# Local average treatment effect

The local average treatment effect (LATE), also known as the complier average causal effect (CACE), was first introduced into the econometrics literature by Guido W. Imbens and Joshua D. Angrist in 1994.[1] It is the treatment effect for the subset of the sample that takes the treatment if and only if they were assigned to the treatment, otherwise known as the compliers. It is not to be confused with the average treatment effect (ATE), which is the average subject-level treatment effect; the LATE is only the ATE among the compliers. 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. Since the LATE may not be equivalent to the ATE, we should be cautious of extrapolation.

## General definition

The ATE is the difference between the expected value of the treatment group and the expected value of the control group. In an experimental setting, random assignment allows us to assume that the treatment group and control group have the same expected potential outcomes when treated (or untreated). This can be expressed as:

${\displaystyle E[Y_{i}(1)|D_{i}=1]=E[Y_{i}(1)|D_{i}=0]=E[Y_{i}(1)]}$

${\displaystyle E[Y_{i}(0)|D_{i}=1]=E[Y_{i}(0)|D_{i}=0]=E[Y_{i}(0)]}$

In an ideal experiment, all subjects assigned to treatment are treated, while those that are assigned to control will remain untreated. In reality, however, the compliance rate is never perfect, 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 and notation

Let ${\displaystyle Y_{i}(d)}$ denotes the potential outcome of subject i, where d is the binary indicator of subject ${\displaystyle i}$’s treatment status. ${\displaystyle Y_{i}(1)}$ denotes the treated potential outcome for subject i, while ${\displaystyle Y_{i}(0)}$ denotes the untreated potential outcome. The causal effect of the treatment on subject ${\displaystyle i}$ is ${\displaystyle Y_{i}(1)-Y_{i}(0)}$. However, we can never observe both ${\displaystyle Y_{i}(1)}$and ${\displaystyle Y_{i}(0)}$ for the same subject. At any given time, we can only observe a subject in its treated ${\displaystyle Y_{i}(1)}$ or untreated ${\displaystyle Y_{i}(0)}$ state.

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 treatment group is the same as that of the control group. The random assignment assumption thus allows us 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:

${\displaystyle ATE=E[Y_{i}(1)-Y_{i}(0)]=E[Y_{i}(1)]-E[Y_{i}(0)]=E[Y_{i}(1)|D_{i}=1]-E[Y_{i}(0)|D_{i}=0]}$

### Noncompliance framework

Quite often will researchers encounter noncompliance problems in their experiments, whereby subjects fail to comply with their experimental assignments. Some subjects will not take the treatment when assigned to the treatment group, so their potential outcome of ${\displaystyle Y_{i}(1)}$ will not be revealed, while some subjects assigned to the control group will take the treatment, so they will not reveal their ${\displaystyle Y_{i}(0)}$.

Given noncompliance, the population in an experiment can be divided into four subgroups: compliers, always-takers, never-takers and defiers. We then introduce ${\displaystyle z}$ as a binary indicator of experimental assignment, such that when ${\displaystyle z_{i}=1}$, subject ${\displaystyle i}$ is assigned to treatment, and when ${\displaystyle z_{i}=0}$, subject ${\displaystyle i}$ is assigned to control. Thus, ${\displaystyle d_{i}(z)}$represents whether subject ${\displaystyle i}$ is actually treated or not when treatment assignment is ${\displaystyle z_{i}}$.

Compliers are subjects who will take the treatment if and only if they were assigned to the treatment group, i.e. the subpopulation with ${\displaystyle d_{i}(1)=1}$ and ${\displaystyle d_{i}(0)=0}$.

Noncompliers 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 ${\displaystyle d_{i}(z)=1}$
• Never-takers are subjects who will never take the treatment even if they were assigned to the treatment group, i.e. the subpopulation with ${\displaystyle d_{i}(z)=0}$
• Defiers are subjects who will do the opposite of their treatment assignment status, i.e. the subpopulation with ${\displaystyle d_{i}(1)=0}$ and ${\displaystyle d_{i}(0)=1}$

Non-compliance can take two forms. 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 ${\displaystyle d_{i}(0)=0}$for all ${\displaystyle i}$, while ${\displaystyle d_{i}(1)=}$ 0 or 1. 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 ${\displaystyle d_{i}(0)}$ and ${\displaystyle d_{i}(1)}$ can be 0 or 1.

Given non-compliance, we require certain assumptions to estimate the LATE. Under one-sided non-compliance, we assume non-interference and excludability. Under two-sided non-compliance, we assume non-interference, excludability, and monotonicity.

