Causal Trees
Definition
A causal tree is a decision tree designed to estimate heterogeneous treatment effects (HTEs) — that is, how the causal effect of a treatment varies across different subgroups defined by features.
- Traditional decision trees split the data to improve prediction accuracy.
- Causal trees split the data to improve treatment effect estimation.
In short: instead of predicting outcomes, a causal tree predicts causal effects within each leaf.
Formal Idea
We want to estimate the Conditional Average Treatment Effect (CATE):
$\tau(x) = E[Y(1) – Y(0) \mid X = x]$
where:
- $Y(1)$: Potential outcome if treated
- $Y(0)$: Potential outcome if not treated
- $X$: Features
A causal tree partitions the feature space into subgroups (leaves) and estimates $\tau(x)$ within each subgroup.
How It Differs from Standard Trees
- Standard tree: Splits to minimize outcome prediction error (e.g., mean squared error).
- Causal tree: Splits to maximize differences in treatment vs control outcomes between groups.
- Leaves represent subpopulations with similar treatment effects.
Training Process (Simplified)
- Input: Dataset with features $X$, treatment indicator $T$, and outcome $Y$.
- At each split:
- Evaluate candidate splits by how well they separate different treatment effects.
- Example splitting criterion: maximize variance in estimated treatment effects between child nodes.
- Stop splitting when nodes are too small or effect estimates become unreliable.
- Output: A tree structure where each leaf contains an estimate of the local treatment effect.
Example (Marketing)
- Treatment: Send promotional email.
- Outcome: Customer purchases or not.
- Features: Age, income, browsing history.
A causal tree might discover:
- Leaf 1: Young, low-income → uplift = –2% (negative effect, don’t target).
- Leaf 2: Middle-aged, high-income → uplift = +15% (strong positive effect, best target).
- Leaf 3: Older customers → uplift ≈ 0% (neutral effect).
This shows which subgroups benefit most from treatment.
Advantages
- Intuitive, interpretable (“if-then” rules).
- Directly targets heterogeneous treatment effects.
- Helps with personalized decision-making (who to treat, who not to treat).
Limitations
- Can be unstable (like regular decision trees).
- Treatment effect estimates in small leaves can be noisy.
- Solution: Causal Forests (ensemble of causal trees) for stability.
In short:
Causal trees are decision trees adapted for causal inference, partitioning data into subgroups to estimate how treatment effects vary across populations. They’re the building blocks for causal forests.