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Binary treatments can be seen as special case of a much broader class of causal models

The dummy treatment variable and the measures it gives rise to such as ATE average treatment effect are very convenient and some of the most frequently used ideas from causality in practice. This is also the setting that is most often used to introduce the potential outcomes framework [Rubin 1974]. They can, however, be seen as a special case of (linear) structural causal models (SCM) [Pearl 2000] which allow for all kinds of causal relationships (see also this question).

Imagine for example the following causal model $X\to Y$ with exogenous noise terms $N$:

$X = N_X;\quad Y = X + N_Y$.

If $X$ only takes the values 0 and 1, it is very useful to define the ATE $E(Y\mid X=1) - E(Y\mid X=0)$. In your application, $X=1$ could indicate the presence of UNPKO, and $X=0$ its absence.

If you suspect that there is a relationship between the number of deployed troops and outcome $Y$, we can redefine $X$ to be the number of deployed troops (e.g. in 10,000), and estimate the functional dependency between $X$ and $Y$, for example using a linear model:

$Y = \beta X + N_Y$,

where $\beta$ is now the target parameter of our estimation. How to correctly estimate $\beta$ in more complex causal models is one of the primary questions in causal inference (e.g. by finding appropriate adjustment sets). Such a setup is standard practice in disciplines such es econometrics and economic history. Looking at some of those papers (e.g. this classic by Acemoglu et al.) might be a good source of inspiration and also of the difficulties that arise in these kind of statistical designs.

In principal, such models would not have to be linear, and the noise need not be additive, etc.. The statistical theory of structural causal models is very well developed for more general model classes (see e.g. Section 3 in the Elements of Causal Inference). In practice however, simplifying assumptions like linearity or additive and homogenous noise are common.

Binary treatments can be seen as special case of a much broader class of causal models

The dummy treatment variable and the measures it gives rise to such as ATE average treatment effect are very convenient and some of the most frequently used ideas from causality in practice. This is also the setting that is most often used to introduce the potential outcomes framework [Rubin 1974]. They can, however, be seen as a special case of (linear) structural causal models [Pearl 2000] which allow for all kinds of causal relationships (see also this question).

Imagine for example the following causal model $X\to Y$ with exogenous noise terms $N$:

$X = N_X;\quad Y = X + N_Y$.

If $X$ only takes the values 0 and 1, it is very useful to define the ATE $E(Y\mid X=1) - E(Y\mid X=0)$. In your application, $X=1$ could indicate the presence of UNPKO, and $X=0$ its absence.

If you suspect that there is a relationship between the number of deployed troops and outcome $Y$, we can redefine $X$ to be the number of deployed troops (e.g. in 10,000), and estimate the functional dependency between $X$ and $Y$, for example using a linear model:

$Y = \beta X + N_Y$,

where $\beta$ is now the target parameter of our estimation. How to correctly estimate $\beta$ in more complex causal models is one of the primary questions in causal inference (e.g. by finding appropriate adjustment sets). Such a setup is standard practice in disciplines such es econometrics and economic history. Looking at some of those papers (e.g. this classic by Acemoglu et al.) might be a good source of inspiration and also of the difficulties that arise in these kind of statistical designs.

In principal, such models would not have to be linear, and the noise need not be additive, etc.. The statistical theory of structural causal models is very well developed for more general model classes (see e.g. Section 3 in the Elements of Causal Inference). In practice however, simplifying assumptions like linearity or additive and homogenous noise are common.

Binary treatments can be seen as special case of a much broader class of causal models

The dummy treatment variable and the measures it gives rise to such as ATE average treatment effect are very convenient and some of the most frequently used ideas from causality in practice. This is also the setting that is most often used to introduce the potential outcomes framework [Rubin 1974]. They can, however, be seen as a special case of (linear) structural causal models (SCM) [Pearl 2000] which allow for all kinds of causal relationships (see also this question).

Imagine for example the following causal model $X\to Y$ with exogenous noise terms $N$:

$X = N_X;\quad Y = X + N_Y$.

If $X$ only takes the values 0 and 1, it is very useful to define the ATE $E(Y\mid X=1) - E(Y\mid X=0)$. In your application, $X=1$ could indicate the presence of UNPKO, and $X=0$ its absence.

If you suspect that there is a relationship between the number of deployed troops and outcome $Y$, we can redefine $X$ to be the number of deployed troops (e.g. in 10,000), and estimate the functional dependency between $X$ and $Y$, for example using a linear model:

$Y = \beta X + N_Y$,

where $\beta$ is now the target parameter of our estimation. How to correctly estimate $\beta$ in more complex causal models is one of the primary questions in causal inference (e.g. by finding appropriate adjustment sets). Such a setup is standard practice in disciplines such es econometrics and economic history. Looking at some of those papers (e.g. this classic by Acemoglu et al.) might be a good source of inspiration and also of the difficulties that arise in these kind of statistical designs.