### 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.[2]
• The first part of this assumption stipulates that the actual treatment status, ${\displaystyle d_{i}}$, of subject ${\displaystyle i}$ depends only on the subject's own treatment assignment status, ${\displaystyle z_{i}}$. The treatment assignment status of other subjects will not affect the treatment status of subject ${\displaystyle i}$. Formally, if ${\displaystyle z_{i}=z_{i}'}$, then ${\displaystyle D_{i}(\mathbf {z} )=D_{i}(\mathbf {z} ')}$, where ${\displaystyle \mathbf {z} }$ denotes the vector of treatment assignment status for all individuals.[3]
• The second part of this assumption stipulates that subject ${\displaystyle i}$'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 ${\displaystyle i}$'s outcomes. Formally, if ${\displaystyle z_{i}=z_{i}'}$ and ${\displaystyle d_{i}=d_{i}'}$, then ${\displaystyle Y_{i}(z,d)=Y_{i}(z',d)}$.
• 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, ${\displaystyle d_{i}}$, not treatment assignment, ${\displaystyle z_{i}}$. Formally ${\displaystyle Y_{i}(z,d)=Y_{i}(d)}$. So under this assumption, only ${\displaystyle d}$matters.[4] 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 all subject ${\displaystyle i}$, ${\displaystyle d_{i}(1)\geq d_{i}(0)}$. This states that whenever a subject moves from the control to treatment group, ${\displaystyle d_{i}}$ either remains unchanged or increases. The monotonicity assumption rules out defiers, since their potential outcomes are characterized by ${\displaystyle d_{i}(1).[1] It should be noted that 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 ${\displaystyle LATE={\frac {ITT}{ITT_{D}}}}$, whereby

${\displaystyle ITT=E[Y_{i}(z=1)]-E[Y_{i}(z=0)]}$

${\displaystyle ITT_{D}=E[d_{i}(z=1)]-E[d_{i}(z=0)]}$

The ${\displaystyle ITT}$ measures the average effect of experimental assignment on outcomes without accounting for the proportion of the group that was actually treated (i.e. average of those assigned to treatment minus the average of those assigned to control). In experiments with full compliance, the ${\displaystyle ITT=ATE}$.

The ${\displaystyle ITT_{D}}$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. ${\displaystyle ITT_{D}}$= the share of compliers.

### Proof

Under one-sided noncompliance , all subjects assigned to control group will not take the treatment, therefore:[3] ${\displaystyle E[d_{i}(z=0)]=0}$,

so that ${\displaystyle ITT_{D}=E[d_{i}(z=1)]==P[d_{i}(1)=1]}$

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

{\displaystyle {\begin{aligned}{\displaystyle E[Y_{i}(z=1)]=E[Y_{i}(d(1),z=1)]}=E[Y_{i}(z=1,d=1)|d_{i}(1)=1]*P[d_{i}(1)=1]&\\+E[Y_{i}(z=1,d=0)|d_{i}(1)=0]*(1-P[d_{i}(1)=1])\end{aligned}}}

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

{\displaystyle {\begin{aligned}{\displaystyle E[Y_{i}(z=0)]=E[Y_{i}(d=0,z=0)]}=E[Y_{i}(z=0,d=0)|d_{i}(1)=1]*P[d_{i}(1)=1]&\\+E[Y_{i}(z=0,d=0)|d_{i}(1)=0]*(1-P[d_{i}(1)=1])\end{aligned}}}

Through substitution, we can express the ITT as a weighted average of the ITT among the two subpopulations (compliers and never-takers), such that

{\displaystyle {\begin{alignedat}{2}ITT=E[Y_{i}(z=1)]-E[Y_{i}(z=0)]=E[Y_{i}(z=1,d=1)-Y_{i}(z=0,d=0)|d_{i}(1)=1]*P[d_{i}(1)=1]+&\\E[Y_{i}(z=1,d=1)-Y_{i}(z=0,d=0)|d_{i}(1)=0]*P[d_{i}(1)=0]\end{alignedat}}}

Given the exclusion and monotonicity assumption, the second half of this equation should be zero.

As such,

${\displaystyle {\frac {ITT}{ITT_{D}}}={\frac {E[Y_{i}(z=1,d=1)-Y_{i}(z=0,d=0)|d_{i}(1)=1]*P[d_{i}(1)=1]}{P[d_{i}(1)=1]}}=E[Y_{i}(d=1)-Y_{i}(d=0)|d_{i}(1)=1]=LATE}$

## Application: hypothetical schedule of 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 ${\displaystyle Y_{i}(d=1)-Y_{i}(d=0)}$

