A Worked Causal Forest Analysis

From a Study Question to Forest Outputs

A causal forest can return a different estimated treatment effect for people with different measured characteristics. The practical challenge is to carry the analysis from a defensible covariate set to predictions and summaries without changing the quantity being estimated along the way. In particular, a predicted effect is an average for people with similar covariates, not a measurement of what treatment did to one specific person.

We will follow a fully hypothetical simulated study of a treatment intended to reduce a symptom score measured at follow-up. Lower scores mean fewer symptoms. Treatment is randomized, and the score is measured in points, so a negative treated-minus-control contrast represents a reduction in symptoms under treatment. All sample sizes and forest outputs below are illustrative.

The target in this example is the CATE for people represented in the simulated study. Identification requires consistency (the observed outcome agrees with the potential outcome for the treatment received), no interference between units, and treatment assignment independent of potential outcomes. Randomization with positive probability of assignment to both arms supplies the latter condition here. For observational data, conditional exchangeability given \(X\) and positivity would instead be needed; a forest does not make those assumptions true.

Preparing Covariates and Fitting the Forest

Suppose the simulated study contains 4,000 participants, with 2,000 assigned to treatment and 2,000 to control. The outcome \(Y\) is the symptom score at follow-up. The baseline covariates are age in years, symptom severity on a 0–10 scale, and the number of prior episodes. In the simulation, treatment is assigned with probability one-half. The outcome-generating pattern has larger average symptom reductions at higher baseline severity, but the forest is not told that pattern.

Covariate preparation starts by defining the prediction target and time order. Age, baseline severity, and prior episodes are measured before assignment, so they can be used as \(X\). Follow-up adherence and the follow-up symptom score are not baseline covariates; adding them as predictors would condition on post-treatment information and change the analysis. Categorical baseline variables, if present, should be encoded consistently. Missing values need an explicit, defensible handling rule, rather than silently treating a missing value as a meaningful numerical value. Tree-based methods generally do not require predictors to be put on a common measurement scale, but clear units and coding still matter for interpretation and reproducibility.

For fitting, use the causal-forest procedure appropriate to the target CATE and its treatment-assignment design. In this randomized example, the known assignment probability is \(e(x)=P(W=1\mid X=x)=0.5\), where \(W\) indicates treatment. A forest uses covariate-dependent neighborhoods and treatment-outcome information to estimate conditional contrasts; its predictions are not obtained by simply fitting a separate treatment effect to each person. Follow the procedure's honesty or sample-separation options as intended, and record the covariates, treatment coding, outcome definition, forest settings, and random seed. These choices make the analysis auditable, but do not replace the causal assumptions.

Reading CATE Predictions

Imagine that one illustrative forest fit returns a predicted CATE of \(-1.42\) symptom-score points at age 45, baseline severity 9, and 2 prior episodes. The value is an estimated conditional average, based on the fitted forest and the available study data. It is not a guarantee that a participant with those covariates will improve by 1.42 points, nor does it describe the difference between that participant's two unobserved potential outcomes.

It is also important to separate an estimated CATE at a covariate value from a subgroup effect. A point prediction refers to a conditional mean at \(x\); a subgroup effect averages that conditional mean over a defined population. The subgroup definition and its target population must be explicit. A threshold selected after inspecting many forest predictions is a data-adaptive discovery, not automatically a prespecified subgroup result.

Summarizing Effects Over a Subgroup

For a subgroup \(G\) defined using baseline covariates, its average treatment effect is the average of \(\tau(X)\) among members of \(G\). The relationship follows directly from iterated expectations: $$ E[Y(1)-Y(0)\mid X\in G] =E\{E[Y(1)-Y(0)\mid X]\mid X\in G\} =E[\tau(X)\mid X\in G]. $$ Thus subgroup summaries can be formed by averaging CATE predictions for the subgroup's covariate profiles. The average must use the subgroup's target-population weights; an unweighted average of arbitrary profile predictions need not represent that population.

Uncertainty and Support Checks

A point prediction can look precise while being based on little relevant information. Before interpreting a pattern, check that the covariate profiles or subgroups occur in the data and have treated and control observations. In this randomized example, treatment probability is one-half by design, but sparse combinations of severity, age, and prior episodes may still yield unstable local estimates. A forest cannot manufacture information for profiles far outside the observed covariate support.

If the implementation reports a standard error for a prediction, record it with the prediction and check its scale. Its meaning and inferential validity depend on the estimator and the procedure used to calculate it. Reasonable sensitivity checks include refitting with a different random seed or modestly varied forest settings and examining whether the broad pattern persists. These checks are diagnostics, not substitutes for formal inference or a validation analysis.

Why the Workflow Matters

A causal-forest analysis is most useful when each output is tied to its estimand. Covariate preparation determines which conditional comparisons are being made. The fitted forest supplies estimated CATEs; subgroup averages describe those estimates for a stated population; uncertainty checks indicate how cautiously to read them. The causal interpretation rests on design and assumptions, not on the complexity of the algorithm.

One short derivation helps keep subgroup summaries straight. Because \(\tau(X)\) is the conditional expectation of \(Y(1)-Y(0)\), conditioning on membership in \(G\) and taking an expectation gives $$ E[\tau(X)\mid X\in G] =E\{E[Y(1)-Y(0)\mid X]\mid X\in G\} =E[Y(1)-Y(0)\mid X\in G]. $$ This is why the target distribution within the subgroup matters: changing which covariate profiles receive weight changes the subgroup average even when the conditional effect function is unchanged.