LATE Versus ATE Under Treatment-Effect Heterogeneity

Why the Two Effects Can Differ

An instrumental-variable analysis can identify a causal effect even when treatment is not assigned directly. But under the usual assumptions for a binary instrument, the effect identified is generally an effect for compliers: people whose treatment status changes with the instrument. The population average treatment effect includes everyone, including people who would take treatment regardless of the instrument and people who would not take it either way. If treatment effects vary across these groups, the two averages need not be equal.

The distinction matters for interpretation. A complier effect can answer a useful causal question, but it does not automatically describe the effect of treating everyone, or even the average effect in the study population. Comparing the two estimands makes clear which additional facts about treatment-effect heterogeneity would be needed to connect them.

Definitions and Identification Conditions

Let \(Z\) be a binary instrument, with values 0 and 1, and let \(D\) be a binary treatment. For each person, \(D(z)\) is the treatment they would receive if the instrument were set to \(z\), and \(Y(d)\) is their outcome if treatment were set to \(d\). The individual treatment effect is \(Y(1)-Y(0)\). Assume consistency and no interference: the observed treatment and outcome agree with the potential outcomes under the values actually received, and one person's assignment does not change another person's outcome.

Definition: The population average treatment effect (ATE) is \(E[Y(1)-Y(0)]\), averaging the individual treatment effect over the target population. The complier average causal effect (CACE), also called the local average treatment effect (LATE), is \(E[Y(1)-Y(0)\mid D(1)>D(0)]\), averaging over people whose treatment status is changed from 0 to 1 by the instrument.

With a binary instrument and treatment, compliance type is determined by the pair \((D(0),D(1))\). Always-takers have \((1,1)\), never-takers have \((0,0)\), compliers have \((0,1)\), and defiers have \((1,0)\). The standard identification result for the Wald ratio requires instrument relevance, independence of the instrument from the potential outcomes and potential treatments, exclusion of any effect of the instrument on the outcome except through treatment, and monotonicity \(D(1)\geq D(0)\). Relevance means the instrument changes the treatment probability; monotonicity rules out defiers. Under these conditions, with a nonzero first stage, the Wald ratio identifies the CACE. The ATE itself is defined whether or not these IV assumptions hold; the assumptions concern what the IV contrast identifies.

\text{Wald ratio} = \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 identification result does not say that the CACE equals the ATE. It says which population subgroup's average effect is identified by the ratio. To compare that effect with the ATE, we must account for treatment effects among the other compliance types.

How the Population Effect Is Weighted

Under monotonicity, the population consists of compliers, always-takers, and never-takers. Let their population proportions be \(p_C\), \(p_A\), and \(p_N\), respectively, so \(p_C+p_A+p_N=1\). Write \(\tau_C\), \(\tau_A\), and \(\tau_N\) for the average treatment effects within each type. The CACE is \(\tau_C\); the ATE averages all three type-specific effects, weighted by their population proportions.

\text{ATE}=p_C\tau_C+p_A\tau_A+p_N\tau_N.

Subtracting the CACE gives a direct expression for the gap. This is a decomposition of the ATE, not an additional IV identification claim: it follows by partitioning the population into the three compliance types and averaging within each type.

\begin{aligned} \text{ATE}-\text{CACE} &=p_C\tau_C+p_A\tau_A+p_N\tau_N-\tau_C\\ &=p_A(\tau_A-\tau_C)+p_N(\tau_N-\tau_C), \end{aligned}

The compliers' contribution cancels because their effect is the CACE itself. The gap depends on how the average effect among always-takers and never-takers differs from the complier average, and on how common those groups are. Individual effects can vary substantially while the two averages still agree: positive and negative differences among the noncomplier groups may offset one another.

Key result: The CACE equals the ATE if the weighted differences among always-takers and never-takers sum to zero. Equal average effects across compliance types are sufficient for equality, but not necessary. If everyone has the same individual treatment effect, equality also follows, but equality of the two averages does not establish that treatment effects are homogeneous.

Worked Comparisons

Worked Example: A Larger Effect Among Compliers

Suppose a hypothetical study population has 20% always-takers, 50% compliers, and 30% never-takers. The hypothetical average treatment effects are 8 points among always-takers, 4 points among compliers, and 1 point among never-takers on an outcome score measured in points. Assume the instrument satisfies the identification conditions stated above.

The CACE is the average effect among compliers, so it is \(4\) points. The ATE includes all three groups:

\begin{aligned} \text{ATE} &=(0.20)(8)+(0.50)(4)+(0.30)(1)\\ &=1.60+2.00+0.30\\ &=3.90\text{ points}. \end{aligned}

Thus the CACE exceeds the ATE by \(4.00-3.90=0.10\) points. Equivalently, the decomposition gives \((0.20)(8-4)+(0.30)(1-4)=0.80-0.90=-0.10\) for ATE minus CACE. The sign is negative because the noncomplier groups' weighted differences from the complier effect sum to a negative value. The IV result supports a 4-point average effect for compliers; it does not, by itself, establish a 4-point average effect for the whole population.

