Reporting a LATE Without Overclaiming Population Effects

Why the Scope of a LATE Must Be Reported

An instrumental-variable estimate can be numerically clear and still be reported too broadly. A statement such as “the program improves outcomes for participants” may imply an effect for everyone who received the program. A LATE does not generally support that claim. It concerns a particular population, an explicitly defined contrast in an instrument, a particular treatment and outcome, and the people whose treatment status changes across that instrument contrast.

This distinction matters because the instrument does not necessarily change treatment for every person. Some people take the treatment regardless of the instrument, some do not take it regardless, and some change their treatment status in response. Under the assumptions for a LATE, the causal effect is averaged over this last group: the compliers. A careful report makes that scope visible rather than letting a broad label such as “patients” or “participants” stand in for it.

Definition: Let $Z$ be a binary instrument, $D$ a binary treatment, and $Y$ an outcome. Write $D(z)$ for the treatment a person would receive under instrument value $z$, and $Y(d)$ for the outcome under treatment value $d$. For a specified population, the local average treatment effect (LATE) for the contrast $Z=1$ versus $Z=0$ is $$ \operatorname{LATE}=E\left[Y(1)-Y(0)\mid D(1)>D(0)\right]. $$ It is the average effect of treatment among the compliers in that population: people who would receive treatment if $Z=1$ and would not receive it if $Z=0$.

What a Complete LATE Statement Specifies

A report should let a reader answer four questions without guessing: Who is the target population? What instrument values are being contrasted? What counts as treatment? What outcome and scale are being analyzed? It should then say that the effect applies to compliers for that specific contrast, not automatically to all members of the population.

Population: Define the eligible or sampled group, the setting, and any relevant time period. “Patients” is usually too vague; “adults eligible for the clinic’s referral program during the study period” is more informative.
Instrument contrast: Name what $Z=1$ and $Z=0$ mean. The contrast might be an offer versus no offer, or assignment under one rule versus another. Do not describe it only as “the effect of the instrument.”
Treatment: State what $D=1$ and $D=0$ mean, including relevant timing or treatment definition. A change in treatment receipt is not necessarily a change in every aspect of care.
Outcome: Name the outcome, its measurement time, and its scale. A difference in mean days, a risk difference, and a difference in a score are not interchangeable descriptions.
Complier group: Explain that the target is the people whose treatment receipt would differ across the specified instrument values. The group is defined by potential responses to the instrument; it is not generally identifiable person by person from observed data.

The binary-instrument Wald ratio makes the connection between the stated contrast and the LATE explicit. Under the assumptions below, it is the outcome contrast divided by the treatment-receipt contrast for $Z=1$ versus $Z=0$:

\frac{E[Y\mid Z=1]-E[Y\mid Z=0]} {E[D\mid Z=1]-E[D\mid Z=0]} = E[Y(1)-Y(0)\mid D(1)>D(0)].

The numerator and denominator must use the same population and the same direction of instrument contrast. Reversing the order in both differences leaves the ratio unchanged, but reversing only one changes its sign. The ratio is not defined when the instrument does not change treatment receipt in the population.

Assumptions Behind the Interpretation

A numerical Wald ratio is not, by itself, a causal LATE. The following conditions give it that interpretation for the specified binary instrument and treatment. They should be justified from the study design and substantive knowledge, not inferred merely because the ratio can be calculated.

Consistency and well-defined variables: observed treatment and outcome agree with the potential treatment and outcome under the values actually received. The treatment and instrument contrasts must be described precisely enough for these potential outcomes to be meaningful.
Instrument independence: within the stated population, instrument assignment is independent of the potential treatments and potential outcomes. Random assignment can support this condition; an observational instrument requires a defensible design-based argument.
Exclusion: the instrument affects the outcome only through the treatment as defined. If the instrument changes another pathway to the outcome, the LATE interpretation may fail.
Monotonicity: $D(1)\geq D(0)$ for every person under the chosen coding. Thus there are no defiers who would take treatment under $Z=0$ but not under $Z=1$.
Relevance: the instrument changes treatment receipt in the stated population, so $E[D\mid Z=1]-E[D\mid Z=0]\ne 0$.

