Transporting LATE Across Populations

Why a Complier Effect May Not Travel

An instrumental-variable estimate is local to the people whose treatment choices are changed by the instrument. If the same instrument is used in two populations, it need not move treatment for the same fraction of people in each, and the compliers in the two populations need not have the same covariate composition. Even when covariate-specific effects are stable, the populations’ overall local average treatment effects (LATEs) can therefore differ.

Transporting LATE requires making the target population explicit. The key question is not simply whether a source estimate is precise, but whether the conditional effects and complier contributions that generated it also apply in the target population. We will separate two sources of change: shifts in the shares of compliers within covariate groups, and shifts in the mix of groups or their treatment effects.

Conditional Compliers and Population LATE

Let \(S=0\) denote the source population and \(S=1\) the target population. For each person, let \(Z\) be a binary instrument, \(D\) the binary treatment, and \(Y\) the outcome. Write \(D(z)\) for the treatment that would be received under instrument value \(z\), and \(Y(d)\) for the outcome under treatment value \(d\). A complier has \(D(1)=1\) and \(D(0)=0\). Let \(X\) denote measured covariates used to distinguish groups.

Within each population, the usual binary-instrument conditions are required for a covariate-specific Wald ratio to identify a complier effect: consistency and no interference; exclusion, so \(Z\) affects \(Y\) only through \(D\); conditional independence of \(Z\) from the potential treatments and outcomes given \(X\) and \(S\); monotonicity, \(D(1)\ge D(0)\); and relevance, so the instrument changes treatment probability. These conditions apply within the population in which the conditional effect is identified. They do not by themselves guarantee that an effect identified in the source population also holds in the target.

Definition: For population \(s\) and covariate value \(x\), the conditional complier share is \(\pi_s(x)=P(D(1)>D(0)\mid X=x,S=s)\). The conditional LATE, \(\tau_s(x)\), is the mean effect \(Y(1)-Y(0)\) among those compliers. Under the stated binary-instrument conditions and a positive first stage, \(\tau_s(x)\) equals the conditional Wald ratio: the difference in mean outcomes across instrument values divided by the difference in treatment probabilities across instrument values, conditional on \(X=x,S=s\).

The population-level LATE is the mean treatment effect among that population’s compliers—not the mean effect among everyone in the population. Its value is an average of conditional LATEs, but the weights are not generally the population proportions in the covariate groups. Groups with more compliers contribute more to the average.

\tau_s = E[Y(1)-Y(0)\mid D(1)>D(0),S=s] = \frac{\int \tau_s(x)\pi_s(x)\,dF_s(x)} {\int \pi_s(x)\,dF_s(x)}.

Here \(F_s\) is the distribution of \(X\) in population \(s\). The denominator is the population’s overall complier share, and the numerator accumulates each group’s conditional effect in proportion to its contribution of compliers.

Why the Weighting Formula Holds

Partition the population’s compliers by their covariate values. Among people with \(X=x\), a fraction \(\pi_s(x)\) are compliers and their mean treatment effect is \(\tau_s(x)\). Thus the contribution of that group to the sum of treatment effects over all compliers, per member of the population, is \(\pi_s(x)\tau_s(x)\). Averaging over \(F_s\) gives the total contribution per population member. The fraction of population members who are compliers is \(\int\pi_s(x)\,dF_s(x)\). Dividing the first quantity by the second gives the mean effect per complier.

\begin{aligned} E[(Y(1)-Y(0))\,\mathbf{1}\{D(1)>D(0)\}\mid S=s] &= \int \tau_s(x)\pi_s(x)\,dF_s(x),\\ P(D(1)>D(0)\mid S=s) &= \int \pi_s(x)\,dF_s(x),\\ E[Y(1)-Y(0)\mid D(1)>D(0),S=s] &= \frac{\int \tau_s(x)\pi_s(x)\,dF_s(x)} {\int \pi_s(x)\,dF_s(x)}. \end{aligned}

This derivation also explains why a simple average of group-specific LATEs can be wrong: it gives each group equal weight, or weights it by population frequency, rather than by the number of compliers it contributes. The relevant covariate distribution among compliers is proportional to \(\pi_s(x)dF_s(x)\).

