In econometrics and related empirical fields, the local average treatment effect (LATE), also known as the complier average causal effect (CACE), is the effect of a treatment for subjects who comply with the experimental treatment assigned to their sample group. It is not to be confused with the average treatment effect (ATE), which includes compliers and non-compliers together. Compliance refers to the human-subject response to a proposed experimental treatment condition. Similar to the ATE, the LATE is calculated but does not include non-compliant parties. If the goal is to evaluate the effect of a treatment in ideal, compliant subjects, the LATE value will give a more precise estimate. However, it may lack external validity by ignoring the effect of non-compliance that is likely to occur in the real-world deployment of a treatment method. The LATE can be estimated by a ratio of the estimated intent-to-treat effect and the estimated proportion of compliers, or alternatively through an instrumental variable estimator.
The LATE was first introduced in the econometrics literature by Guido W. Imbens and Joshua D. Angrist in 1994, who shared one half of the 2021 Nobel Memorial Prize in Economic Sciences. As summarized by the Nobel Committee, the LATE framework "significantly altered how researchers approach empirical questions using data generated from either natural experiments or randomized experiments with incomplete compliance to the assigned treatment. At the core, the LATE interpretation clarifies what can and cannot be learned from such experiments."
The phenomenon of non-compliant subjects (patients) is also known in medical research.Moerbeek, M., & Schie, S. van. (2019). What are the statistical implications of treatment non‐compliance in cluster randomized trials: A simulation study. In Statistics in Medicine (Vol. 38, Issue 26, pp. 5071–5084). Wiley. https://doi.org/10.1002/sim.8351 In the biostatistics literature, Baker and Lindeman (1994) independently developed the LATE method for a binary outcome with the paired availability design and the key monotonicity assumption. Baker, Kramer, Lindeman (2016) summarized the history of its development. Various papers called both Imbens and Angrist (1994) and Baker and Lindeman (1994) seminal.
An early version of LATE involved one-sided noncompliance (and hence no monotonicity assumption). In 1983 Baker wrote a technical report describing LATE for one-sided noncompliance that was published in 2016 in a supplement. In 1984, Bloom published a paper on LATE with one-sided compliance. For a history of multiple discoveries involving LATE see Baker and Lindeman (2024).
General definition
The typical terminology of the Rubin causal model is used to measure the LATE, with units indexed and a binary treatment indicator, for unit . The term is used to denote the potential outcome of unit under treatment .
In an ideal experiment, all subjects assigned to the treatment will comply with the treatment, while those that are assigned to control will remain untreated. In reality, however, the compliance rate is often imperfect, which prevents researchers from identifying the ATE. In such cases, estimating the LATE becomes the more feasible option. The LATE is the average treatment effect among a specific subset of the subjects, who in this case would be the compliers.
Potential outcome framework
The LATE is defined within the potential outcomes framework of causal inference. The treatment effect for subject is . It is impossible to simultaneously observe and for the same subject. At any given time, only a subject in its treated or untreated state can be observed.
Through random assignment, the expected untreated potential outcome of the control group is the same as that of the treatment group, and the expected treated potential outcome of the treatment group is the same as that of the control group. The random assignment assumption thus allows one to take the difference between the average outcome in the treatment group and the average outcome in the control group as the overall average treatment effect, such that:
Non-compliance framework Researchers frequently encounter non-compliance problems in their experiments, whereby subjects fail to comply with their experimental assignments. In an experiment with non-compliance, the subjects can be divided into four subgroups: compliers, always-takers, never-takers and defiers. The term represents the treatment that subject actually takes when their treatment assignment is .
Compliers are subjects who will take the treatment if and only if they were assigned to the treatment group, i.e., the subpopulation with and .
Non-compliers are composed of the three remaining subgroups:
Always-takers are subjects who will always take the treatment even if they were assigned to the control group, i.e., the subpopulation with Never-takers are subjects who will never take the treatment even if they were assigned to the treatment group, i.e., the subpopulation with Defiers are subjects who will do the opposite of their treatment assignment status, i.e., the subpopulation with and
Non-compliance can take two forms: one-sided (always-takers and never-takers) and two-sided (defiers). In the case of one-sided non-compliance, a number of the subjects who were assigned to the treatment group remain untreated. Subjects are thus divided into compliers and never-takers, such that for all , while or . In the case of two-sided non-compliance, a number of the subjects assigned to the treatment group fail to receive the treatment, while a number of the subjects assigned to the control group receive the treatment. In this case, subjects are divided into the four subgroups, such that both and can be 0 or 1.