In principal, such models would not have to be linear, and the noise need not be additive, etc.. The statistical theory of structural causal models is very well developed for more general model classes (see e.g. Section 3 in the Elements of Causal Inference). In practice however, simplifying assumptions like linearity or additive and homogenous noise are common.

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Binary treatments can be seen as special case of a much broader class of causal models

The dummy treatment variable and the measures it gives rise to such as ATE average treatment effect are very convenient and some of the most frequently used ideas from causality in practice. This is also the setting that is most often used to introduce the potential outcomes framework [Rubin 1974]. They can, however, be seen as a special case of (linear) structural causal models [Pearl 2000] which allow for all kinds of causal relationships (see also this question).

Imagine for example the following causal model $X\to Y$ with exogenous noise terms $N$:

$X = N_X;\quad Y = X + N_Y$.

If $X$ only takes the values 0 and 1, it is very useful to define the ATE $E(Y\mid X=1) - E(Y\mid X=0)$. In your application, $X=1$ could indicate the presence of UNPKO, and $X=0$ its absence.

If you suspect that there is a relationship between the number of deployed troops and outcome $Y$, we can redefine $X$ to be the number of deployed troops (e.g. in 10,000), and estimate the functional dependency between $X$ and $Y$, for example using a linear model:

$Y = \beta X + N_Y$,

where $\beta$ is now the target parameter of our estimation. ThisHow to correctly estimate $\beta$ in more complex causal models is one of the primary questions in causal inference (e.g. by finding appropriate adjustment sets). Such a setup is standard practice in disciplines such es econometrics and economic history. Looking at some of those papers (e.g. this classic by Acemoglu et al.) might be a good source of inspiration and also of the difficulties that arise in these kind of statistical designs.

In principal, such models would not have to be linear, and the noise need not be additive, etc.. The statistical theory of structural causal models is very well developed for more general model classes (see e.g. Section 3 in the Elements of Causal Inference). In practice however, simplifying assumptions like linearity or additive and homogenous noise are common.

Binary treatments can be seen as special case of a much broader class of causal models

The dummy treatment variable and the measures it gives rise to such as ATE average treatment effect are very convenient and some of the most frequently used ideas from causality in practice. This is also the setting that is most often used to introduce the potential outcomes framework [Rubin 1974]. They can, however, be seen as a special case of (linear) structural causal models [Pearl 2000] which allow for all kinds of causal relationships (see also this question).

Imagine for example the following causal model $X\to Y$ with exogenous noise terms $N$:

$X = N_X;\quad Y = X + N_Y$.

If $X$ only takes the values 0 and 1, it is very useful to define the ATE $E(Y\mid X=1) - E(Y\mid X=0)$. In your application, $X=1$ could indicate the presence of UNPKO, and $X=0$ its absence.

If you suspect that there is a relationship between the number of deployed troops and outcome $Y$, we can redefine $X$ to be the number of deployed troops (e.g. in 10,000), and estimate the functional dependency between $X$ and $Y$, for example using a linear model:

$Y = \beta X + N_Y$,

where $\beta$ is now the target parameter of our estimation. This is standard practice in disciplines such es econometrics and economic history. Looking at some of those papers (e.g. this classic by Acemoglu et al.) might be a good source of inspiration and also of the difficulties that arise in these kind of statistical designs.

In principal, such models would not have to be linear, and the noise need not be additive, etc.. The statistical theory of structural causal models is very well developed for more general model classes (see e.g. Section 3 in the Elements of Causal Inference). In practice however, simplifying assumptions like linearity or additive and homogenous noise are common.

Binary treatments can be seen as special case of a much broader class of causal models

The dummy treatment variable and the measures it gives rise to such as ATE average treatment effect are very convenient and some of the most frequently used ideas from causality in practice. This is also the setting that is most often used to introduce the potential outcomes framework [Rubin 1974]. They can, however, be seen as a special case of (linear) structural causal models [Pearl 2000] which allow for all kinds of causal relationships (see also this question).

Imagine for example the following causal model $X\to Y$ with exogenous noise terms $N$:

$X = N_X;\quad Y = X + N_Y$.

If $X$ only takes the values 0 and 1, it is very useful to define the ATE $E(Y\mid X=1) - E(Y\mid X=0)$. In your application, $X=1$ could indicate the presence of UNPKO, and $X=0$ its absence.

If you suspect that there is a relationship between the number of deployed troops and outcome $Y$, we can redefine $X$ to be the number of deployed troops (e.g. in 10,000), and estimate the functional dependency between $X$ and $Y$, for example using a linear model:

$Y = \beta X + N_Y$,

where $\beta$ is now the target parameter of our estimation. How to correctly estimate $\beta$ in more complex causal models is one of the primary questions in causal inference (e.g. by finding appropriate adjustment sets). Such a setup is standard practice in disciplines such es econometrics and economic history. Looking at some of those papers (e.g. this classic by Acemoglu et al.) might be a good source of inspiration and also of the difficulties that arise in these kind of statistical designs.

