We unmask heterogeneity, finding out how CATE learners help target interventions to those who will benefit most

The Journey Beyond Average Effects

In the vast landscape of causal inference, we’ve long relied on a simple compass: the Average Treatment Effect (ATE). Like ancient mariners navigating by a single star, researchers across disciplines have used this average to guide important decisions. But what if I told you that this single metric—this lone star—only reveals a fraction of the story?

Imagine prescribing the same medication to everyone because clinical trials showed a positive average effect. Some patients would thrive, others might see no benefit, and some could even be harmed. The average masks this crucial variation—the heterogeneity that matters most when making decisions for individuals.

This is where our hero enters the narrative: the Conditional Average Treatment Effect (CATE). CATE doesn’t just tell us whether a treatment works on average; it reveals for whom it works and by how much. It’s the difference between a blurry photograph and a high-definition image—both capture the same scene, but one reveals the crucial details that make all the difference.

Let’s embark on this journey together, exploring how modern machine learning methods have revolutionized our ability to uncover these heterogeneous effects, transforming how we approach everything from personalized medicine to targeted policy interventions.

The Causal Inference Landscape: Setting the Stage

Before we dive into the world of CATE, let’s establish our foundation. Causal inference attempts to answer the fundamental question: “What is the effect of X on Y?” Not merely the association, but the causal impact—what happens when we intervene?

The gold standard for causal inference has long been the randomized controlled trial (RCT), where subjects are randomly assigned to treatment or control groups. This randomization helps ensure that the only systematic difference between groups is the treatment itself, allowing us to attribute differences in outcomes to the treatment.

In the classical framework, we define the potential outcomes for each individual i:

• $Y_i(1)$: The outcome if the individual receives treatment
• $Y_i(0)$: The outcome if the individual does not receive treatment

The causal effect for individual i is then:

But here lies the “fundamental problem of causal inference”—we can never observe both potential outcomes for the same individual. We either treat them or we don’t; we can’t do both and compare.

This limitation led researchers to focus on average effects:

While useful, the ATE treats everyone as identical, averaging across potentially important differences between individuals. This brings us to CATE, which acknowledges that treatment effects may vary based on observable characteristics $X$:

CATE asks: “What is the expected treatment effect for individuals with characteristics $X = x$?” This simple yet profound shift opens the door to a more nuanced understanding of causal effects.

The Birth of CATE Learners: From Classical Statistics to Machine Learning

The story of CATE estimation begins in traditional statistics but gains momentum with the rise of machine learning. Classical approaches to capturing effect heterogeneity included:

1. Subgroup analysis: Estimating effects separately for different subgroups
2. Interaction terms: Including interaction terms between treatment and covariates in regression models

For example, a linear model with interaction terms might look like:

Where $\beta_3$ captures how the treatment effect varies with $X$.

These methods, while intuitive, face significant limitations:

• They require pre-specifying which interactions to include
• They struggle with high-dimensional data and complex non-linear relationships
• They risk overfitting and false discoveries when testing multiple subgroups

Enter machine learning, with its ability to handle complex, high-dimensional data and automatically discover patterns. The marriage of causal inference and machine learning gave birth to a new generation of methods specifically designed to estimate CATE.

The Meta-Learners: A Framework for CATE Estimation

Meta-learners have become a cornerstone of modern CATE estimation. These techniques serve as frameworks that leverage existing machine learning algorithms (called base learners) to estimate CATE, either using a single base learner with the treatment indicator as a feature, or multiple base learners separately for treatment and control groups.

Let’s explore the most prominent meta-learners:

S-Learner: The Single Model Approach

The S-Learner (Single Learner) uses one model for both treated and control groups, including the treatment as a feature:

The CATE is then estimated as:

This approach is simple but has limitations. When treatment and control groups differ substantially in their covariate distributions, a single model may struggle to capture the different relationships between covariates and outcomes across groups.

T-Learner: The Two-Model Approach

The T-Learner (Two Learner) is an intuitive two-step approach that fits separate models for treatment and control groups:

The CATE is then:

This approach allows different relationships between covariates and outcomes across groups but may suffer from higher variance in finite samples.