Given non-compliance, certain assumptions are required to estimate the LATE. Under one-sided non-compliance, non-interference and excludability is assumed. Under two-sided non-compliance, non-interference, excludability, and monotonicity is assumed.
Assumptions under one-sided non-compliance The non-interference assumption, otherwise known as the Stable Unit Treatment Value Assumption (SUTVA), is composed of two parts. The first part of this assumption stipulates that the actual treatment status, , of subject depends only on the subject's own treatment assignment status, . The treatment assignment status of other subjects will not affect the treatment status of subject . Formally, if , then , where denotes the vector of treatment assignment status for all individuals. The second part of this assumption stipulates that subject 's potential outcomes are affected by its own treatment assignment, and the treatment it receives as a consequence of that assignment. The treatment assignment and treatment status of other subjects will not affect subject 's outcomes. Formally, if and , then . The plausibility of the non-interference assumption must be assessed on a case-by-case basis. The excludability assumption requires that potential outcomes respond to treatment itself, , not treatment assignment, . Formally . So under this assumption, only matters. The plausibility of the excludability assumption must also be assessed on a case-by-case basis.
Assumptions under two-sided non-compliance
All of the above, and: The monotonicity assumption, i.e., for each subject , . This states that if a subject were moved from the control to treatment group, would either remain unchanged or increase. The monotonicity assumption rules out defiers, since their potential outcomes are characterized by . Monotonicity cannot be tested, so like the non-interference and excludability assumptions, its validity must be determined on a case-by-case basis.
Identification The , whereby
The measures the average effect of experimental assignment on outcomes without accounting for the proportion of the group that was actually treated (i.e., an average of those assigned to treatment minus the average of those assigned to control). In experiments with full compliance, the .
The measures the proportion of subjects who are treated when they are assigned to the treatment group, minus the proportion who would have been treated even if they had been assigned to the control group, i.e., = the share of compliers.
Proof Under one-sided noncompliance, all subjects assigned to control group will not take the treatment, therefore: ,
so that
If all subjects were assigned to treatment, the expected potential outcomes would be a weighted average of the treated potential outcomes among compliers, and the untreated potential outcomes among never-takers, such that
If all subjects were assigned to control, however, the expected potential outcomes would be a weighted average of the untreated potential outcomes among compliers and never-takers, such that
Through substitution, the ITT is expressed as a weighted average of the ITT among the two subpopulations (compliers and never-takers), such that
Given the exclusion and monotonicity assumption, the second half of this equation should be zero.
As such,
Application: hypothetical schedule of the potential outcome under two-sided noncompliance The table below lays out the hypothetical schedule of potential outcomes under two-sided noncompliance.
The ATE is calculated by the average of + Hypothetical Schedule of Potential Outcome under Two-sided NoncomplianceObservationType147301Complier235200Never-taker315401Complier458311Always-taker5410601Complier628600Never-taker7610401Complier859401Complier925311Always-taker
LATE is calculated by ATE among compliers, so
ITT is calculated by the average of ,
so
is the share of compliers
Others: LATE in instrumental variable framework LATE can be thought of through an IV framework. Treatment assignment is the instrument that drives the causal effect on outcome through the variable of interest , such that only influences through the endogenous variable , and through no other path. This would produce the treatment effect for compliers.
In addition to the potential outcomes framework mentioned above, LATE can also be estimated through the Structural Equation Modeling (SEM) framework, originally developed for econometric applications.
SEM is derived through the following equations:
The first equation captures the first stage effect of on , adjusting for variance, where
The second equation captures the reduced form effect of on ,
The covariate adjusted IV estimator is the ratio
Similar to the nonzero compliance assumption, the coefficient in first stage regression needs to be significant to make a valid instrument.
However, because of SEM’s strict assumption of constant effect on every individual, the potential outcomes framework is in more prevalent use today.
Generalizing LATE The primary goal of running an experiment is to obtain causal leverage, and it does so by randomly assigning subjects to experimental conditions, which sets it apart from observational studies. In an experiment with perfect compliance, the average treatment effect can be obtained. However, many experiments are likely to experience either one-sided or two-sided non-compliance. In the presence of non-compliance, the ATE can no longer be recovered. Instead, what is recovered is the average treatment effect for a certain subpopulation known as the compliers, which is the LATE.