In principal, such models would not have to be linear, and the noise need not be additive, etc.. The statistical theory of structural causal models is very well developed for more general model classes (see e.g. Section 3 in the Elements of Causal Inference). In practice however, simplifying assumptions like linearity or additive and homogenous noise are common.

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Scriddie
Scriddie
  • 2.4k
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  • 15

Binary treatments can be seen as special case of a much broader class of causal models

The dummy treatment variable and the measures it gives rise to such as ATE average treatment effect are very convenient and some of the most frequently used ideas from causality in practice. This is also the setting that is most often used to introduce the potential outcomes framework [Rubin 1974]. They can, however, be seen as a special case of (linear) structural causal models [Pearl 2000] which allow for all kinds of causal relationships (see also this question).

Imagine for example the following causal model $X\to Y$ with exogenous noise terms $N$:

$X = N_X;\quad Y = X + N_Y$.

If $X$ only takes the values 0 and 1, it is very useful to define the ATE $E(Y\mid X=1) - E(Y\mid X=0)$. In your application, $X=1$ could indicate the presence of UNPKO, and $X=0$ its absence.

If you suspect that there is a relationship between the number of deployed troops and outcome $Y$, we can redefine $X$ to be the number of deployed troops (e.g. in 10,000), and estimate the functional dependency between $X$ and $Y$, for example using a linear model:

$Y = \beta X + N_Y$.,

where $\beta$ is now the target parameter of our estimation. This is standard practice in disciplines such es econometrics and economic history. Looking at some of those papers (e.g. this classic by Acemoglu et al.) might be a good source of inspiration and also of the difficulties that arise in these kind of statistical designs.

In principal, such models would not have to be linear, and the noise need not be additive, etc.. The statistical theory of structural causal models is very well developed for more general model classes (see e.g. Section 3 in the Elements of Causal Inference). In practice however, simplifying assumptions like linearity or additive and homogenous noise are common.

Binary treatments can be seen as special case of a much broader class of causal models

The dummy treatment variable and the measures it gives rise to such as ATE average treatment effect are very convenient and some of the most frequently used ideas from causality in practice. This is also the setting that is most often used to introduce the potential outcomes framework [Rubin 1974]. They can, however, be seen as a special case of (linear) structural causal models [Pearl 2000] which allow for all kinds of causal relationships (see also this question).

Imagine for example the following causal model $X\to Y$ with exogenous noise terms $N$:

$X = N_X;\quad Y = X + N_Y$.

If $X$ only takes the values 0 and 1, it is very useful to define the ATE $E(Y\mid X=1) - E(Y\mid X=0)$. In your application, $X=1$ could indicate the presence of UNPKO, and $X=0$ its absence.

If you suspect that there is a relationship between the number of deployed troops and outcome $Y$, we can redefine $X$ to be the number of deployed troops (e.g. in 10,000), and estimate the functional dependency between $X$ and $Y$, for example using a linear model:

$Y = \beta X + N_Y$.

This is standard practice in disciplines such es econometrics and economic history. Looking at some of those papers (e.g. this classic by Acemoglu et al.) might be a good source of inspiration and also of the difficulties that arise in these kind of statistical designs.

In principal, such models would not have to be linear, and the noise need not be additive, etc.. The statistical theory of structural causal models is very well developed for more general model classes (see e.g. Section 3 in the Elements of Causal Inference). In practice however, simplifying assumptions like linearity or additive and homogenous noise are common.

Binary treatments can be seen as special case of a much broader class of causal models

The dummy treatment variable and the measures it gives rise to such as ATE average treatment effect are very convenient and some of the most frequently used ideas from causality in practice. This is also the setting that is most often used to introduce the potential outcomes framework [Rubin 1974]. They can, however, be seen as a special case of (linear) structural causal models [Pearl 2000] which allow for all kinds of causal relationships (see also this question).

Imagine for example the following causal model $X\to Y$ with exogenous noise terms $N$:

$X = N_X;\quad Y = X + N_Y$.

If $X$ only takes the values 0 and 1, it is very useful to define the ATE $E(Y\mid X=1) - E(Y\mid X=0)$. In your application, $X=1$ could indicate the presence of UNPKO, and $X=0$ its absence.

If you suspect that there is a relationship between the number of deployed troops and outcome $Y$, we can redefine $X$ to be the number of deployed troops (e.g. in 10,000), and estimate the functional dependency between $X$ and $Y$, for example using a linear model:

$Y = \beta X + N_Y$,

where $\beta$ is now the target parameter of our estimation. This is standard practice in disciplines such es econometrics and economic history. Looking at some of those papers (e.g. this classic by Acemoglu et al.) might be a good source of inspiration and also of the difficulties that arise in these kind of statistical designs.

In principal, such models would not have to be linear, and the noise need not be additive, etc.. The statistical theory of structural causal models is very well developed for more general model classes (see e.g. Section 3 in the Elements of Causal Inference). In practice however, simplifying assumptions like linearity or additive and homogenous noise are common.

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Scriddie
Scriddie
  • 2.4k
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