Let’s visualize this with a diagram:

X-Learner: The Cross-Estimation Approach

The X-Learner (Cross-Learner) was introduced to address scenarios where the number of units in one treatment group is much larger than in the other. It can exploit structural properties of the CATE function and achieve better performance when there’s imbalance between treatment and control groups.

The X-Learner works in multiple stages:

1. Estimate separate outcome models for treatment and control groups:

   $$\hat{\mu}_1(x) = E[Y|X=x, T=1]$$
   $$\hat{\mu}_0(x) = E[Y|X=x, T=0]$$

2. Compute “imputed treatment effects” for each individual:

   $$D_i^1 = Y_i - \hat{\mu}_0(X_i)$$

   for treated individuals

   $$D_i^0 = \hat{\mu}_1(X_i) - Y_i$$

   for control individuals

3. Estimate the treatment effect functions using these imputed effects:

   $$\hat{\tau}_1(x) = E[D^1|X=x]$$
   $$\hat{\tau}_0(x) = E[D^0|X=x]$$

4. Combine these estimates using the propensity score:

   $$\hat{\tau}(x) = g(x)\hat{\tau}_0(x) + (1-g(x))\hat{\tau}_1(x)$$

   where $g(x)$ is often chosen as the propensity score.

R-Learner: The Residual-on-Residual Approach

The R-Learner is a meta-algorithm that leverages a powerful insight: if we know the propensity score $e(x)$ and the outcome prediction function $m(x)$, we can recover the CATE by regressing the residuals of the outcome on the residuals of treatment.

The R-Learner uses the following loss function:

where:

• $m(X_i) = E[Y|X=x]$ is the marginal outcome regression
• $e(X_i) = P(W=1|X=x)$ is the propensity score
• $\tau(X_i)$ is the CATE function

This approach is particularly effective when the CATE function is simpler than the outcome or propensity functions.

DR-Learner: The Doubly-Robust Approach

The DR-Learner (Doubly-Robust Learner) estimates the CATE via cross-fitting a doubly-robust score function in two stages. This approach randomly splits the data into three partitions, using different partitions to fit propensity score models, outcome regression models, and finally the CATE model.

The DR-Learner uses the doubly-robust score function:

This approach is more robust to misspecification than other meta-learners, as it only requires either the propensity score model or the outcome regression model to be correctly specified, not both.

Specialized ML Methods for CATE

In addition to meta-learners, several specialized machine learning methods have been developed specifically for CATE estimation:

Causal Forests: Adapting Random Forests for Causal Inference

Causal forests are a modification of random forests designed to estimate heterogeneous treatment effects. The standard approach for building decision trees is adapted to focus on estimating treatment effects rather than predicting outcomes.

In conventional random forests, splits are chosen to maximize the reduction in outcome variance. In causal forests, splits are chosen to maximize the variance in treatment effects between leaves, identifying subgroups with different responses to treatment.

Let’s visualize the basic idea behind causal forests:

Causal BART: Bayesian Additive Regression Trees for Causal Inference

Causal BART (Bayesian Additive Regression Trees) is a nonparametric Bayesian regression approach that uses dimensionally adaptive random basis elements. It’s a “sum-of-trees” model where each tree is constrained by a regularization prior to be a weak learner, and fitting and inference are accomplished via an iterative Bayesian backfitting MCMC algorithm.

The BART approach provides several advantages for CATE estimation:

1. It can naturally incorporate uncertainty in both the outcome and treatment effect estimates
2. It automatically handles complex, non-linear relationships
3. It provides full posterior distributions for inference
4. It can handle high-dimensional feature spaces effectively

BART builds many small decision trees and combines them to form a powerful ensemble. The trees are kept simple through prior constraints, preventing any single tree from dominating the model and promoting diversity in the ensemble.

Neural Network Approaches for CATE

Recent advancements have also brought neural network-based approaches to CATE estimation. These methods leverage the powerful representation learning capabilities of neural networks to address challenges in causal inference.

