Outcome regression

In this post, we consider the task of estimating treatment effects. This is the basic problem in causal inference, and it arises in a many areas of science and engineering. As a running example, we consider the task of estimating the efficacy of a vaccine booster. We begin by mathematically defining treatment effects using the potential outcomes framework.

To keep things simple, we focus on estimating the effect of a binary treatment (e.g. booster vs no booster). We define two potential outcomes $Y_i(1)$ and $Y_i(0)$ for each subject in the study. In the running example, $Y_i(1)$ is the viral load in the $i$-subject if the subject got the booster, and $Y_i(0)$ is the viral load if the subject did not get the booster. The effect of the treatment on the $i$-th subject is

$${\mathrm{\Delta }}_i\triangleq Y_i(1)-Y_i(0).$$

The fundamental challenge in causal inference is only one treatment can be assigned to a subject, so only one of $Y_i(1)$ and $Y_i(0)$ can be observed. Thus ${\mathrm{\Delta }}_i$ is never observed. Nevertheless, it is possible (as we shall see) to estimate the average treatment effect (ATE)

$\tau \triangleq \mathbf{E}[{\mathrm{\Delta }}_i]=\mathbf{E}[Y_i(1)]-\mathbf{E}[Y_i(0)]$ by performing randomized experiments.

In a randomized experiment, we randomly assign treatments to the subjects and record the outcomes. Let $W_i\in \{0,1\}$ and $Y_i$ be the treatment assignment and observed outcome of the $i$-th subject. In the running example, $W_i$ indicates whether the $i$-th subject got the booster and $Y_i$ is the (observed) viral load in the $i$-th subject. Mathematically, in a randomized experiment, we have

$$\begin{array}{rlr}{Y}_i=Y_i(W_i)& & \text{(SUTVA),}\\ (Y_i(1),Y_i(0))\perp \perp W_i& & \text{(random treatment assignment).}\end{array}$$

The first condition (SUTVA) relates the observed outcomes to the potential outcomes: the observed outcome of the $i$-subject $Y_i$ is $Y_i(1)$ (resp $Y_i(0)$) if $W_i=1$ (resp $W_i=0$). The second condition says treatments are assigned in a way that does not depend on the potential outcomes. It implies the distribution of potential outcomes in the treated and untreated groups are identical:

$$(Y_i(1),Y_i(0))\mid \{W_i=1\}\stackrel{d}{=}(Y_i(1),Y_i(0))\mid \{W_i=0\}.$$

In practice, treatments are often assigned randomly (e.g. by flipping a coin) to satisfy this condition.

Difference-in-means

A simple estimate of the ATE in a randomized experiment is the difference between the (sample) mean outcomes in treated and untreated subjects:

$${\hat{\tau }}_{\text{DM}}=\frac{1}{n_1}\sum _{i=1}^n{Y}_i\mathbf{1}\{W_i=1\}-\frac{1}{n_0}\sum _{i=1}^n{Y}_i\mathbf{1}\{W_i=0\},$$

where $n_w\triangleq \sum _{i=1}^n\mathbf{1}\{W_i=w\}$ is the number of subjects assigned treatment $w\in{0,1}$. This is called the difference-in-means estimator, and it is motivated by the observation that the (sample) mean outcome in a treatment group is an unbiased estimate of the expected potential outcome in a randomized experiment:

$$\begin{array}{rl}& \mathbf{E}[\frac{1}{n_w}\sum _{i=1}^n{Y}_i\mathbf{1}\{W_i=w\}]\\ & =\mathbf{E}[Y_i\mid W_i=w]\\ & =\mathbf{E}[Y_i(w)\mid W_i=w]& & \text{(SUTVA)}\\ & =\mathbf{E}[Y_i(w)]& & \text{(random treatment assignment)}.\end{array}$$

In light of this observation, it is not hard to see that the difference-in-means estimator is unbiased:

$$\begin{array}{rl}\mathbf{E}[{\hat{\tau }}_{\text{DM}}]& =\mathbf{E}[\frac{1}{n_1}\sum _{i=1}^n{Y}_i\mathbf{1}\{W_i=1\}]-\mathbf{E}[\frac{1}{n_0}\sum _{i=1}^n{Y}_i\mathbf{1}\{W_i=0\}]\\ & =\mathbf{E}[Y_i(1)]-\mathbf{E}[Y_i(0)]\\ & =\tau .\end{array}$$

We leave as an exercise to show that ${\hat{\tau }}_{\text{DM}}$ is asymptotically normal:

$$\sqrt{n_1+n_0}({\hat{\tau }}_{\text{DM}}-\tau )\stackrel{d}{\to }N(0,\frac{{\sigma }_1^2}{{\pi }_1}+\frac{{\sigma }_0^2}{{\pi }_0}),$$

where ${\sigma }_w^2\triangleq \mathbf{v}\mathbf{a}\mathbf{r}[Y_i(w)]$ and ${\pi }_w\triangleq \mathbf{P}\{W_i=w\}$ for $w\in \{0,1\}$. This result allows us to form confidence intervals and test hypothesis regarding the ATE.

Posted on December 11, 2021 from Ann Arbor, MI