Why the Ratio Averages Complier Effects

A short derivation shows why reporting the complier group is essential. With a binary treatment and monotonicity, each person is an always-taker, never-taker, or complier. Always-takers have $D(1)-D(0)=0$, as do never-takers. Compliers have $D(1)-D(0)=1$. Independence lets us express the observed instrument contrasts using these potential responses, and exclusion means the instrument has no direct effect on $Y$.

For the treatment contrast, only compliers contribute. Therefore,

E[D\mid Z=1]-E[D\mid Z=0] =E[D(1)-D(0)] =P(C),

where $C$ denotes membership in the complier group. For the outcome contrast, exclusion and independence give the average difference in treatment-specific outcomes. Always-takers and never-takers contribute zero to this contrast because their treatment does not change with $Z$. Compliers contribute their treatment effect:

E[Y\mid Z=1]-E[Y\mid Z=0] =E\left[(Y(1)-Y(0))\,\mathbb{1}(C)\right] =P(C)E[Y(1)-Y(0)\mid C].

Here $\mathbb{1}(C)$ equals 1 for compliers and 0 otherwise. Dividing the outcome contrast by the treatment contrast cancels $P(C)$, provided relevance holds. The result is the average treatment effect among compliers, not an average over everyone in the population.

Worked Example: Reporting a Mean Outcome Effect in Days

Suppose, hypothetically, a clinic studies adults eligible for a six-week rehabilitation program during one calendar year. The binary instrument is a randomly assigned offer of an appointment: $Z=1$ means offered and $Z=0$ means not offered. Treatment $D=1$ means attending the program, and the outcome is days alive and out of hospital during the following 90 days. Assume the design supports instrument independence, the offer affects this outcome only through attendance, monotonicity is plausible, and the offer changes attendance.

In the hypothetical population, 62% of those offered attend, compared with 30% of those not offered. The mean outcome is 74.0 days for those offered and 73.2 days for those not offered. The treatment-receipt contrast is $0.62-0.30=0.32$, and the outcome contrast is $74.0-73.2=0.8$ days.

The Wald estimate is

\widehat{\operatorname{LATE}} = \frac{74.0-73.2\text{ days}}{0.62-0.30} = \frac{0.8\text{ days}}{0.32} = 2.50\text{ days}.

A suitably scoped report is: “Among adults eligible for this clinic program during the study year, the estimated LATE of attending the six-week program on days alive and out of hospital over 90 days was 2.50 days for compliers whose attendance was changed by receiving an appointment offer rather than no offer.” This statement names the population, instrument contrast, treatment, outcome, time period, and relevant subgroup. It does not claim a 2.50-day effect for every attendee or every eligible adult.

Interpret the Scale and Direction Carefully

A LATE inherits the scale of the outcome contrast. If $Y$ is measured in days, the effect is in days. If $Y$ is a binary outcome coded 0 or 1, its mean is a risk, and the LATE is a risk difference among compliers. If an outcome is coded so that larger values indicate worse health, a positive estimate has a different substantive meaning than when larger values indicate better health. State the coding and direction rather than relying on a reader to infer them.

Worked Example: Reporting a Risk Difference

Consider a hypothetical population of students eligible for a course-completion support service. The instrument is assignment to receive an invitation ($Z=1$) versus no invitation ($Z=0$); the treatment is completing the service; and the outcome is dropping out before the end of the academic year ($Y=1$ for dropout). Suppose completion rates are 0.55 under invitation and 0.31 without invitation. The dropout risks are 0.14 and 0.20, respectively. Assume the instrument conditions, including exclusion and monotonicity, are credible for this specific setup.

The instrument-induced treatment-receipt difference is $0.55-0.31=0.24$. The outcome difference, in the stated direction, is $0.14-0.20=-0.06$. Thus,

\widehat{\operatorname{LATE}} = \frac{-0.06}{0.24} = -0.25.

