Short Tutorial

Constructing the Sequence of Groupings

To construct the sequence of optimal groupings, we first need to estimate the CATEs. Here we use the causal forest estimator. To achieve valid inference about the GATEs, we split the sample into a training sample and an honest sample of equal sizes. The forest is built using only the training sample.

## Sample split.
splits <- sample_split(length(y), training_frac = 0.5)
training_idx <- splits$training_idx
honest_idx <- splits$honest_idx

y_tr <- y[training_idx]
D_tr <- D[training_idx]
X_tr <- X[training_idx, ]

y_hon <- y[honest_idx]
D_hon <- D[honest_idx]
X_hon <- X[honest_idx, ]

## Estimate the CATEs. Use only training sample.
library(grf)
forest <- causal_forest(X_tr, y_tr, D_tr) 
cates <- predict(forest, X)$predictions

Now we use the build_aggtree function to construct the sequence of groupings. This function approximates the estimated CATEs by a decision tree using only the training sample and computes node predictions (i.e., the GATEs) using only the honest sample. build_aggtree allows the user to choose between two GATE estimators:

1. If we set method = "raw", the GATEs are estimated by taking the differences between the mean outcomes of treated and control units in each node. This is an unbiased estimator (only) in randomized experiments;
2. If we set method = "aipw", the GATEs are estimated by averaging doubly-robust scores (see Appendix below) in each node. This is an unbiased estimator also in observational studies under particular conditions on the construction of the scores.

The doubly-robust scores can be estimated separately and passed in by the scores argument. Otherwise, they are estimated internally. Notice the use of the is_honest argument, a logical vector denoting which observations we allocated to the honest sample. This way, build_aggtree knows which observations must be used to construct the tree and compute node predictions.

## Construct the sequence. Use doubly-robust scores.
groupings <- build_aggtree(y, D, X, method = "aipw", 
                           cates = cates, is_honest = 1:length(y) %in% honest_idx)

## Print.
print(groupings)

## Plot.
plot(groupings) # Try also setting 'sequence = TRUE'.

Further Analysis

Now that we have a whole sequence of optimal groupings, we can pick the grouping associated with our preferred granularity level and call the inference_aggtree function. This function does the following:

1. It gets standard errors for the GATEs by estimating via OLS appropriate linear models using the honest sample. The choice of the linear model depends on the method we used when we called build_aggtree (see Appendix below);
2. It tests the null hypotheses that the differences in the GATEs across all pairs of groups equal zero. Here, we account for multiple hypotheses testing by adjusting the \(p\)-values using Holm’s procedure;
3. It computes the average characteristics of the units in each group.

To report the results, we can print nice LATEX tables.

## Inference with 4 groups.
results <- inference_aggtree(groupings, n_groups = 4)

## LATEX.
print(results, table = "diff")
print(results, table = "avg_char")

Appendix

The point of estimating the linear models is to get standard errors for the GATEs. Under an honesty condition, we can use the estimated standard errors to conduct valid inference as usual, e.g., by constructing conventional confidence intervals. Honesty is a subsample-splitting technique that requires that different observations are used to form the subgroups and estimate the GATEs. inference_aggtree always uses the honest sample to estimate the linear models below (unless we called build_aggtree without using the honesty settings).

If we set method = "raw", inference_aggtree estimates via OLS the following linear model:

\[\begin{equation} Y_i = \sum_{l = 1}^{|\mathcal{T_{\alpha}}|} L_{i, l} \, \gamma_l + \sum_{l = 1}^{|\mathcal{T}_{\alpha}|} L_{i, l} \, D_i \, \beta_l + \epsilon_i \end{equation}\]

with \(|\mathcal{T}_{\alpha}|\) the number of leaves of a particular tree \(\mathcal{T}_{\alpha}\), and \(L_{i, l}\) a dummy variable equal to one if the \(i\)-th unit falls in the \(l\)-th leaf of \(\mathcal{T}_{\alpha}\). Exploiting the random assignment to treatment, we can show that each \(\beta_l\) identifies the GATE in the \(l\)-th leaf. Under honesty, the OLS estimator \(\hat{\beta}_l\) of \(\beta_l\) is root-\(n\) consistent and asymptotically normal.

If we set method = "aipw", inference_aggtree estimates via OLS the following linear model:

\[\begin{equation} \widehat{\Gamma}_i = \sum_{l = 1}^{|\mathcal{T}_{\alpha}|} L_{i, l} \, \beta_l + \epsilon_i \end{equation}\]

where \(\Gamma_i\) are the following doubly-robust scores:

\[\begin{equation*} \Gamma_i = \mu \left( 1, X_i \right) - \mu \left( 0, X_i \right) + \frac{D_i \left[ Y_i - \mu \left( 1, X_i \right) \right]}{p \left( X_i \right)} - \frac{ \left( 1 - D_i \right) \left[ Y_i - \mu \left( 0, X_i \right) \right]}{1 - p \left( X_i \right)} \end{equation*}\]

with \(\mu \left(D_i, X_i \right) = \mathbb{E} \left[ Y_i | D_i, Z_i \right]\) the conditional mean of \(Y_i\) and \(p \left( X_i \right) = \mathbb{P} \left( D_i = 1 | X_i \right)\) the propensity score. These scores are inherited by the scores used in the build_aggtree call. As before, we can show that each \(\beta_l\) identifies the GATE in the \(l\)-th leaf, this time even in observational studies. Under honesty, the OLS estimator \(\hat{\beta}_l\) of \(\beta_l\) is root-\(n\) consistent and asymptotically normal, provided that the \(\Gamma_i\) are cross-fitted and that the product of the convergence rates of the estimators of the nuisance functions \(\mu \left( \cdot, \cdot \right)\) and \(p \left( \cdot \right)\) is faster than \(n^{1/2}\).