# Causal inference learners

Given covariates $$X$$, treatment indicator $$W$$ and a binary outcome $$Y \in \{0,1\}$$.

## Inverse probability of treatment weights

Fit a model on observations $$X$$ to predict the treatment $$W$$, $$p_{w_i}(x_i)=P(W=w_i|X=x_i)$$ and use the probability of being treated as a sample weight to predict the outcome $$Y$$.

$IPTW_i=\frac{1}{p_{w_i}}$

## S-learner

A single model is trained on observations and treatments.

$\begin{eqnarray} \hat{\mu} = M(Y \sim (X,W))\\ \hat{\tau}(x)=\hat{\mu}(x,1) - \hat{\mu}(x,0) \end{eqnarray}$

## T-learner

Two models, one model per treatment, are trained on the observations only.

$\begin{eqnarray} \hat{\mu}_0 = M_0(Y^0 \sim X^0)\\ \hat{\mu}_1 = M_1(Y^1 \sim X^1)\\ \hat{\tau}(x)=\hat{\mu}_1(x)-\hat{\mu}_0(x) \end{eqnarray}$

## X-learner

The X-learner first estimates the response function ($$\hat{\mu}_0$$ and $$\hat{\mu}_1$$), then coomputes the imputed treatment effects ($$\tilde{D}_{i}^{1}$$ and $$\tilde{D}_{i}^{0}$$), then estimates the conditional average treatment effect for treated and controls ($$\hat{\tau}_1$$ and $$\hat{\tau}_0$$), and finally averages the estimates.

$\begin{eqnarray} \hat{\mu}_0 = M_1(Y^0 \sim X^0)\\ \hat{\mu}_1 = M_2(Y^1 \sim X^1)\\ \tilde{D}_{i}^{1}=Y_i^1 - \hat{\mu}_0(X_i^1)\\ \tilde{D}_{i}^{0}=\hat{\mu}_1(X_i^0) - Y_i^0\\ \hat{\tau}_1=M_3(\tilde{D}^{1} \sim X^1)\\ \hat{\tau}_0=M_4(\tilde{D}^{0} \sim X^0)\\ \hat{\tau}(x)=g(x)\hat{\tau}_0(x)+(1-g(x))\hat{\tau}_1(x) \end{eqnarray}$

$$g(x)\in[0,1]$$ is a weighting function which is chosen to minimize the variance of $$\hat{\tau}(x)$$. $$g(x)$$ can be estimated by the propsensity score or set to a constant and equal o the ratio of treated to untreated samples.

## G-Computation

Parameters we can derive from the G-Computation:

• For a binary outcome, $$Y\in\{0,1\}$$, $$P(Y=1)=E[Y]$$.
• $$E[Y^1]-E[Y^0]$$ is the causal risk difference due to treatment
• $$E[Y^1]/E[Y^0]$$ is the causal relative risk
• $$E[Y^1](1-E[Y^0])/E[Y^0](1-E[Y^1])$$ is the causal odds ratio due to treatment