What Instrument Independence Says
An instrument is an assignment or encouragement that may change whether a person receives a treatment. The independence assumption concerns the instrument’s relationship to what each person would experience under the possible assignments and treatment states. It does not say that people assigned to different instrument values must receive the same treatment, or have the same observed outcomes. Indeed, an instrument is useful when assignment changes treatment receipt.
For a binary instrument \(Z_i\) and binary treatment \(D_i\), let \(D_i(z)\) denote person \(i\)’s treatment receipt if the instrument were set to \(z\), and let \(Y_i(z,d)\) denote the outcome if the instrument were set to \(z\) and treatment to \(d\), for \(z,d\in\{0,1\}\). The complete potential-response schedule records all these counterfactual values. Including \(Y_i(z,d)\) for both instrument values allows the schedule to represent direct as well as treatment-mediated effects of assignment; independence alone makes no claim that those effects are absent.
The independence claim is about assignment and counterfactuals, not about assignment and observed treatment or outcomes. Observed values are linked to assignment through the response schedule: \(D_i=D_i(Z_i)\) and \(Y_i=Y_i(Z_i,D_i(Z_i))\). Consequently, \(Z_i\) can be associated with observed \(D_i\), and with observed \(Y_i\), even when it is independent of \(V_i\). The schedule is not fully observed for any one person, so the assumption cannot generally be checked by directly comparing all potential outcomes across instrument groups.
Why Randomized Encouragement Can Justify Independence
In a randomized encouragement design, people are assigned to receive or not receive an offer, invitation, reminder, or other encouragement. The assignment \(Z_i\) is randomized; actual treatment receipt \(D_i\) is not necessarily randomized. A person can decline an offer, and a person assigned no offer may obtain the treatment another way. If the assignment mechanism is independent of each person’s pre-existing potential-response schedule, then it provides a design-based reason for instrument independence.
The justification is a property of the randomization procedure, not a claim that the realized groups will have exactly identical observed characteristics. Consider complete random assignment in a finite study population, with \(n_1\) of \(N\) people assigned \(Z=1\). Treat each person’s schedule \(V_i\) as fixed before assignment. For any measurable set \(B\) of schedules, the number of people whose schedules fall in \(B\) is fixed, and each person has the same assignment probability \(n_1/N\). In the randomization distribution,
When \(P(V_i\in B)>0\), dividing both sides by \(P(V_i\in B)\) gives
Because this holds for every measurable \(B\), assignment is independent of the schedule for that person under the randomization distribution. Complete randomization makes different people’s assignments dependent on one another because the total number assigned is fixed; that dependence does not invalidate this individual-level argument. With Bernoulli assignment, each assignment is generated by a random mechanism with a specified probability, and the same justification applies when that mechanism is independent of the potential-response schedules.
This is a probability statement about the design. In one realized experiment, chance imbalances can occur. Randomization does not guarantee that the treated-assignment group and comparison group will match exactly on every baseline feature, much less on every potential response.
Worked Example: Independence Across Potential-Response Types
Suppose a hypothetical population contains 100 people divided into four categories according to their full potential-response schedules. The categories might differ in potential treatment receipt and outcomes. An encouragement is assigned so that, within each category, each person has probability \(0.5\) of \(Z=1\). The table gives the population shares and the assignment probability for each category.
| Schedule category | Population share | \(P(Z=1\mid\text{category})\) | Joint probability of category and \(Z=1\) |
|---|---|---|---|
| A | 0.40 | 0.50 | \(0.40\times0.50=0.20\) |
| B | 0.30 | 0.50 | \(0.30\times0.50=0.15\) |
| C | 0.20 | 0.50 | \(0.20\times0.50=0.10\) |
| D | 0.10 | 0.50 | \(0.10\times0.50=0.05\) |
The overall probability of assignment to encouragement is $$ P(Z=1)=0.20+0.15+0.10+0.05=0.50. $$ For category A, for example, \(P(Z=1\mid A)=0.50=P(Z=1)\). The same equality holds for each category. Since these categories partition the potential-response schedules in this hypothetical example, assignment is independent of category. In an actual setting, potential schedules need not fall into a small number of observed categories; the formal definition therefore uses all measurable sets of schedules, not just a list of discrete types.
The example’s conclusion follows from the stated assignment mechanism, not from the fact that the table looks balanced. If assignment probabilities had differed by category, the instrument would not be independent of these schedules.
Assignment Is Not Treatment Receipt
A randomized encouragement can be independent of potential responses while actual treatment receipt is associated with them. Receipt depends on the assignment and on a person’s own response to that assignment. The following hypothetical numbers illustrate the distinction. Suppose 40% of people would take treatment whether encouraged or not, 40% would take it only if encouraged, and 20% would not take it under either assignment. These are response patterns for illustration; no additional assumption about which patterns exist in a real study is implied.
Worked Example: Randomized Encouragement With Incomplete Uptake
Let assignment be randomized with \(P(Z=1)=P(Z=0)=0.50\), independently of the three potential-receipt patterns. The treatment probability in the encouraged group is $$ P(D=1\mid Z=1)=0.40(1)+0.40(1)+0.20(0)=0.80. $$ In the group not assigned encouragement, it is $$ P(D=1\mid Z=0)=0.40(1)+0.40(0)+0.20(0)=0.40. $$ Thus encouragement changes the treatment-receipt rate by \(0.80-0.40=0.40\), or 40 percentage points.