Because the outcome is coded 1 for dropout, the estimate is a 0.25 lower probability of dropout, or 25 percentage points lower, among compliers whose service completion was changed by the invitation. It is not a 25% reduction in dropout risk: that would be a relative, not absolute, comparison. Nor does the estimate say that every student who completed the service had a 25-percentage-point reduction.

A LATE Belongs to Its Instrument Contrast

The complier group depends on how the instrument is defined and contrasted. A different offer, assignment rule, or comparison can change which people alter their treatment receipt. Consequently, a LATE for one instrument contrast is not automatically the effect for compliers under another contrast, even when the treatment and outcome have the same names. Likewise, a result in one eligible population should not be transported to a broader population without additional justification.

Worked Example: Avoiding a Population-Wide Claim

Suppose, hypothetically, a housing authority evaluates a priority-letter policy among applicants with a particular eligibility status in one district. The contrast is receiving the priority letter versus not receiving it; treatment is entering a supported-housing program within three months; and the outcome is the number of emergency shelter nights in the next six months. Assume the stated IV conditions are plausible for this population and contrast.

Suppose the letter changes program entry by 0.20, and the mean shelter-night outcome is 1.0 night lower under the letter. In the order “letter versus no letter,” the Wald estimate is

\frac{-1.0\text{ night}}{0.20} = -5.00\text{ nights}.

A defensible interpretation is: “For applicants with this eligibility status in this district, the estimated LATE of entering the supported-housing program on shelter nights over six months was 5.00 fewer nights among compliers whose program entry was changed by the priority-letter policy.” The phrase “whose program entry was changed” ties the result to the relevant subgroup and instrument contrast.

The broader statement “supported housing reduces shelter use by five nights for all applicants” is not justified by this estimate alone. It changes both the population and the group receiving the causal interpretation. People who would enter regardless of the letter and people who would not enter under either condition are not the complier group in this LATE.

A Practical Reporting Template

Before interpreting an IV estimate, fill in each part of a sentence like this:

Reporting template: “Among [defined population], comparing [instrument value 1] with [instrument value 0], the estimated LATE of [defined treatment] on [outcome and scale] was [estimate and units] for compliers whose treatment status was changed by that instrument contrast.”

The template is not a substitute for defending the assumptions. It makes the intended causal claim explicit and can reveal missing information: an undefined population, a vague instrument comparison, an ambiguous treatment, or an outcome with no stated scale. If the report cannot identify these pieces, its interpretation is not yet sufficiently bounded.

Common Mistakes

Claiming an effect for all treated people. “Treatment caused this outcome for participants” treats the complier estimate as if it were an average for everyone treated. Instead, say that the estimate applies to compliers under the named instrument contrast.
Claiming an effect for the entire population. The LATE is not generally the ATE. Do not remove the complier qualification unless additional assumptions justify that broader interpretation.
Leaving the instrument contrast implicit. “The effect of the instrument” does not tell readers what $Z=1$ and $Z=0$ represent. State both conditions and their order.
Reporting a number without its scale. A value of $-0.25$ could mean a risk difference, a change in a continuous score, or something else. Name the outcome coding and units.
Equating relevance with validity. A nonzero first-stage contrast allows a Wald ratio to be calculated, but it does not establish independence, exclusion, or monotonicity.

A clear LATE report is deliberately local: it says which people, instrument change, treatment, and outcome support the interpretation. That precision is a strength. It prevents a valid but limited causal estimate from being presented as an answer to a broader question it was not designed to answer.

Check Your Understanding

Use the population, contrast, treatment, outcome, and complier group to assess each interpretation.

In a randomized encouragement study, what does the complier group mean in terms of $D(1)$ and $D(0)$?
A report gives a LATE for an offer-versus-no-offer contrast among eligible adults. Why is it not automatically an effect for all adults who received treatment?
If the outcome is coded 1 for hospitalization, what does a LATE of $-0.10$ mean on the risk-difference scale?
Why should the population and instrument contrast be stated even when the treatment and outcome are clearly named?
Suppose the instrument changes treatment receipt by zero in the target population. What happens to the Wald ratio, and which assumption is not met?