Worked Example: Heterogeneity but Equal ATE and CACE

In a second hypothetical population, 30% are always-takers, 40% are compliers, and 30% are never-takers. Suppose their average treatment effects on a score measured in points are 8, 5, and 2 points, respectively. Assume the same IV identification conditions hold.

The CACE is \(5\) points. The ATE is

\begin{aligned} \text{ATE} &=(0.30)(8)+(0.40)(5)+(0.30)(2)\\ &=2.40+2.00+0.60\\ &=5.00\text{ points}. \end{aligned}

The two averages agree even though the type-specific average effects differ by as much as 6 points. The reason is exact cancellation: always-takers contribute \(0.30(8-5)=0.90\) points to ATE minus CACE, while never-takers contribute \(0.30(2-5)=-0.90\) points. Their sum is zero. This equality is a fact about these particular weighted averages, not proof that the effect is the same for each person or for each type.

Worked Example: A High Complier Proportion Does Not Guarantee Equality

Consider a third hypothetical population with 4% always-takers, 90% compliers, and 6% never-takers. Suppose the type-specific average effects on a score measured in points are 8 points for always-takers, 6 points for compliers, and 0 points for never-takers. Assume the IV identification conditions hold.

The CACE is \(6\) points. The ATE is

\begin{aligned} \text{ATE} &=(0.04)(8)+(0.90)(6)+(0.06)(0)\\ &=0.32+5.40+0\\ &=5.72\text{ points}. \end{aligned}

The ATE is \(6.00-5.72=0.28\) points lower than the CACE. Only 10% of the population are noncompliers, so the difference is modest in this example, but it is not zero. A high complier proportion can make the population average closer to the complier average when noncomplier effects are not extreme; it does not ensure equality. The direction and size of the gap still depend on the noncompliers' average effects.

What the Comparison Does—and Does Not—Establish

The CACE and ATE answer different questions. The CACE concerns people whose treatment receipt is changed by the instrument. The ATE concerns the average effect if treatment were set to 1 rather than 0 for the target population. In particular, always-takers and never-takers contribute to the ATE even though their treatment status does not change when the instrument changes. Their treatment effects therefore matter to the ATE, but are not supplied by the basic Wald identification result.

The relationship can be summarized through the weighted gap. If noncompliers are rare and their effects are close to the complier effect, the ATE may be close to the CACE. If noncompliers are numerous or have very different effects, the gap may be substantial. Neither conclusion follows from instrument relevance alone: relevance concerns the change in treatment probability, not how treatment effects compare across types.

This distinction is especially important when translating an IV result into a policy claim. An effect for compliers may be directly relevant to an intervention that moves treatment uptake among similar people. A claim about treating the full population requires an additional basis for generalizing beyond compliers, such as information about treatment effects in the other groups or a defensible assumption connecting their effects to the complier effect. Without such support, report the CACE as a complier effect rather than labeling it the population effect.

Common Mistake

Incorrect reasoning: “The instrument is valid, so the Wald estimate is the average effect of treatment for everyone.” Validity assumptions support a particular identification result: the Wald ratio identifies the CACE under the stated conditions. They do not make the instrument change treatment for always-takers or never-takers, and they do not imply that those groups have the same treatment effects as compliers.

Correction: First identify the target of the estimand. Then compare it with the desired population quantity. The ATE equals the CACE only when the weighted noncomplier differences cancel, for example when average effects are equal across compliance types. The IV assumptions alone do not guarantee that condition.

Key takeaway: Under the standard binary-IV assumptions, the Wald estimand is local to compliers. The ATE also averages effects among always-takers and never-takers. Treatment-effect heterogeneity, together with the proportions of the compliance types, determines whether those two averages differ.

Check Your Understanding

Use the definitions and weighted decomposition above to answer these questions.

A population is 25% always-takers, 50% compliers, and 25% never-takers. Their average treatment effects are 6, 4, and 2 outcome units, respectively. Calculate the CACE and ATE.
Can the ATE equal the CACE when average effects differ across compliance types? Explain how the weighted differences can produce equality.
If the complier proportion is 95%, must the ATE equal the CACE? Give a reason for your answer.
Which population does the Wald ratio identify under the standard binary-IV assumptions, and why does that not automatically identify the ATE?
Suppose the CACE is reported as 3 outcome points. What additional feature of the population would need to be understood before interpreting 3 points as the ATE?