Hypothetical Schedule of Potential Outcome under Two-sided Noncompliance
Obeservation ${\displaystyle Y_{i}(1)}$ ${\displaystyle Y_{i}(0)}$ ${\displaystyle Y_{i}(1)-Y_{i}(0)}$ ${\displaystyle d_{i}(z=0)}$ ${\displaystyle d_{i}(z=1)}$ Type
1 4 7 3 0 1 Complier
2 3 5 2 0 0 Never-taker
3 1 5 4 0 1 Complier
4 5 8 3 1 1 Always-taker
5 4 10 6 0 1 Complier
6 2 8 6 0 0 Never-taker
7 6 10 4 0 1 Complier
8 5 9 4 0 1 Complier
9 2 5 3 1 1 Always-taker

${\displaystyle ATE={\frac {3+2+4+4+6+6+4+4+3}{9}}={\frac {35}{9}}=3.9}$

LATE is calculated by ATE among compliers, so

${\displaystyle LATE={\frac {3+4+6+4+4}{5}}=4.2}$

ITT is calculated by the average of ${\displaystyle Y_{i}(z=1)-Y_{i}(z=0)}$,

so ${\displaystyle ITT={\frac {3+0+4+0+6+0+4+4+0}{9}}={\frac {21}{9}}=2.3}$

${\displaystyle ITT_{D}}$ is the share of compliers

${\displaystyle ITT_{D}={\frac {5}{9}}}$ ${\displaystyle {\frac {ITT}{ITT_{D}}}={\frac {21/9}{5/9}}={\frac {21}{5}}=4.2=LATE}$

## Others: LATE in instrumental variable framework

We can also think of  LATE through an IV framework.[5] Treatment assignment ${\displaystyle z_{i}}$ is the instrument that drives the causal effect on outcome ${\displaystyle Y_{i}}$ through the variable of interest ${\displaystyle d_{i}}$, such that ${\displaystyle z_{i}}$ only influences ${\displaystyle Y_{i}}$ through the endogenous variable ${\displaystyle d_{i}}$, and through no other path. This would produce the treatment effect for compliers.

Except for the potential outcome framework mentioned about, LATE can also be estimated thorough Structural Equation Framework, which is built on specific equation with parameters and variables. SEM was first developed in the field of econometrics and has grown an enormous literature in the field.

SEM is derived through the following equations:

${\displaystyle D_{1}=\alpha _{0}+\alpha _{1}Z_{i}+\xi _{1i}}$

${\displaystyle Y_{i}=\beta _{0}+\beta _{1}Z_{i}+\xi _{2i}}$

The first equation captures the first stage effect of ${\displaystyle z_{i}}$on ${\displaystyle d_{i}}$, adjusting for variance, where

${\displaystyle \alpha _{1}=Cov(D,Z)/var(Z)}$

The second equation ${\displaystyle \beta _{1}}$ captures the reduced form effect of ${\displaystyle z_{i}}$ on ${\displaystyle Y_{i}}$,

${\displaystyle \beta _{1}=Cov(Y,Z)/var(Z)}$

The covariate-adjusted IV estimator is  the ratio ${\displaystyle \tau _{LATE}={\frac {\alpha _{1}}{\beta _{1}}}={\frac {Cov(Y,Z)/Var(Z)}{Cov(D,Z)/Var(Z)}}={\frac {Cov(Y,Z)}{Cov(D,Z)}}}$

Similar to the nonzero compliance assumption, the coefficient ${\displaystyle \alpha _{1}}$ in first stage regression needs to be significant to make ${\displaystyle z}$ a valid instrument.

However, because of SEM’s strict assumption of constant effect on every individual, potential outcome model is of more prevalent use today.

## References

1. ^ a b Imbens, Guido W.; Angrist, Joshua D. (March 1994). "Identification and Estimation of Local Average Treatment Effects". Econometrica. 62 (2): 467. doi:10.2307/2951620. ISSN 0012-9682. JSTOR 2951620.
2. ^ Rubin, Donald B. (January 1978). "Bayesian Inference for Causal Effects: The Role of Randomization". The Annals of Statistics. 6 (1): 34–58. doi:10.1214/aos/1176344064. ISSN 0090-5364.
3. ^ a b Angrist, Joshua D.; Imbens, Guido W.; Rubin, Donald B. (June 1996). "Identification of Causal Effects Using Instrumental Variables". Journal of the American Statistical Association. 91 (434): 444–455. doi:10.1080/01621459.1996.10476902. ISSN 0162-1459.
4. ^ Imbens, G. W.; Rubin, D. B. (1997-10-01). "Estimating Outcome Distributions for Compliers in Instrumental Variables Models". The Review of Economic Studies. 64 (4): 555–574. doi:10.2307/2971731. ISSN 0034-6527. JSTOR 2971731.
5. ^ Hanck, Christoph (2009-10-24). "Joshua D. Angrist and Jörn-Steffen Pischke (2009): Mostly Harmless Econometrics: An Empiricist's Companion". Statistical Papers. 52 (2): 503–504. doi:10.1007/s00362-009-0284-y. ISSN 0932-5026.