Some notable approaches include:

1. Representation Learning: These methods learn representations of the covariates that balance the treatment and control groups, attempting to approximate the counterfactual outcomes.

2. Targeted Regularization: Adding regularization terms to the loss function that encourage the model to learn representations that are predictive of the outcome but not predictive of the treatment assignment.

3. Adversarial Training: Using adversarial networks to ensure that the learned representations cannot distinguish between treatment and control groups.

Let’s look at a simple example of a neural network architecture for CATE estimation:

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# pseudocode for a simple neural network CATE estimator
import tensorflow as tf

def build_cate_model(input_dim, hidden_layers=[64, 32]):
    # Shared representation network
    inputs = tf.keras.Input(shape=(input_dim,))
    x = inputs
    
    # Hidden layers for shared representation
    for units in hidden_layers:
        x = tf.keras.layers.Dense(units, activation='relu')(x)
    
    # Separate output heads for treatment and control
    treatment_output = tf.keras.layers.Dense(1)(x)
    control_output = tf.keras.layers.Dense(1)(x)
    
    # CATE is the difference
    cate = tf.keras.layers.Subtract()([treatment_output, control_output])
    
    # Create the model
    model = tf.keras.Model(inputs=inputs, outputs=[treatment_output, control_output, cate])
    
    return model

# Training would involve predicting outcomes for treated and control samples
# and optimizing the network's parameters

Putting CATE Into Action

Now that we’ve covered the theoretical foundations and methods, let’s explore how these techniques are implemented in practice.

Key Libraries and Tools

Several libraries have been developed to make CATE estimation accessible:

1. EconML: Microsoft’s toolkit for heterogeneous treatment effect estimation with a focus on economic applications.

2. CausalML: A Python package that provides implementations of various meta-learners and specialized methods for estimating heterogeneous treatment effects.

3. grf: The generalized random forests package in R, which includes implementations of causal forests.

4. BART: The Bayesian Additive Regression Trees implementation in R.

Example: Using Meta-Learners with CausalML

Here’s a simplified example of using the CausalML library to estimate CATE:

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# Sample code for using CausalML
import causalml as cm
import numpy as np
from sklearn.ensemble import RandomForestRegressor

# Generate synthetic data
n = 1000
p = 5
X = np.random.normal(0, 1, size=(n, p))
# Generate treatment based on covariates
propensity = 1 / (1 + np.exp(-X[:, 0] - 0.5 * X[:, 1]))
T = np.random.binomial(1, propensity)
# Generate outcomes with heterogeneous treatment effects
tau = X[:, 0] + X[:, 1]  # True CATE is a function of X0 and X1
y_control = 0.5 * X[:, 0] + 0.5 * X[:, 1] + np.random.normal(0, 1, size=n)
y = y_control + tau * T

# Split the data
from sklearn.model_selection import train_test_split
X_train, X_test, T_train, T_test, y_train, y_test = train_test_split(
    X, T, y, test_size=0.3, random_state=42)

# Initialize a meta-learner (S-learner in this case)
s_learner = cm.meta.SLearner(learner=RandomForestRegressor(n_estimators=100, max_depth=5))

# Train the model
s_learner.fit(X=X_train, treatment=T_train, y=y_train)

# Get the treatment effect estimates
tau_hat = s_learner.predict(X_test)

# Evaluate performance (if true effects are known)
from causalml.metrics import plot_gain
plot_gain(tau_test, tau_hat)

Example: Using Causal Forests

Here’s an example of using causal forests with the EconML package in Python:

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# Python sample code for causal forests using EconML
import numpy as np
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.ensemble import RandomForestRegressor
from econml.dml import CausalForestDML

# Load the crime data (similar to the R example in the blog post)
url = 'https://vincentarelbundock.github.io/Rdatasets/csv/Ecdat/Crime.csv'
df = pd.read_csv(url)

# Set the categorical variables
cat_vars = ['year', 'region', 'smsa']

# Transform the categorical variables to dummies and add them back in
xf = pd.get_dummies(df[cat_vars])
df = pd.concat([df.drop(cat_vars, axis=1), xf], axis=1)
cat_var_dummy_names = list(xf.columns)

