The Gold Standard · How X Works
Random assignment identifies an offer before it identifies the effect of uptake
Some participants do not take the treatment they were assigned, while others access it outside their assigned group. Randomisation still protects the assignment comparison. Estimating the effect of treatment received needs additional assumptions, a precise target and careful reporting.
Choose a reading route
- Start from first principles — define the unit and the threat.
- Follow the worked example — make the hidden arithmetic visible.
- Test transfer — change the context and check the rule.
- Audit the evidence — inspect records, assumptions and failure points.
- Use the practical checklist — apply the method to new work.
Chapter contents
Foundations: Chapters 1–3
Method: Chapters 4–6
Transfer and limits: Chapters 7–9
Connections and use: Chapters 10–12
1. Assignment and receipt are different variables
Intention-to-treat compares outcomes by random assignment, regardless of treatment received. It estimates the effect of being assigned or offered the intervention under the trial’s delivery conditions. Noncompliance means treatment received differs from assignment.
A comparison of receivers versus non-receivers is usually confounded because uptake reflects motivation, access, prognosis or preference. Randomisation can sometimes serve as an instrumental variable for treatment received, but the resulting effect is not the same as the overall assignment effect.
First principle
State whether the target is the assignment effect, treatment-receipt effect or effect in a defined compliance subgroup.
2. Compliance types are counterfactual
Under a simple binary assignment and treatment setting, principal strata describe how a person would take treatment under either assignment: compliers follow assignment, always-takers receive treatment either way, never-takers do not receive it, and defiers do the opposite.
A person’s full type is not observed because only one assignment is seen. The “complier” group is therefore latent, not a subgroup that can be selected directly from observed adherers. This is why causal identification requires assumptions.
| Type | If assigned control | If assigned treatment |
|---|---|---|
| Complier | No treatment | Treatment |
| Always-taker | Treatment | Treatment |
| Never-taker | No treatment | No treatment |
| Defier | Treatment | No treatment |
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3. Instrumental-variable assumptions
Random assignment is a candidate instrument when it changes treatment receipt, is independent of potential outcomes through randomisation, and affects the outcome only through the treatment pathway relevant to the estimand. A common monotonicity assumption rules out defiers.
Exclusion restriction can fail when assignment itself changes behaviour, expectations, access or co-interventions even without receipt. The instrument also needs a meaningful first stage: if assignment barely changes uptake, the complier estimate becomes unstable.
- Relevance: assignment changes receipt.
- Independence: assignment is random as implemented.
- Exclusion: no direct assignment pathway under the model.
- Monotonicity: no units move opposite assignment.
- Stable treatment: receipt has a defined meaning.
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4. Worked example: an offered programme
A fictional trial assigns 100 people an offer and 100 to control access. Treatment uptake is 70% in the offer group and 10% in control, a first-stage difference of 0.60. Mean outcome is 62 under assignment to offer and 54 under control, so the intention-to-treat difference is 8 points.
The simple Wald estimate is 8 ÷ 0.60 = 13.3 points. Under the required assumptions, this estimates the average treatment effect among compliers—people whose receipt is changed by the offer.
| Quantity | Calculation | Value |
|---|---|---|
| Assignment effect | 62 − 54 | 8 |
| Receipt first stage | 0.70 − 0.10 | 0.60 |
| Wald complier effect | 8 ÷ 0.60 | 13.3 |
The 13.3-point estimate is not the effect for every participant and should not be described as such.
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5. Keep the estimands separate
The assignment effect answers a policy question about offering the programme. The complier effect answers a local question about those whose uptake responds to assignment. A per-protocol or as-treated contrast answers neither automatically.
Report both assignment and receipt patterns. Define what counts as treatment, timing, partial exposure and crossover. If dose varies continuously, a binary compliance story may be inadequate.
| Estimand | Comparison | Main meaning |
|---|---|---|
| Intention-to-treat | Assigned offer vs control | Effect of assignment policy |
| Complier effect | Latent assignment-responsive subgroup | Effect of receipt under assumptions |
| As-treated | Observed receivers vs non-receivers | Generally confounded |
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6. Diagnose weak and broken instruments
Show the first-stage difference with uncertainty. A weak instrument makes the ratio noisy and sensitive to small imbalances. Check whether assignment changed other services, contact or expectations. Document contamination routes and post-assignment eligibility changes.
Sensitivity analysis should challenge exclusion, monotonicity or missing-outcome assumptions rather than merely change software commands. When assumptions are doubtful, present bounds or a limited interpretation instead of a confident local effect.
Practical check
Write a causal pathway from assignment to receipt to outcome, then list every route that bypasses receipt.
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7. Transfer test: a tutoring invitation
Move the design to schools where students are randomly invited to optional tutoring. Some invited students do not attend; some controls obtain similar tutoring elsewhere. Assignment is a clear invitation effect, while attendance is self-selected.
An instrumental-variable analysis can target students whose attendance changes because of the invitation if the invitation affects outcomes only through the defined tutoring and other assumptions hold. If the invitation also sends study materials or changes teacher attention, exclusion is questionable.
Transfer succeeds when the intervention received includes every pathway the assignment creates.
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8. Misconceptions and limits
- “Noncompliance destroys randomisation.” Assignment remains random; receipt is not.
- “Compliers are the people who attended.” Compliance type depends on both potential assignments.
- “Divide ITT by uptake.” Use the difference in receipt rates, not one arm’s uptake alone.
- “IV gives the treatment effect for everyone.” It usually targets an instrument-specific local effect.
- “A strong first stage proves validity.” Relevance does not establish exclusion or monotonicity.
Complier effects can be hard to transport because a different encouragement may create a different complier group. Ethical reporting should avoid labelling people as disobedient when access, side effects or constraints shape receipt.
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9. Evidence audit
Audit allocation, receipt definitions, crossover, timing, missing outcomes and analysis code. Reconcile the first-stage denominator with all randomised participants. Verify that the instrument was implemented as assigned and that outcome collection did not depend on receipt.
| Audit question | Evidence | Warning sign |
|---|---|---|
| Was assignment random? | Sequence and concealment | Predictable allocation |
| Is receipt defined? | Exposure records | Any contact counted as treatment |
| Is exclusion plausible? | Delivery map | Assignment changes extra support |
| Is the first stage strong? | Receipt rates by arm | Tiny or uncertain difference |
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10. Questions readers often ask
Why not analyse only people who complied?
Observed compliers in one arm are not the same latent group across arms. Restriction usually breaks the randomised comparison.
Can random assignment always be an instrument?
No. It must change receipt and satisfy assumptions appropriate to the target effect.
What happens with missing outcomes?
Missingness requires its own assumptions and sensitivity analysis; instrumental-variable methods do not automatically repair it.
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11. Reading routes and ecosystem connections
Begin with the Gold Standard of Experimental Design. Use How Randomisation Works, Causal Inference and Biostatistics for the supporting logic.
The companion guides on treatment fidelity and estimands, attrition and missing outcomes and cluster randomisation cover adjacent post-assignment problems. Continue through learning that transfers into life, The Gold Standard and the How X Works hub.
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12. Sources and practical checklist
Authoritative and primary references include research on methods for complier average causal effects, guidance on estimating effects under noncompliance, and the International Council for Harmonisation’s E9(R1) estimand and sensitivity-analysis framework.
- Define assignment, receipt and timing.
- Report intention-to-treat first.
- Show receipt rates by assignment.
- State every IV assumption.
- Calculate the first stage correctly.
- Test assumption sensitivity.
- Label the complier effect as local, not universal.