# ATE, ATET, LATE, and MATE in Casual Inferences

ATE, ATET, LATE, and MATE are all measures used in causal inference to estimate the effect of a treatment or intervention on a specific outcome. Each measure focuses on a different population or subgroup and uses different statistical methods to estimate the treatment effect.

### ATE (Average Treatment Effect)

The Average Treatment Effect (ATE) is the average effect of a treatment or intervention on an outcome for a given population. It estimates the difference in outcomes between a group that receives the treatment and a group that does not receive the treatment. The ATE assumes that the treatment is randomly assigned and there are no confounding factors affecting the outcome.

For example, if a new medication is given to a randomly selected group of patients and a placebo is given to another randomly selected group of patients, the ATE would estimate the average difference in health outcomes between the two groups.

$ATE=E[Y_i^1-Y_i^0]$

### ATET (Average Treatment Effect for the Treated)

The Average Treatment Effect for the Treated (ATET) estimates the average effect of a treatment or intervention on an outcome for a specific subgroup of the population who received the treatment. It focuses on individuals who have a specific binary covariate, such as being diagnosed with a certain medical condition or having a certain genetic marker.

For example, if a new medication is given to patients diagnosed with a specific medical condition, the ATET would estimate the average difference in health outcomes between those patients who received the medication and those who did not, while controlling for other factors that may affect the outcome.

Why ATET?

Suppose we perform a simple difference between the outcome of those who receive treatment and who do not receive treatment.

$\Delta = E[Y^1|D=1]-E[Y^0|D=0]$

$\Delta = E[Y^1|D=1]-E[Y^0|D=0]+E[Y^0|D=1]-E[Y^0|D=1]$

$\Delta = E[Y^1|D=1]-E[Y^0|D=1]+E[Y^0|D=1]-E[Y^0|D=0]$

$\Delta = ATET + Selection Bias$

Selection Bias is the difference between the counterfactual for treated individuals and the observed outcome for the untreated individuals.

### LATE (Local Average Treatment Effect)

The Local Average Treatment Effect (LATE) estimates the average effect of a treatment or intervention on an outcome for individuals who comply with the assigned treatment. It assumes that there is a binary instrument that determines whether an individual receives the treatment or not, and that compliance with the instrument is affected by a set of observed covariates.

For example, if a study offers a scholarship to students who live in a specific area and attend a specific school, the LATE would estimate the average difference in educational outcomes between students who receive the scholarship and those who do not, while controlling for other factors that may affect the outcome, such as socioeconomic status.

LATE is given by

$\frac{E[Y_i|Z_i=1]-E[Y_i|Z_i=0]}{E[D_i|Z_i=1]-E[D_i|Z_i=0]}$

Provided that Z is a binary variable.

A general version that does not require that $Z_i$ is a dummy is

$\frac{Cov(Z_i,Y_i)}{Cov(Z_i,D_i)}$

Derivation of above formula

$y_i=\beta_i+\beta_2 x_{2,i}+u_i---(1)$

$x_{2,i}=\alpha_1+\alpha_2 z_{2,i}+\epsilon_i---(2)$

Now, writing equation (1) in terms of mean deviation

$y_i-E(y_i)=\beta_2 (x_{2,i}-E(x_{2,i}))+u_i---(3)$

Now, multiply both sides by $(z_i-E(z_i))$ and impose expectations on both sides.

$E[(z_i-E(z_i))(y_i-E(y_i))]=\beta_2 E[(x_{2,i}-E(x_{2,i}))(z_i-E(z_i))]+E[u_i(z_i-E(z_i))]---(4)$

In equation (4),

$E[(z_i-E(z_i))(y_i-E(y_i))] =Cov(z,y)$

$E[(x_{2,i}-E(x_{2,i}))(z_i-E(z_i))]=Cov(z,x_2)$

$E[u_i(z_i-E(z_i))]=Cov(u,z)$

So, equation (4) becomes

$Cov(z,y)=\beta_2 Cov(z,x_2)+Cov(u,z)$

We know, $Cov(u,z)=0$ so,

$Cov(z,y)=\beta_2 Cov(z,x_2)$

$\beta_2 = \frac{Cov(z,y)}{Cov(z,x_2)}$

### MATE (Marginal Average Treatment Effect)

The Marginal Average Treatment Effect (MATE) estimates the average effect of a treatment or intervention on an outcome for a specific subgroup of the population defined by a set of covariates. It measures how the treatment effect varies across different values of these covariates.

For example, if a study examines the effect of a new medication on blood pressure, the MATE would estimate the average treatment effect for individuals with a specific age range, such as 50-60 years old, while controlling for other covariates such as gender, race, and initial blood pressure levels.

In summary, ATE, ATET, LATE, and MATE are all important measures for estimating the effect of a treatment or intervention on a specific outcome. They differ in their focus, assumptions, and statistical methods used to estimate the treatment effect, and each measure may be more appropriate for different research questions and populations of interest.

### Key Differences

 Aspect ATE ATET LATE MATE Definition The average treatment effect of a treatment or intervention on a population The average treatment effect for a specific subgroup of the population defined by a binary covariate The average treatment effect for compliers only The average treatment effect for a specific subgroup of the population defined by a set of covariates Focus Population Binary covariate Compliers only Specific subgroups defined by covariates Causal inference Estimates causal effect of treatment on outcome Estimates causal effect of treatment on the outcome, conditional on binary covariate Estimates the causal effect of treatment on the outcome, for compliers only Estimates the causal effect of treatment on the outcome, for specific subgroups defined by covariates Treatment assignment Assumes random assignment of treatment Assumes random or quasi-random assignment of treatment Assumes treatment is assigned based on compliance with the assignment Assumes treatment is assigned based on observed covariates Handling of confounding factors Assumes no unmeasured confounding factors Assumes no unmeasured confounding factors Assumes no unmeasured confounding factors, and only considers compliers Controls for other covariates that may affect the outcome Subgroup effects Provides an average effect for the entire population Does not account for subgroup effects Does not account for subgroup effects Accounts for subgroup effects Practical use Provides overall measure of treatment effectiveness Useful for evaluating treatment effectiveness in a binary subgroup Useful for evaluating treatment effectiveness for those who comply with the assignment Useful for targeting interventions to specific subgroups based on observed covariates