I saw ATE and ATT in some discussions regarding DID settings recently. ATE is the Average Treatment Effect while ATT is Average Treatment Effects on Treated.

I am wondering the difference between these two terms and is there any example to clarify such a difference?


4 Answers 4


Treatment effects are causal effects of a binary treatment. Because the treatment is binary, individuals are either treated or they are not treated. For the sake of example assume that the treatment is participation in a money making course - the course is claimed to make you better at making money.

Obviously, the causal effect of such a course could very well be differ from person to person (this is referred to as treatment heterogeneity). Some people may learn a lot from the course and actually improve at making money while others will be bored by the content of the course and experience a zero effect. As usual when important quantitative measures vary across observational units a canonical summary statistic is the average. The Average Treatment Effect (ATE) is simply that: The average of the individual treatment effects of the population under consideration. And the Average Treatment Effect of the Treated (ATT) is simply the average of the individual treatment effects of those treated (hence not the entire population).

To make it formally more clear what the causal effect of treatment is, it is often assumed that for each individual $i$ there exists an amount of money $Y_i^0$ individual $i$ will make without taking the training course. And there also exists an amount of money $Y^1_i$ that individual $i$ will make if she takes course. The causal effect for individual $i$ of participation in the course is then defined as

$$\tau_i := Y_i^1- Y^0_i,$$

the difference in outcome with and without treatment.

For the sake of example consider the following table for 6 individuals:

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It is clear from the table that individual $i=1,2,3$ are treated $D_i=1$ while $i=4,5,6$ are not treated. For those who are treated the observed amount of money made by the individual $Y_i$ is equal to $Y_i^1$. For those not treated the observed amount of money made $Y_i$ is equal to $Y_i^0$. In general this is written as

$$Y_i = D_i Y_i^1 + (1-D_i)Y_i^0.$$

An important part of the setup is therefore that while $Y_i^1$ and $Y_i^0$ are assumed to exist they are not assumed to be observed.

However, getting back to ATT and ATE. In the above example the ATE can be calculated as

$$ATE := \frac{1}{N} \sum_i \tau_i = \frac{1}{N} \sum_i (Y_i^1 - Y_i^0) = \frac{1+1+1+0+1-1}{6} = 0.5,$$

and the average treatment effect of those treated is calculated as

$$ATT := \frac{1}{N_1} \sum_i \tau_i = \frac{1}{N_1} \sum_i (Y_i^1 - Y_i^0) = \frac{1+1+1}{3} = 1.0,$$

where $N_1 = \sum_i D_i = 3$.

In this example ATE and ATT are numerically the same but as you can see they are averages of different sets of individual causal effects. As such they are not necessarily expected to be the same. Try to construct an example yourself where they are different simply by changing the group of treated individuals.

The average treatment effect (ATE) is used when we are interested in the average treatment of the entire population, whereas the average treatment effect of the treated (ATT) is used when we are only interested in the average treatment effect of those treated.

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    $\begingroup$ but I mean that is my understanding based on what is written in MHE, so that is the reference for me. I mean Rubin directly does not use ATE terminology either (you would not find that word printed in his seminal article). What is wrong with the example I given? And also I am willing to accept that I might be wrong here to be honest you made me unsure of if I understand it correctly, but I cant see a outright mistake in the example I given you $\endgroup$
    – 1muflon1
    Jun 6, 2021 at 16:57
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    $\begingroup$ Left hand side of identity I buy $ATE := \mathbb E[Y_1-Y_0] = \mathbb E[Y_1] - \mathbb E[Y_0]$ but then it stops because $\bar Y_{i1} - \bar Y_{i0} = ...$ is a difference between sample averages that at best can be a consistent estimator for some poplation moments. $\endgroup$ Jun 6, 2021 at 17:02
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    $\begingroup$ Obviously, what these sample moments consistently estimate - their difference - is $\mathbb E[Y\lvert D=1] - \mathbb E[Y \lvert D= 0]$ but exactly because this difference is not necessarily equal to $ \mathbb E[Y_1] - \mathbb E[Y_0] = ATE$ and not necessarily equal to $\mathbb E[Y_1\lvert D=1] - \mathbb E[Y_0 \lvert D= 1]:=ATT$ we have an estimation problem. $\endgroup$ Jun 6, 2021 at 17:02
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    $\begingroup$ ok but in the sample $E[Y|D=1] = E[Y_1]$ as in the example above, or wait do you by $E[Y_1]$ mean expectationof all potential outcomes as opposed to those that are observed? $\endgroup$
    – 1muflon1
    Jun 6, 2021 at 17:05
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    $\begingroup$ The 6 people in my table are the population .... and yes $\mathbb E[Y_1]$ is expectation of all potential outcomes under treatment $\mathbb E[Y_0]$ under non-treatment. $\endgroup$ Jun 6, 2021 at 17:06