# Define regressors (control variables)
regressors = ['prbarr', 'prbconv', 'prbpris', 'avgsen', 'polpc', 
              'density', 'taxpc', 'pctmin', 'wcon']

# Add in the dummy names to the list of regressors
regressors = regressors + cat_var_dummy_names

# Drop rows with missing values
df = df.dropna()

# Split into train and test sets
train, test = train_test_split(df, test_size=0.2, random_state=42)

# Isolate the "treatment" variable (equivalent to pctymle in the R example)
T_train = train['pctymle'].values.reshape(-1, 1)  # Treatment needs to be 2D
T_test = test['pctymle'].values.reshape(-1, 1)

# Isolate the outcome variable (equivalent to crmrte in the R example)
Y_train = train['crmrte'].values
Y_test = test['crmrte'].values

# Create feature matrix X (control variables)
X_train = train[regressors].values
X_test = test[regressors].values

# Initialize a causal forest model
# Note: CausalForestDML combines Double Machine Learning with Causal Forest
cf = CausalForestDML(
    n_estimators=2000,          # Number of trees
    min_samples_leaf=5,         # Minimum samples in leaf nodes
    max_depth=10,               # Maximum depth of trees
    verbose=0,                  # Verbosity level
    random_state=42             # For reproducibility
)

# Fit the model
cf.fit(Y_train, T_train, X=X_train)

# Get predicted treatment effects for each observation in test set
effects = cf.effect(X_test)

# Get confidence intervals for the predictions
lower, upper = cf.effect_interval(X_test)

# Calculate feature importance
feature_importance = cf.feature_importances_

# Print feature importance
print("Feature importance:")
for i, feature in enumerate(regressors):
    print(f"{feature}: {feature_importance[i]:.4f}")

# Print some example predictions with confidence intervals
print("\nSample predictions (first 5):")
for i in range(5):
    print(f"Effect: {effects[i]:.4f}, 95% CI: [{lower[i]:.4f}, {upper[i]:.4f}]")

# You can also get the average treatment effect
print(f"\nAverage Treatment Effect: {np.mean(effects):.4f}")

# Plot the distribution of treatment effects
import matplotlib.pyplot as plt

plt.figure(figsize=(10, 6))
plt.hist(effects, bins=30, edgecolor='black')
plt.title('Distribution of Estimated Treatment Effects')
plt.xlabel('Treatment Effect')
plt.ylabel('Frequency')
plt.axvline(x=np.mean(effects), color='red', linestyle='--', 
            label=f'Mean: {np.mean(effects):.4f}')
plt.legend()
plt.show()

# For a more advanced visualization, you can plot treatment effects against a feature
plt.figure(figsize=(10, 6))
plt.scatter(X_test[:, 0], effects, alpha=0.5)  # Using first feature as example
plt.title('Treatment Effects vs. Feature')
plt.xlabel(regressors[0])
plt.ylabel('Treatment Effect')
plt.show()

Challenges and Considerations in CATE Estimation

While CATE estimation offers powerful insights, several challenges remain:

1. The Fundamental Problem of Causal Inference

The fundamental problem—that we never observe both potential outcomes for the same individual—means that we can never directly validate CATE estimates at the individual level. This makes evaluation inherently difficult, often requiring simulation studies where the true treatment effect is known.

2. Confounding

In observational studies, treatment assignment is not random but influenced by covariates. If some confounders are unobserved, CATE estimates may be biased. Methods like multi-accurate post-processing can help address unknown covariate shifts between observational and randomized datasets.

3. High-Dimensional Covariates

Modern datasets often contain many potential covariates. In high-dimensional settings, machine learning methods such as boosting or random forests work well for estimating propensity scores, as they excel at prediction problems with many covariates.

4. Small Sample Sizes and Imbalance

Small sample sizes or imbalance between treatment and control groups can pose challenges. Some research suggests that sample-splitting and cross-fitting are beneficial in large samples for bias reduction and efficiency of meta-learners, whereas full-sample estimation may be preferable in small samples.

