Treatment Effect¶

Effects¶

Let \(\tau = y^1 - y^0\).

\(\tau\) will have a distribution because it is a random variable (\(\tau_1, \tau_2, \dots, \tau_i\))

$$ \begin{aligned} \pi_C + \pi_A &= p(T=1 \vert \text{Treatment group}) \ \pi_A + \pi_D &= p(T=1 \vert \text{Control group}) \ \pi_N + \pi_D &= p(T=0 \vert \text{Treatment group}) \ \pi_N + \pi_C &= p(T=0 \vert \text{Control group}) \end{aligned} $$ Usually, we assume that \(\pi_D=0\)

Why are there 3 different average variables? The people in each group is different. So, \(\tau\) for the entire group, treated and untreated groups are different, due to ‘selection effect’. This is like people who go to uni vs don’t. $$ \begin{aligned} y &= y^0 \cdot I(x=0) \times y^1 \cdot I(x=1)\ &\text{where \(I\) means if}\

\implies y &= f(x, y^0, y^1) \end{aligned} $$

ATE¶

• ATE = difference of the mean = mean of the difference $$ \begin{aligned} \text{ATE} =& E[y^1] - E[y^0] \ =& E[y^1 - y^0] \end{aligned} $$
• ATE = weighted average of ATT and ATU $$ \begin{aligned} \text{ATE} =& \text{ATT} \cdot P(x=1) + \text{ATU} \cdot P(x=0) \ =& E(y^1 - y^0 | x = 1) \cdot P(x=1) \ &+ E(y^1 - y^0 | x = 0) \cdot P(x=0) \ =& \int E[y^1 - y^0 \vert s] \cdot p(s) \cdot ds \end{aligned} $$ This reminds me of the total probability like in Bayes’ conditional probability. But here, we are taking expectation \(E\) (mean), because it’ll more accurate than taking one value from the PDF, as \(\tau\) is a random variable
• ATE = Weighted average of CATE, given unconfoundedness: proving that treatment randomly assigned within each group \(z_i\)

$$ \widehat{\text{ATE}} = \sum_i \widehat{\text{CATE}}_i \cdot P(Z=z_i) $$ - ATE = ATT + Selection Bias - Randomization makes selection bias 0

ATE = Causal Effect $$ \begin{aligned} E(y \vert \text{do}(x), s) &= E(y \vert x, s) \ \implies \widehat{\text{ATE}}(x, s) &=\dfrac{d}{dx} \hat E[y \vert \text{do}(x)] \ &=\dfrac{d}{dx} \hat E[y \vert \text{do}(x), s] \ &= \dfrac{d}{dx} E_s \Big[ \hat E[y \vert \text{do}(x), s] \Big] \ &= \dfrac{d}{dx} E_s \Big[ \hat E[y \vert x, s] \Big] \ &= E_s \left[ \dfrac{\partial}{\partial x} \hat E[y \vert x, s] \right] \end{aligned} $$

HTE = Heterogeneous Treatment Effect = ATE with high dimensional \(s\)

If \(x\) is binary $$ \begin{aligned} x &\in { 0, 1 } \ \implies \text{ATE}(x, s) &= E[ y \vert \text{do}(x=1) ] - E[ y \vert \text{do}(x=0) ] \ &= E_s \Bigg[ E[y \vert x=1, s ] - E[ y \vert x=0, s] \Bigg] \ \end{aligned} $$

IDK¶

	\(E[y^1 - y^0]\)	\(E[y \vert x=1] - E[y \vert x=0]\)
Compares what would happen if the __ sample receives treatment \(x=1\) vs \(x=0\)	same	2 different
Provides	Average causal effect	Average difference in outcome b/w sub populations defined by treatment group

IDK¶

$$ \begin{aligned}

\text{ATT} &= E(y^1 - y^0 | x = 1) \ &= \underbrace{E(y^1 | x = 1)}{E(y | x = 1)} - \underbrace{E(y^0 | x = 1)} \ \text{ATU} &= E(y^1 - y^0 | x = 0) \ &= \underbrace{E(y^1 | x = 0)}}{\text{Cannot be estimated}} - \underbrace{E(y^0 | x = 0)} \ %{ %%\text{Similarly,} &\ %%\text{ATE} %%&= \underbrace{E(y^1 | x = 0)}{\text{Cannot be estimated}} - %%\underbrace{E(y^0 | x = 0)} %} \end{aligned} $$

Solution: Randomized Treatment

PDF Graph¶

$$ \begin{aligned} \text{ATE}(x) &= \int \tau(x, y) P(y) dy \ &= \frac{ dE[y| \text{ do}(x)] }{dx} \

T(x, y) &= \frac{ \partial P(y | \text{do}(x)) }{ \partial x } \ \end{aligned} $$

We could also interpret this entire distribution as a 3 variable joint PDF of the form \(P(x, y^0, y^1)\)

IDK¶

Here

• Modelling the ITE is correct
• Modelling \(y\) vs \(x\) is incorrect

Probabilistic¶

Causal effect of a treatment is a probability distribution: it is not the same for every individual.

• Learning the individual-level is nearly impossible
• Learning the pdf of the effect is hard

Usually, the Average Treatment Effect is used

But better to compare the distributions of the treatment and control group

Heterogeneous Treatment Effects¶

If we consider effect modifier \(s\) $$ \begin{aligned} \text{ATE} &= \dfrac{\partial}{\partial x} E[ y \vert \text{do}(x), s] \ &= \int \limits_s E[y^1 - y^0 | s] \cdot P(s) \cdot ds \ \end{aligned} $$

where \(P(s)\) is the distribution of effect modifier.

Then, the result of the randomized test actually gives us \(E[\tilde \tau]\), which may not be equal to \(E[\tau]\)

Example¶

For example, the yield depends on the season and the crop for which was grown previous year. Let’s take the example of a a randomized test of a fertilizer used in the summer.

If no crop was grown the previous year, then we

Comments