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Average Treatment Effect (ATE) is a common measure to determine if a treatment (e.g. taking a particular medicine, you either take it or not) exert an effect on a particular target variable (e.g. sickness status). Additionally, as it seems to me a big part of causal inference literature, ATE considers binary treatments.

I was wondering if there is any similar concept in the case of ordinal non-binary treatments for binary outcomes. One such example may be to quantify the effect that your school final grade (finite, ordinal, and non-binary) has on whether or not you gain more than 100k$ per year (binary).

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  • $\begingroup$ ATE is just the difference between the expected potential outcomes in the treated / non-treated case. All ATE methods (?) simply compute the two expected potential outcomes individually and then subtract them. You can use the same methods to compute expected potential outcomes for more than two treatment options, and then do whatever you want with them? $\endgroup$
    – Eike P.
    Commented Jun 29, 2023 at 22:42
  • $\begingroup$ I thought about that as well, but I wanted to be sure if there was any other way to measure the concept of "causal effect" with ordinal variables, maybe taking into account distance between treatments. Think for example as a outcome that is linear w.r.t. the treatment (e.g. Y(x) = 2*x), and let's suppose we want to study the effect for treatment values {1,4,5}. If you just do the difference you find that the ATE going from 1 to 4 is different that one from 4 to 5, but the "relative" increase is the same $\endgroup$
    – DaSim
    Commented Jun 30, 2023 at 15:13

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The ATE is specific to binary treatments, but causal effect estimates are a much broader concept

The average treatment effect (ATE) is built specifically for the binary treatment scenario which is common in certain domains (e.g. medicine). The more general quantity of interest is the expected value of a target $Y$ given some treatment $T$, written as $E(Y\mid do(T=t))$. When $T$ is binary, the ATE (given as $E(Y\mid do(T=1)) - E(Y\mid do(T=0))$ is a useful concept. But in fact the binary case is just one particular case. How to estimate $E(Y\mid do(T=t))$ depends on $Y$, $T$, and the functional dependency between them. The example you gave in the comment is that of a linear function, in which case one could perform linear regression. This is a commonly considered scenario. When interpreting a regression analysis of a continuous treatment and continuous outcome, the slope (in your example it is 2) conveys the causal effect and can be thought of as somewhat "analogous" to the ATE as it describes the change in the effect variable per unit of change in the treatment variable.

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  • $\begingroup$ Blame my ignorance, but could you tell what do is in the expressions? $\endgroup$
    – User1865345
    Commented Jul 10, 2023 at 8:25
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    $\begingroup$ the $do$ operator denotes the act of setting a random variable to a value (which is not the same as observing it to have a certain value – this distinction is what causality is all about). You could have a look at this Causal Inference tutorial for some examples connecting it to the ATE. If you're new to Causal Inference, the video is part of a series that may be a good place to start with. $\endgroup$
    – Scriddie
    Commented Jul 10, 2023 at 8:32
  • $\begingroup$ Thanks for that! It's beyond my field of specialization. But nice to know that. $\endgroup$
    – User1865345
    Commented Jul 10, 2023 at 8:37

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