5. Model Selection and Hyperparameter Tuning

With multiple methods available, selecting the appropriate approach and tuning its hyperparameters becomes crucial. Simulation studies suggest that meta-learners tend to have higher variance than modified ML methods like causal BART and causal forest, with the T-learner showing the highest variance.

CATE vs. Traditional Causal Inference Methods

As we dive deeper into causal inference, it’s valuable to understand how Conditional Average Treatment Effect (CATE) estimation compares to well-established approaches like Difference-in-Differences (DiD) and Propensity Score Matching (PSM). Each method has distinct strengths and applications, making them complementary tools in the causal inference toolkit rather than competing alternatives.

The Quest for Different Causal Questions

At their core, these methodologies address fundamentally different questions about causal effects. CATE estimation focuses on understanding heterogeneity—how treatment effects vary across different subpopulations based on observable characteristics. The central question it answers is not simply whether a treatment works, but for whom it works and by how much. This stands in contrast to DiD, which primarily answers questions about average treatment effects while controlling for time trends. DiD excels at identifying the average effect of a treatment after accounting for time-invariant confounders and common time trends, but offers less insight into effect variation across subgroups.

Propensity Score Matching occupies yet another niche in the causal inference landscape. Rather than directly focusing on effect heterogeneity, PSM addresses selection bias by creating balanced comparison groups. It answers questions about average treatment effects after matching individuals with similar probabilities of receiving treatment, essentially creating a quasi-experimental setting from observational data.

Divergent Data Requirements and Assumptions

These methods also differ substantially in their data requirements and underlying assumptions. CATE estimation works flexibly with either cross-sectional or panel data and particularly thrives when rich covariate information is available. Its primary limitation lies in the unconfoundedness assumption—the requirement that all factors affecting both treatment assignment and outcomes are observed.

DiD, by contrast, requires panel data with observations both before and after treatment implementation. Its validity hinges on the parallel trends assumption—that treatment and control groups would follow parallel paths in the absence of treatment. This approach shines in natural experiment settings where a policy or intervention affects some groups but not others at a specific point in time.

PSM works primarily with cross-sectional data but requires sufficient overlap in characteristics between treated and control groups. Like CATE, it relies on unconfoundedness, but additionally requires sufficient overlap in propensity scores between treatment and control groups to create valid matches.

Methodological Foundations and Computational Approaches

The methodological underpinnings of these approaches reflect their different origins and objectives. CATE estimation employs flexible machine learning techniques to adaptively discover patterns of treatment effect heterogeneity without imposing rigid structural assumptions about the form these patterns might take. This data-driven approach contrasts with DiD, which typically uses linear regression models with interaction terms to estimate average treatment effects in a more structured framework.

PSM represents yet another paradigm—a two-stage approach that first estimates propensity scores, then matches or weights observations based on these scores. This approach directly addresses selection bias but traditionally has done so using parametric models with limited flexibility.

High-Dimensional Challenges and Opportunities

The methods also differ dramatically in their ability to handle high-dimensional data. CATE estimation methods like causal forests excel with high-dimensional covariates due to their machine learning foundation. They can navigate complex feature spaces to identify meaningful patterns of treatment effect heterogeneity without succumbing to the curse of dimensionality that plagues traditional approaches. DiD methods, rooted in linear modeling traditions, can become unwieldy with many covariates or interaction terms. The model complexity grows exponentially as researchers attempt to capture richer heterogeneity structures within the DiD framework. Traditional PSM implementations similarly struggle with high-dimensional data, though modern variants increasingly incorporate machine learning techniques to overcome these limitations.

Selecting the Right Causal Tool

The choice between these methods should be guided by the research question, available data, and plausibility of underlying assumptions. CATE estimation proves most valuable when heterogeneity is of primary interest, when rich covariate information is available, and when the goal is personalized decision-making. Its main limitations stem from the strong assumptions about unconfoundedness and the greater complexity involved in implementation and interpretation.

