D_i(0)\), so \(D_i(0)=0\), making the patient a never-taker.
The other two patterns remain ambiguous even with monotonicity. A patient with \(Z_i=0,D_i=0\) may be a never-taker or a complier, because only \(D_i(0)\) is observed. A patient with \(Z_i=1,D_i=1\) may be an always-taker or a complier, because only \(D_i(1)\) is observed. Monotonicity narrows the possibilities; it does not reveal everyone’s type.
Worked Example: Reading Uptake Counts Under Monotonicity
Consider a hypothetical randomized study of 100 patients, with 50 assigned an offer of a medication-management program and 50 assigned no offer. The offer is not the treatment: \(D=1\) records actual receipt of the program. Suppose the observed counts are as follows.
| Assignment | Received treatment, \(D=1\) | Did not receive treatment, \(D=0\) | Total |
|---|---|---|---|
| Offered, \(Z=1\) | 34 | 16 | 50 |
| Not offered, \(Z=0\) | 8 | 42 | 50 |
Assume monotonicity for this interpretation. Among the 8 patients with \(Z=0,D=1\), each has \(D_i(0)=1\). Monotonicity then implies \(D_i(1)=1\), so those 8 observed patients are always-takers. Among the 16 patients with \(Z=1,D=0\), each has \(D_i(1)=0\). Monotonicity implies \(D_i(0)=0\), so those 16 observed patients are never-takers.
The 42 patients with \(Z=0,D=0\) could be never-takers or compliers: the observed data establish \(D_i(0)=0\), but not \(D_i(1)\). The 34 patients with \(Z=1,D=1\) could be always-takers or compliers: the observed data establish \(D_i(1)=1\), but not \(D_i(0)\). Thus the counts 8 and 16 classify observed patients under monotonicity; the other two cells are mixtures of compatible types.
Random assignment matters for interpreting group comparisons and for using the observed groups to learn about the study population. It does not let the researcher see both potential receipts for an individual. In particular, one should not report 34 as the number of compliers or subtract a count in one assignment group from a count in another as if that matched individual patients across conditions.
Why the Type Classification Matters
The classification separates an offer’s effect on treatment receipt from treatment receipt itself. A change in offer status can change receipt for compliers, but not for always-takers or never-takers. Defiers, if present, move in the opposite direction. This is why the assumptions about compliance types matter for interpreting an instrumental-variable design. The classification alone, however, does not calculate an outcome effect or establish that the offer is a valid instrument.
It is also important to distinguish a type from observed adherence. “Accepted the offer” describes a realized event. “Complier” describes a pair of potential responses. A patient who accepts an offer could be an always-taker or a complier; without the no-offer potential receipt, the observed acceptance alone cannot distinguish them. Similarly, a patient who refuses an offer could be a never-taker or a defier unless monotonicity is assumed.
Specify what \(Z_i=1\) and \(D_i=1\) mean in the medical setting; do not use “offered” and “treated” interchangeably.
List \(D_i(0)\) first and \(D_i(1)\) second, so the ordering of the type labels is unambiguous.
Use \((1,1)\) for always-taker, \((0,0)\) for never-taker, \((0,1)\) for complier, and \((1,0)\) for defier.
Use the assigned condition to identify the one observed potential receipt. Apply monotonicity only if it is justified, and do not claim it reveals the missing potential receipt for every patient.
Diagnostics, Alternatives, and Limitations
A useful diagnostic is to inspect whether the verbal account of the offer agrees with the potential-receipt coding. If an offer is described as encouraging treatment, but the coding makes \(Z=1\) mean “no offer,” conclusions about compliers and defiers will be reversed. Another warning sign is calling every patient who takes up treatment a complier. Check whether the statement concerns a realized receipt or the pair of potential receipts.
Monotonicity is substantive, not a consequence of random assignment and not something established merely by finding a higher average uptake rate among offered patients. A higher rate is compatible with some individuals responding in the opposite direction. The assumption rules out every defier, so it requires a defensible account of why offering the program would not deter treatment for any patient who would otherwise receive it. If the offer could discourage some patients—for example, by displacing another route to care—defiers may be possible.
Observed uptake tables are useful for describing receipt by offer condition, but they do not provide each patient’s compliance type. A treatment-received comparison groups patients by their realized treatment and is not the same classification exercise as grouping by potential responses. Likewise, an offer-based comparison preserves the distinction between assignment and receipt but does not identify each individual type. In practice, type-specific statements depend on the design and assumptions, not on directly observing both potential receipts.
The main limitation is fundamental: for each patient, only \(D_i(Z_i)\) is observed. No amount of relabeling the observed \(Z,D\) table supplies that patient’s missing potential receipt. Monotonicity can eliminate one of the four logical types and can identify some observed patients’ types, but it does not turn the unobserved counterfactual into data. Treating the resulting classifications as known without stating the assumption overstates what the study shows.
Check Your Understanding
Answer each question by distinguishing the observed receipt from the unobserved potential receipt. Solutions follow the questions.
- A patient would receive treatment whether or not offered. What is the patient’s type?
- A patient is offered treatment and does not receive it. Without monotonicity, what types are compatible with this observation?
- Under monotonicity, what type is identified when a patient is not offered treatment but receives it? Explain using the potential receipts.
- In a randomized medical-offer study, 40% of offered patients take up treatment. Can the 40% be interpreted as the proportion of compliers? Why or why not?
- What does monotonicity assert in terms of \(D_i(0)\) and \(D_i(1)\), and which type does it rule out?
Solutions. (1) Always-taker, with pair \((1,1)\). (2) \(Z_i=1,D_i=0\) reveals \(D_i(1)=0\); the patient may be a never-taker \((0,0)\) or a defier \((1,0)\). (3) The observation gives \(D_i(0)=1\). Monotonicity requires \(D_i(1)\geq D_i(0)=1\), so \(D_i(1)=1\); the patient is an always-taker. (4) No. The 40% is the observed uptake proportion among offered patients. It may include always-takers as well as compliers, and the uptake rate alone does not reveal each patient’s missing \(D_i(0)\). (5) Monotonicity asserts \(D_i(1)\geq D_i(0)\) for every patient. It rules out defiers, whose pair is \((1,0)\).