Transporting Conditional Effects

A useful transport strategy first estimates \(\tau_0(x)\) in the source population and then combines those conditional effects with the target population’s covariate distribution and complier shares. This requires the target’s covariate values to be represented in the source data (common support) and a substantive transport assumption: conditional on \(X=x\), the complier treatment effect is the same across populations, so \(\tau_1(x)=\tau_0(x)\) over the target’s support. This is an assumption about effect stability among compliers, not a consequence of instrument validity within the source.

If target first-stage information is available, \(\pi_1(x)\) can be estimated in the target under its own valid-instrument conditions. Then the target LATE can be standardized using source conditional effects and target complier shares. If both conditional effects and conditional complier shares are assumed stable, source estimates of both can be combined with the target covariate distribution. Which strategy is justified depends on which conditional quantities can be measured and which invariance assumptions are credible.

\tau_1 = \frac{\int \tau_0(x)\pi_1(x)\,dF_1(x)} {\int \pi_1(x)\,dF_1(x)} \qquad \text{if } \tau_1(x)=\tau_0(x) \text{ on the target support.}

The target distribution \(F_1\) matters even when both conditional quantities are stable: a different mix of covariate groups changes their contributions. Conversely, a change in \(\pi_1(x)\) changes the relative weight on groups even if their population frequencies stay fixed. A source overall LATE can be carried over unchanged only under additional conditions that preserve the aggregate weighting and effects; conditional effect stability alone is not enough.

Worked Example: The Complier Share Changes Across Groups

Suppose \(X\) has two groups, A and B. In both populations, the instrument is randomized within each group, affects outcomes only through treatment, and satisfies monotonicity. The instrument has a positive first stage in both groups. Suppose the conditional complier shares are \(\pi(A)=0.8\) and \(\pi(B)=0.2\), and the source conditional LATEs are \(\tau(A)=2\) and \(\tau(B)=8\), in outcome units. Assume these conditional shares and effects are stable across populations.

In the source, groups A and B each make up one-half of the population. The complier-weighted numerator is \(0.5(0.8)(2)+0.5(0.2)(8)=0.8+0.8=1.6\). The overall complier share is \(0.5(0.8)+0.5(0.2)=0.4+0.1=0.5\). Therefore,

\tau_0=\frac{1.6}{0.5}=3.2 \quad\text{outcome units.}

Now suppose the target population is 20% group A and 80% group B. Its weighted numerator is \(0.2(0.8)(2)+0.8(0.2)(8)=0.32+1.28=1.60\). Its overall complier share is \(0.2(0.8)+0.8(0.2)=0.16+0.16=0.32\). Thus,

\tau_1=\frac{1.60}{0.32}=5.00 \quad\text{outcome units.}

The conditional effects and conditional complier shares have not changed, yet the population LATE rises from 3.2 to 5.00 because group B, with the larger effect, contributes a greater fraction of target compliers. The target overall LATE is not the source overall LATE transported unchanged.

Separate Population Composition from Effect Heterogeneity

The previous example involved different complier shares across groups as well as a different population mix. A second comparison isolates effect heterogeneity. If conditional complier shares are equal across groups, then changes in the covariate distribution affect the overall LATE only when conditional effects differ. If effects are constant across groups, changing the group mix cannot change this weighted average, provided the relevant groups have positive complier shares.

Worked Example: A Different Covariate Mix with Equal Complier Shares

Suppose both groups have a 0.5 complier share in both populations, and the conditional LATEs are 1 outcome unit in group A and 7 outcome units in group B. Assume the conditional effects and shares are stable, and that each population satisfies the instrument conditions within groups. Group A makes up 80% of the source and 20% of the target; group B makes up the remaining fraction.