This difference does not contradict instrument independence. Independence concerns \(Z\) and the potential receipt schedules before assignment; the randomized assignment has the same probability in every schedule category. The observed \(D\), by contrast, is \(D(Z)\), so its distribution differs by assignment. In particular, the people who actually receive treatment need not be comparable to those who do not: receipt reflects their responses to encouragement as well as the assignment. Treating receipt as if it were randomized would confuse the intervention being assigned with the behavior that follows it.
The same reasoning applies to observed outcomes. Instrument independence is not the statement \(Z\perp Y\) for observed \(Y\); assignment may change treatment receipt and thereby change the observed outcome. Nor does independence alone establish that assignment affects outcomes only through treatment. That is a separate assumption.
When Independence Is Not Plausible
If an instrument is not randomized, its independence from potential responses needs a substantive justification. For example, an encouragement might be offered more often to people judged more likely to benefit, or to people with greater barriers to treatment. In that case, assignment may reveal information about their potential treatment receipt or outcomes.
Worked Example: Assignment Related to a Potential Outcome
Imagine a hypothetical population with equal numbers of people at high and low baseline risk. Let the indicator \(V=1\) denote a high-risk potential-outcome profile and \(V=0\) a low-risk profile. Suppose an encouragement is offered to 80% of high-risk people but to 20% of low-risk people. Then $$ P(Z=1)=0.50(0.80)+0.50(0.20)=0.50. $$ Among those assigned encouragement, the high-risk proportion is $$ P(V=1\mid Z=1) =\frac{P(Z=1\mid V=1)P(V=1)}{P(Z=1)} =\frac{0.80(0.50)}{0.50}=0.80. $$ Among those not assigned encouragement, it is $$ P(V=1\mid Z=0) =\frac{P(Z=0\mid V=1)P(V=1)}{P(Z=0)} =\frac{0.20(0.50)}{0.50}=0.20. $$ Because the high-risk share differs between assignment groups, \(Z\) and this potential-outcome profile are not independent. The overall assignment probability of 0.50 does not fix the problem: independence requires the same assignment probability across schedule categories, not merely a particular marginal probability.
This example describes a possible failure mechanism, not a test that can be applied to all potential outcomes. In practice, a researcher may observe baseline risk factors and use them to assess whether the assignment process seems to have targeted particular kinds of people. Unobserved potential-response differences may remain.
Diagnostics, Limitations, and Comparisons
With a genuine randomized assignment, the design is the primary support for independence. Comparisons of pre-assignment characteristics across \(Z\) groups can reveal implementation problems or an unusual chance imbalance. They are diagnostics, not proofs: observed balance cannot establish independence from unobserved features or the full potential-response schedule. Conversely, a statistically significant difference on one baseline measure can arise by chance, particularly when many measures are examined. Interpret such checks in light of the randomization procedure and the study as a whole.
Several failures deserve attention. If staff override assignments based on prognosis, the delivered instrument may no longer follow the intended randomization. If inclusion in the analyzed sample or outcome observation depends jointly on assignment and a person’s potential responses, independence in the original randomized population need not carry over to the selected sample. Assignment should also be defined at the level and time at which it was randomized; confusing a randomized offer with later access, uptake, or adherence changes the assumption being assessed.
The key comparison is between design-based and observational justification. Random assignment provides a direct reason that \(Z_i\) is unrelated to the fixed potential-response schedule in the randomization distribution. For an observational instrument, independence is instead a substantive claim about how the instrument was generated and why it does not sort people by their potential responses. Neither a strong relationship between the instrument and treatment receipt nor a balanced set of measured covariates, by itself, establishes independence.
The assumption is also limited in what it establishes. It says assignment is unrelated to potential responses; it does not say assignment changes treatment receipt, and it does not rule out an effect of assignment on outcomes that is not mediated by treatment. Those are distinct questions. Keeping the assumptions separate prevents evidence for random assignment from being mistaken for evidence for every other instrumental-variable condition.
Check Your Understanding
Answer each question using the distinction between assignment and potential responses.
- Why should independence be stated for measurable sets of potential-response schedules rather than only for individual values?
- In the incomplete-uptake example, what is \(P(D=1\mid Z=1)\), and why does its difference from \(P(D=1\mid Z=0)\) not contradict instrument independence?
- If a study randomizes exactly half of its participants to encouragement, does that guarantee that the realized groups have identical baseline characteristics?
- In the high-risk assignment example, calculate the high-risk share among those not assigned encouragement and state what it implies.
- Does instrument independence imply that assignment has no direct effect on the outcome? Explain.
Solutions. (1) Potential outcomes can be continuous, so a single value may have probability zero; measurable sets give a non-vacuous statement of independence. (2) It is \(0.80\). Receipt is \(D(Z)\), so it can vary with assignment even when assignment is independent of the potential-receipt schedule. (3) No. Randomization makes the assignment independent of the schedules in the randomization distribution, but chance imbalances can occur in the realized groups. (4) The share is \(0.20(0.50)/0.50=0.20\), so the high-risk share differs by assignment and independence fails for this risk profile. (5) No. Independence concerns assignment’s relationship with potential responses; it does not rule out a direct effect of assignment on outcome.