DiD shines when treatment occurs at a specific time point, in natural experiment settings, and when pre-treatment data is readily available. Its weakness lies in its limited ability to capture treatment effect heterogeneity and its sensitivity to violations of the parallel trends assumption. When these violations occur, even the average treatment effect estimates become suspect.

PSM offers particular advantages in contexts with clear selection mechanisms, when balance between treatment and control groups is crucial, and when the research focus centers on average treatment effects. However, it cannot directly model heterogeneity without additional modifications and remains sensitive to unmeasured confounding factors that might influence both treatment assignment and outcomes.

The Convergence of Causal Methods

The boundaries between these methods are increasingly blurred by innovative approaches that combine their strengths. Machine learning-enhanced PSM uses algorithms like random forests to estimate propensity scores with greater flexibility and accuracy. Heterogeneous DiD approaches combine the time-based identification strategy of DiD with machine learning techniques to capture treatment effect heterogeneity across subgroups. Double/debiased machine learning integrates econometric insights with machine learning to address confounding in high-dimensional settings.

Illuminating Different Facets of Causality

To illustrate these differences, consider the evaluation of a job training program. A CATE-based approach would identify which specific demographics benefit most from the program, enabling targeted refinement of program features. DiD analysis would assess the overall program effect by comparing employment rates before and after implementation relative to control regions. PSM would match program participants with similar non-participants to estimate the average program effect while controlling for selection into treatment. Together, these approaches provide complementary insights, from the broad average impact to the nuanced patterns of who benefits and by how much.

CATE estimation represents an evolution in causal inference that builds upon the foundations established by methods like DiD and PSM, extending them to capture richer patterns of treatment effect heterogeneity using modern computational techniques. While traditional methods remain invaluable tools for certain causal questions, CATE estimation opens new possibilities for personalized decision-making and precision policy in an era of increasing data availability and computational capacity.

The Future of CATE Estimation

The field of CATE estimation continues to evolve rapidly. Some promising directions include:

Integration with Deep Learning

As deep learning advances, integrating these powerful models with causal inference frameworks offers exciting possibilities for capturing complex relationships in high-dimensional data.

Causal Inference with Unstructured Data

Extending CATE estimation to unstructured data like text, images, and time series presents both challenges and opportunities for new methodological developments.

Uncertainty Quantification

Better methods for quantifying uncertainty in CATE estimates will be crucial for decision-making, particularly in high-stakes domains like healthcare and public policy.

Fairness and Ethics

As CATE estimates inform increasingly important decisions, ensuring that these methods don’t encode or amplify biases becomes a critical area of research.

Conclusion: The Path Forward

The journey from average treatment effects to conditional average treatment effects represents more than a technical advancement—it’s a fundamental shift in how we approach causality. By recognizing and modeling heterogeneity, we move from one-size-fits-all solutions to personalized interventions tailored to individual characteristics.

As computational methods continue to advance, the gap between what we can estimate and what we actually need for decision-making narrows. However, these powerful tools must be wielded with care, acknowledging their limitations and the assumptions they entail.

The quest for understanding heterogeneous treatment effects is far from over. Each new method, each application to a real-world problem, brings us closer to uncovering the rich tapestry of causal relationships that govern our world. And in that ongoing quest lies the promise of more effective interventions, more precise policies, and ultimately, better outcomes for individuals across diverse contexts and circumstances.

So as you apply these methods to your own research questions, remember that you’re not just estimating a parameter—you’re uncovering insights that can transform how we make decisions and design interventions in an inherently heterogeneous world.

References and Helpful Resources

References

Athey, S., & Imbens, G. (2016). Recursive partitioning for heterogeneous causal effects. Proceedings of the National Academy of Sciences, 113(27), 7353-7360.

Athey, S., Tibshirani, J., & Wager, S. (2019). Generalized random forests. The Annals of Statistics, 47(2), 1148-1178.

Bruns-Smith, D., Farias, V., Li, H., & Tamkin, A. (2023). Augmented balancing weights and target-independence. arXiv preprint.

Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., & Robins, J. (2018). Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21(1), C1-C68.

