Causal Forest Splitting Criteria

Why Causal Splits Are Different

A conventional regression tree seeks groups with different outcome levels. A causal forest instead seeks groups with different treatment effects. Those goals can point to different splits: a covariate may strongly predict the outcome while having no bearing on how outcomes change under treatment. A useful splitting criterion must therefore focus on estimated treatment effects, not simply on how well a split predicts the observed outcome.

The target is usually the conditional average treatment effect (CATE). Let \(Y(1)\) and \(Y(0)\) denote the potential outcomes under treatment and control, and let \(X\) denote observed covariates. The CATE at covariate value \(x\) is the conditional mean difference between those potential outcomes. A tree represents that function approximately by assigning similar covariate values to the same leaf.

This is a statement about average effects for covariate-defined groups. It does not claim that the forest observes each person's individual treatment effect. Nor does a large split score establish that the covariate causes the effect difference. The score is a data-adaptive measure of how much the estimated CATE varies across the proposed leaves.

Representing Treatment Effects With Pseudo-Outcomes

A practical way to evaluate candidate splits is to construct a treatment-effect pseudo-outcome for each observation. The earlier material on cross-fitting nuisance models for heterogeneity introduced this approach. Here, the important property is that the conditional mean of the pseudo-outcome targets the CATE, so a tree criterion applied to its values targets effect variation rather than raw outcome variation.

For clarity, suppose treatment is binary and the data are generated under consistency, conditional exchangeability, and overlap. Consistency means the observed outcome is \(Y=DY(1)+(1-D)Y(0)\). Conditional exchangeability means \((Y(1),Y(0))\) is independent of \(D\) given \(X\), and overlap means the propensity score \(e(x)=P(D=1\mid X=x)\) is strictly between zero and one. Define \(m_d(x)=E[Y\mid D=d,X=x]\) for \(d\in\{0,1\}\). Under these conditions, \(m_1(x)-m_0(x)=\tau(x)\).

Given nuisance estimates \(\hat m_1,\hat m_0,\hat e\), a commonly used augmented inverse-probability-weighted pseudo-outcome is

With the true nuisance functions, its conditional expectation given \(X=x\) equals \(\tau(x)\). For example, conditional on \(X=x\), the treated residual contribution has mean \(E[D\{Y-m_1(x)\}/e(x)\mid X=x]=0\), because the treated outcome has conditional mean \(m_1(x)\). The control residual contribution also has mean zero; the remaining term is \(m_1(x)-m_0(x)=\tau(x)\). Cross-fitted estimates use nuisance predictions obtained without fitting those nuisance models to the same observation. In a finite sample, estimated pseudo-outcomes remain noisy, so their averages estimate—not reveal exactly—the effects in a leaf.

Deriving the Between-Leaf Split Criterion

Let \(P\) be a parent node with \(n\) observations, divided by a candidate split into children \(L\) and \(R\), with sizes \(n_L\) and \(n_R\). Let \(\bar\Gamma_P,\bar\Gamma_L,\bar\Gamma_R\) be the averages of the pseudo-outcomes in those sets. A simple criterion scores a split by how much it reduces the total squared variation around the leaf means.

To see what this reduction measures, add and subtract each child mean inside its squared deviation. Within either child, the deviations from that child’s mean sum to zero. Thus, for \(j\in\{L,R\}\), \(\sum_{i\in j}(\Gamma_i-\bar\Gamma_P)^2 =\sum_{i\in j}(\Gamma_i-\bar\Gamma_j)^2+n_j(\bar\Gamma_j-\bar\Gamma_P)^2\). Subtracting the two within-child sums from the parent sum leaves \(n_L(\bar\Gamma_L-\bar\Gamma_P)^2+n_R(\bar\Gamma_R-\bar\Gamma_P)^2\). Because \(\bar\Gamma_P=(n_L\bar\Gamma_L+n_R\bar\Gamma_R)/n\), this expression simplifies to the following score:

The score is larger when the child means are farther apart, but it also depends on the child sizes. Equivalently, dividing \(S\) by \(n\) gives the reduction in average squared variation. Within a fixed parent, either scale ranks candidate splits the same way. This derivation explains the role of within-leaf effect variation: a split is useful when it removes variation in pseudo-outcomes by assigning observations with different conditional average effects to different leaves.

Why Outcome-Prediction Splits Can Mislead

A raw-outcome split can look compelling even when treatment effects are constant. For example, a covariate may mark a difference in baseline prognosis: people with higher baseline outcome levels can have higher outcomes under both treatment and control, without having a larger treatment benefit. A tree that splits on the observed outcome alone can prioritize that prognostic difference instead of the effect difference of interest.

Practical Constraints and Common Pitfalls

A split score is only as useful as the information supporting its child-leaf estimates. In finite samples, pseudo-outcomes can be highly variable, especially when estimated propensities are close to zero or one. A small child can also have a high or low mean due to chance. Implementations therefore restrict candidate splits using minimum leaf sizes and treatment-information requirements, and may regularize the search. Such restrictions do not change the target of the criterion; they help prevent a noisy empirical score from dominating the tree.

A split should also be interpreted in the units of the outcome. If \(Y\) is measured in days, the difference \(\bar\Gamma_L-\bar\Gamma_R\) is an estimated difference in treatment effects measured in days, while \(S\) is a squared, size-weighted quantity. Its numerical magnitude depends on the outcome scale and node size, so it should not be interpreted as an effect estimate in its own right.

• Mistake: choosing the split with the biggest difference in raw outcome averages. That rewards prognostic separation, which need not be effect heterogeneity. Correction: compare child averages of treatment-effect scores, such as appropriately constructed pseudo-outcomes.
• Mistake: treating a pseudo-outcome as an observed individual treatment effect. It is a noisy score whose conditional mean targets the CATE under the relevant assumptions. Correction: interpret leaf averages as estimates of group-level conditional mean effects.
• Mistake: preferring a split solely because its child means are far apart. The difference can be unstable if a child has very few observations or inadequate treatment representation. Correction: assess the score together with the tree’s minimum-size and treatment-information constraints.
• Mistake: interpreting the split score as evidence of statistical significance or causation by the splitting covariate. The score ranks candidate partitions; it does not provide a calibrated test. Correction: separate tree construction from later inference and avoid causal claims about the covariate itself.