Let us explain this in a backdrop of simple model used by Burde and Linden (2013) who looked at an effect of building new village schools (as opposed to have children commuting) on students academic outcomes. They estimated the following model:

$$Y_{ijk} = β_0 + β_1T_k + e_{ijk}$$

Where $Y_{ijk}$ is an academic outcome of child $i$ in household $j$ in a village $k$.

Here the $T_k$ is a dummy that signifies whether the village got a school in the first year or not, that can take only two values 1 or 0 .

In this setting we will have 2 possible conditional outcomes

$$T_k = \begin{cases} Y_{ij1}| T_k = 1\implies Y_{ij1} = β_0 + β_{1}1 + e_{ijk} \\ Y_{ij0}| T_k = 0 \implies Y_{ij0} = β_0 + β_10 + e_{ijk} \end{cases}$$

Now finally we can turn our attention to explanation of ATE vs ATT.

Average Treatment Effect

In this setting the average treatment effect is simply:

$$\text{ATE}= E[Y_{ij1} - Y_{ij0}]$$

So it it is the difference between potential outcomes, in this case academic achievement between children who got access to village schools and children who did not get access to village schools.

Average Treatment Effect on Treated

Now when it comes to average treatment effect on treated this is defined as:

$$E[Y_{ij1} − Y_{ij0}|T_k = 1]$$

So in plain English and applied to the case above this is the difference in academic potential outcomes between children that got access to village schools, and children that did not get access to village schools conditionally on the fact that they both are assigned to village schools.

The outcome $Y_{ij0}|T_k = 1$ is essentially a counterfactual for $Y_{ij1}$ in a 'parallel universe' where exactly the same people who where assigned the treatment in this universe would not get the treatment. That is with ATE you are comparing children who got schools with other children who did not get schools, whereas in ATT you are comparing children who got schools, with the same children who got schools from a 'parallel universe' where they did not get schools.

  • $\begingroup$ In order to estimate the ATT, my understanding is that you include all the subjects in the treatment group and compare them to subjects in the control group who most closely resemble the treatment group members. This can be done via a number of methods, like exact matching subjects on characteristics (age, sex, etc.) or by upweighting members in the control group based on their probability of belonging in the treatment group (inverse propensity score weighting). $\endgroup$
    – RobertF
    Nov 22, 2021 at 20:05

In RCT, $ ATE = ATT = ATU$.

In observational study, $ATT\neq ATE$, since $$ ATE = E\{Y(1)-Y(0)\} = E_X\{E(Y|X,A=1)-E(Y|X,A=0)\}= E_X\{E(Y|X,A=1)\}- E_X\{ E(Y|X,A=0)\}\}, $$ $$ ATT = E\{Y(1)-Y(0)|A=1\} = E(Y|A=1) - E_X\{E(Y|X,A=0)\}. $$

  • $\begingroup$ That's incorrect - ATE = ATT = ATU only when treatment is "forced", aka no one can opt to not take the treatment. In many circumstances (e.g. clinical trials), there's non-compliance - you're in the treatment group, but you don't take the treatment. $\endgroup$ Dec 15, 2022 at 14:43

I'm a social science researcher, so my example will come from observational studies. Consider for example, a university offers calculus tutoring to all students taking the course, but only some choose to use this service. You want to estimate the effects of tutoring, so you can do it two ways:

  1. ATE (Average Treatment Effect). This one would measure the effect that participating in tutoring would have on the entire population of students (those who used tutoring (1) and those who did not use tutoring (0)). If you see a positive effect then you can generalize to say that all students who take calculus should use tutoring services.

  2. ATT (Average Treatment Effect of Treated). This one would measure the effect that participating in tutoring would have on the students that used the service (the treated) or (those who used tutoring (1)). You can still compare it to those who did not use tutoring (0), but the treatment effect will tell you how students who used tutoring would have faired had they not used tutoring.


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