Extensions: Recent research

The ideas above are the core of many advances that involve straightforward extensions. This page summarizes a few of my own recent or ongoing projects.

What if the treatment is continuous?

This methodological project is Lundberg and Brand (in progress).

What if the treatment $A$ is continuous? For example, we may want to know how a child’s probability of college enrollment $Y$ would change if their family income $A$ at age 17 took a different value. When the treatment is continuous, each unit has many potential outcomes (under every possible $A$ value) instead of 2 (under treatment and control).

ID	Treatment $A$	$Y^{\$20\text{k}}$	$Y^{\$40\text{k}}$	$Y^{\$60\text{k}}$	$Y^{\$80\text{k}}$
1	$40	?	1	?	?
2	$20	0	?	?	?
3	$80	?	?	?	1

In ongoing work with Jennie Brand, we reason about what causal estimand to study here and how to estimate it.

How would you define the causal estimand here?

We choose an additive shift estimand: how much would the college enrollment rate rise on average under an intervention to increase family income by $\delta = \$10,000$? \[ \text{E}(Y^{A + \$10k}) \]

An advantage of this estimand is that it avoids extrapolation: if person $i$ is observed with income $A_i$, it is likely that we observe other people with incomes near $A_i + \$10,000$.

How would you conduct estimation?

We chose outcome modeling.

Model $Y$ given $A$ and $\vec{X}$
Predict $Y_i$ and $Y_i^{A_i + \delta}$ for each $i$
Difference and average over the sample

We have realized since working on this project that our approach is an application of a general class of estimands developed in biostatistics for modified treatment policies. See Haneuse & Rotnitzky (2013).

What if the treatment is categorical?

This project is Lundberg, Molitor, and Brand (2025).

If a categorical treatment has $k$ values, then every person has $k$ potential outcomes instead of just two. Otherwise, this is just like a binary treatment!

How would you use an outcome model to estimate the outcome if a categorical treatment $A$ takes the value 3?

Here is how we did it:

Define the potential outcomes $Y^1$, $Y^2$, $Y^3$, etc.
Draw a DAG to find a sufficient adjustment set $\vec{X}$
Model $Y$ given $A$ and $\vec{X}$
For each person $i$, set $A_i = 3$
Predict $\hat{Y}_i^3$ and average over people.

One challenge that arises with categorical treatments is that some treatment values may be unlikely to occur in practice for people with some confounder values. For example, if the treatment is being a secondary school teacher then this treatment is impossible for people whose educational attainment was less than high school (unless we also intervened on their educational attainment). The paper above focuses considerably on this problem.

What if the outcome does not exist for some people?

This ongoing project with Soonhong Cho has a preprint (Lundberg & Cho, in progress) and an R package in development which includes pedagogical website: ilundberg.github.io/pstratreg

Some outcomes only exist for some people. We may want to know how motherhood $A$ affects hourly wage $Y$. But people only have a wage if they are employed $M = 1$. Employment is a post-treatment variable that may be caused by motherhood, so we cannot condition on employment. What causal estimand makes sense here?

What estimand would you choose?

We first study the causal effect on outcome existence, \[ \text{E}(M^1-M^0) \] which corresponds here to the causal effect of motherhood on employment. Then, we define a causal estimand that looks a bit complicated:

\[ \text{E}(Y^1-Y^0\mid M^1=M^0=1) \]

This is the average effect of motherhood on wage ($Y^1-Y^0$) among those who would be employed regardless of motherhood ($M^1=M^0=1$). This estimand is difficult because it conditions on potential outcomes that are only partially observed ($M^1$ and $M^0$), so we can only identify this estimand with a set.

This ongoing work builds on principal stratification. To learn more about principal stratification, a few good papers are this introduction to principal stratification (Miratrix et al. 2018) and an application to the study of racially biased policing (Knox et al. 2020). For foundational statistics papers, see Zhang & Rubin (2003), Lee (2009) and Semenova (2025).

What if I am interested in the causes of a disparity?

This methodological project is Lundberg (2024), an application paper Lundberg (2025), and the R package gapclosing.

We usually ask questios about average treatment effects $\text{E}(Y^1-Y^0)$: how would an outcome change on average if a treatment were modified? But there are many other causal questions one can ask with potential outcomes. One is: what gap between subgroups $X = 0$ and $X = 1$ would remain if we set the treatment to the value $A = 1$?

\[ \text{Gap-Closing Estimand:}\qquad \text{E}(Y^1\mid X = 1) - \text{E}(Y^1\mid X = 0)$ \]

Given what you have learned above, you can probably answer the question: how would we estimate that with an outcome modeling estimator?

Click to show answer

Draw a DAG.
Find adjustment set $\vec{Z}$ such that conditional exchangeability holds: $Y^1\unicode{x2AEB} A\mid Z$
Model $Y$ given treatment $A$ and confounders $\vec{Z}$
Predict $\hat{Y}^1$ for everyone
Average within $X$

This project drew on methods developed in biostatistics that decompose disparities by their causal inputs, such as Jackson & Vanderweele (2018).

ID	Treatment \(A\)	\(Y^{\$20\text{k}}\)	\(Y^{\$40\text{k}}\)	\(Y^{\$60\text{k}}\)	\(Y^{\$80\text{k}}\)
1	$40	?	1	?	?
2	$20	0	?	?	?
3	$80	?	?	?	1