For the source, the weighted numerator is \(0.8(0.5)(1)+0.2(0.5)(7)=0.4+0.7=1.1\). The overall complier share is \(0.8(0.5)+0.2(0.5)=0.5\). So the source LATE is \(1.1/0.5=2.20\) outcome units.

For the target, the numerator is \(0.2(0.5)(1)+0.8(0.5)(7)=0.1+2.8=2.9\), and the overall complier share remains \(0.5\). Therefore the target LATE is \(2.9/0.5=5.80\) outcome units. Here the change comes from treatment-effect heterogeneity across groups combined with a changed group mix; the complier shares are the same.

If both groups instead had a conditional LATE of 4 outcome units, each population’s weighted numerator would be 4 times its overall complier share. Dividing by that share would give a LATE of 4 in both populations, regardless of their covariate distributions.

When Conditional Effects Do Not Transport

Standardization cannot repair a failure of the conditional transport assumption. Source and target may have valid instruments within their respective populations, yet the effects among their compliers at the same \(X\) can differ. Differences in treatment versions, context, or the unmeasured characteristics of people induced into treatment are possible reasons. The source conditional LATE then is not the target conditional LATE.

Worked Example: The Source Conditional Effect Is Not Stable

Suppose both populations have two equally common covariate groups, with a complier share of 0.5 in each group. In the source, the conditional LATEs are 2 outcome units in A and 6 in B. In the target, the conditional LATE in A is 4, while the effect in B remains 6. Assume both populations’ instruments satisfy relevance, conditional independence, exclusion, and monotonicity within groups.

Using the source effects as if they transported would predict a target LATE of \(\{0.5(0.5)(2)+0.5(0.5)(6)\}/\{0.5(0.5)+0.5(0.5)\}=(0.5+1.5)/0.5=4.00\) outcome units. But the actual target numerator is \(0.5(0.5)(4)+0.5(0.5)(6)=1.0+1.5=2.5\), with the same denominator, \(0.5\). The target LATE is \(2.5/0.5=5.00\) outcome units.

The source calculation is internally correct for the source, and the target instrument can identify the target effect. The error is assuming conditional effect stability when it does not hold. Without target instrument data or a defensible transport assumption, the source data alone do not reveal this target-specific change.

Practical Checks and a Common Mistake

Before reporting a transported LATE, specify the target estimand and make the weighting choices visible. In particular, report whether the calculation uses source or target conditional complier shares, which distribution of \(X\) is used, and why conditional effects are expected to be stable. Check that every target covariate group used in the calculation has source support; a source effect cannot be estimated for a group that is absent from the source without additional assumptions or extrapolation.

Common mistake: “The source and target have the same conditional LATEs, so the overall LATE transports.” This ignores the group-specific complier shares and covariate distributions.
Correction: Calculate the target numerator and denominator using target group frequencies and target complier shares, then take their ratio.
Common mistake: Average conditional LATEs using the target population’s group proportions alone.
Correction: Weight each group by its population frequency multiplied by its conditional complier share, then normalize by the overall complier share.
Common mistake: Treat instrument validity in the source as evidence that conditional effects are unchanged in the target.
Correction: Instrument validity identifies a local effect within a population; cross-population stability is a separate, substantive assumption.

Key takeaway: A population LATE is a complier-weighted average of covariate-specific LATEs. Transporting it requires attention to both the conditional effects and the target distribution of compliers; a change in either can change the aggregate effect.

Check Your Understanding

For each question, distinguish the conditional effect, the complier share, and the population distribution being used.

In a target population, group A has a larger complier share than group B. If their conditional LATEs differ, why is an unweighted average of the two effects generally not the population LATE?
If conditional complier shares and conditional LATEs are stable across populations, what information is still needed to calculate the target LATE?
Suppose conditional complier shares are the same in both groups and conditional LATEs are constant across groups. What happens to the overall LATE when the covariate distribution changes, and why?
Can valid instruments in both populations establish that \(\tau_1(x)=\tau_0(x)\)? Explain the distinction between within-population identification and transport.
What common-support problem arises if a covariate group occurs in the target but not the source?