Christainsen, M. R., & Jessen, A. J. (2022). Meta-learners for estimation of causal effects: Finite sample cross-fit performance. Scandinavian Journal of Statistics, 49(2), 674-703.

Hahn, P. R., Murray, J. S., & Carvalho, C. M. (2020). Bayesian regression tree models for causal inference: Regularization, confounding, and heterogeneous effects. Bayesian Analysis, 15(3), 965-1056.

Hill, J. L. (2011). Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics, 20(1), 217-240.

Horvitz, D. G., & Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47(260), 663-685.

Imbens, G. W., & Rubin, D. B. (2015). Causal inference for statistics, social, and biomedical sciences: An introduction. Cambridge University Press.

Jacob, D. (2021). CATE meets ML: The conditional average treatment effect and machine learning. Digital Finance, 3, 211-237. https://link.springer.com/article/10.1007/s42521-021-00033-7

Kennedy, E. H. (2020). Optimal doubly robust estimation of heterogeneous causal effects. arXiv preprint arXiv:2004.14497.

Kim, M. P., Reingold, O., & Rothblum, G. N. (2022). Generalization and robustness implications in object-centric learning. International Conference on Machine Learning.

Knaus, M. C., Lechner, M., & Strittmatter, A. (2020). Heterogeneous employment effects of job search programmes: A machine learning approach. Journal of Human Resources, 56(1), 125-159.

Künzel, S. R., Sekhon, J. S., Bickel, P. J., & Yu, B. (2019). Metalearners for estimating heterogeneous treatment effects using machine learning. Proceedings of the National Academy of Sciences, 116(10), 4156-4165.

Nie, X., & Wager, S. (2020). Quasi-oracle estimation of heterogeneous treatment effects. Biometrika, 108(2), 299-319.

Oprescu, M., Syrgkanis, V., & Wu, Z. S. (2019). Orthogonal random forest for causal inference. International Conference on Machine Learning.

Powers, S., Qian, J., Jung, K., Schuler, A., Shah, N. H., Hastie, T., & Tibshirani, R. (2018). Some methods for heterogeneous treatment effect estimation in high dimensions. Statistics in Medicine, 37(11), 1767-1787.

Semenova, V., & Chernozhukov, V. (2021). Debiased machine learning of conditional average treatment effects and other causal functions. The Econometrics Journal, 24(2), C265-C293.

Shalit, U., Johansson, F. D., & Sontag, D. (2017). Estimating individual treatment effect: generalization bounds and algorithms. International Conference on Machine Learning.

Sharma, A., Gupta, S., Sharma, A., & Paruchuri, P. (2020). Heterogeneous treatment effect estimation using machine learning for healthcare applications: Tutorial and benchmark. Journal of Biomedical Informatics, 107, 103472.

Shi, C., Blei, D., & Veitch, V. (2019). Adapting neural networks for the estimation of treatment effects. Advances in Neural Information Processing Systems, 32.

Spirtes, P., Glymour, C., & Scheines, R. (2000). Causation, prediction, and search. MIT Press.

Wager, S., & Athey, S. (2018). Estimation and inference of heterogeneous treatment effects using random forests. Journal of the American Statistical Association, 113(523), 1228-1242.

Wang, G., Li, J., & Hopp, W. J. (2019). An instrumental variable tree approach for detecting heterogeneous treatment effects in observational studies. Biometrika, 106(1), 155-172.

Yang, S., & Ding, P. (2020). Combining multiple observational data sources to estimate causal effects. Journal of the American Statistical Association, 115(531), 1540-1554.

Software Packages and Tools

CausalML. Python package for causal machine learning. https://causalml.readthedocs.io/

EconML. Microsoft’s Python package for estimating heterogeneous treatment effects. https://github.com/microsoft/EconML

grf. R package for generalized random forests. https://github.com/grf-labs/grf

causalToolbox. R package implementing meta-learners for causal inference. https://github.com/saberpowers/causalToolbox

BART. R package for Bayesian Additive Regression Trees. https://cran.r-project.org/web/packages/BART/