build_aggtree: Aggregation Trees

Description

Nonparametric data-driven approach to discovering heterogeneous subgroups in a selection-on-observables framework. The approach constructs a sequence of groupings, one for each level of granularity. Groupings are nested and feature an optimality property. For each grouping, we obtain point estimation and standard errors for the group average treatment effects (GATEs) using debiased machine learning procedures. Additionally, we assess whether systematic heterogeneity is found by testing the hypotheses that the differences in the GATEs across all pairs of groups are zero. Finally, we investigate the driving mechanisms of effect heterogeneity by computing the average characteristics of units in each group.

Usage

build_aggtree(
  Y_tr,
  D_tr,
  X_tr,
  Y_hon = NULL,
  D_hon = NULL,
  X_hon = NULL,
  cates_tr = NULL,
  cates_hon = NULL,
  method = "aipw",
  scores = NULL,
  ...
)
inference_aggtree(object, n_groups, boot_ci = FALSE, boot_R = 2000)

Value

build_aggtree returns an aggTrees object.

inference_aggtree returns an aggTrees.inference object, which in turn contains the aggTrees object used in the call.

Arguments

Y_tr: Outcome vector for training sample.
D_tr: Treatment vector for training sample.
X_tr: Covariate matrix (no intercept) for training sample.
Y_hon: Outcome vector for honest sample.
D_hon: Treatment vector for honest sample.
X_hon: Covariate matrix (no intercept) for honest sample.
cates_tr: Optional, predicted CATEs for training sample. If not provided by the user, CATEs are estimated internally via a causal_forest.
cates_hon: Optional, predicted CATEs for honest sample. If not provided by the user, CATEs are estimated internally via a causal_forest.
method: Either "raw" or "aipw", controls how node predictions are computed.
scores: Optional, vector of scores to be used in computing node predictions. Useful to save computational time if scores have already been estimated. Ignored if method == "raw".
...: Further arguments from rpart.control.
object: An aggTrees object.
n_groups: Number of desired groups.
boot_ci: Logical, whether to compute bootstrap confidence intervals.
boot_R: Number of bootstrap replications. Ignored if boot_ci == FALSE.

Author

Riccardo Di Francesco

Details

Aggregation trees are a three-step procedure. First, the conditional average treatment effects (CATEs) are estimated using any estimator. Second, a tree is grown to approximate the CATEs. Third, the tree is pruned to derive a nested sequence of optimal groupings, one for each granularity level. For each level of granularity, we can obtain point estimation and inference about the GATEs.

To implement this methodology, the user can rely on two core functions that handle the various steps.

Constructing the Sequence of Groupings

build_aggtree constructs the sequence of groupings (i.e., the tree) and estimate the GATEs in each node. The GATEs can be estimated in several ways. This is controlled by the method argument. If method == "raw", we compute the difference in mean outcomes between treated and control observations in each node. This is an unbiased estimator in randomized experiment. If method == "aipw", we construct doubly-robust scores and average them in each node. This is unbiased also in observational studies. Honest regression forests and 5-fold cross fitting are used to estimate the propensity score and the conditional mean function of the outcome (unless the user specifies the argument scores).

The user can provide a vector of the estimated CATEs via the cates_tr and cates_hon arguments. If no CATEs are provided, these are estimated internally via a causal_forest using only the training sample, that is, Y_tr, D_tr, and X_tr.

GATEs Estimation and Inference

inference_aggtree takes as input an aggTrees object constructed by build_aggtree. Then, for the desired granularity level, chosen via the n_groups argument, it provides point estimation and standard errors for the GATEs. Additionally, it performs some hypothesis testing to assess whether we find systematic heterogeneity and computes the average characteristics of the units in each group to investigate the driving mechanisms.

Point estimates and standard errors for the GATEs

GATEs and their standard errors are obtained by fitting an appropriate linear model. If method == "raw", we estimate via OLS the following:

$$Y_i = \sum_{l = 1}^{|T|} L_{i, l} \gamma_l + \sum_{l = 1}^{|T|} L_{i, l} D_i \beta_l + \epsilon_i$$

with L_{i, l} a dummy variable equal to one if the i-th unit falls in the l-th group, and |T| the number of groups. If the treatment is randomly assigned, one can show that the betas identify the GATE of each group. However, this is not true in observational studies due to selection into treatment. In this case, the user is expected to use method == "aipw" when calling build_aggtree. In this case, inference_aggtree uses the scores in the following regression:

$$score_i = \sum_{l = 1}^{|T|} L_{i, l} \beta_l + \epsilon_i$$

This way, betas again identify the GATEs.

Regardless of method, standard errors are estimated via the Eicker-Huber-White estimator.

If boot_ci == TRUE, the routine also computes asymmetric bias-corrected and accelerated 95% confidence intervals using 2000 bootstrap samples. Particularly useful when the honest sample is small-ish.

Hypothesis testing

inference_aggtree uses the standard errors obtained by fitting the linear models above to test the hypotheses that the GATEs are different across all pairs of leaves. Here, we adjust p-values to account for multiple hypotheses testing using Holm's procedure.

Average Characteristics

inference_aggtree regresses each covariate on a set of dummies denoting group membership. This way, we get the average characteristics of units in each leaf, together with a standard error. Leaves are ordered in increasing order of their predictions (from most negative to most positive). Standard errors are estimated via the Eicker-Huber-White estimator.

Caution on Inference

Regardless of the chosen method, both functions estimate the GATEs, the linear models, and the average characteristics of units in each group using only observations in the honest sample. If the honest sample is empty (this happens when the user either does not provide Y_hon, D_hon, and X_hon or sets them to NULL), the same data used to construct the tree are used to estimate the above quantities. This is fine for prediction but invalidates inference.

References

Di Francesco, R. (2022). Aggregation Trees. CEIS Research Paper, 546. tools:::Rd_expr_doi("10.2139/ssrn.4304256").

Examples

Run this code

## Generate data.
set.seed(1986)

n <- 1000
k <- 3

X <- matrix(rnorm(n * k), ncol = k)
colnames(X) <- paste0("x", seq_len(k))
D <- rbinom(n, size = 1, prob = 0.5)
mu0 <- 0.5 * X[, 1]
mu1 <- 0.5 * X[, 1] + X[, 2]
Y <- mu0 + D * (mu1 - mu0) + rnorm(n)

## Training-honest sample split.
honest_frac <- 0.5
splits <- sample_split(length(Y), training_frac = (1 - honest_frac))
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, ]

## Construct sequence of groupings. CATEs estimated internally.
groupings <- build_aggtree(Y_tr, D_tr, X_tr, # Training sample.
                           Y_hon, D_hon, X_hon) # Honest sample.

## Alternatively, we can estimate the CATEs and pass them.
library(grf)
forest <- causal_forest(X_tr, Y_tr, D_tr) # Use training sample.
cates_tr <- predict(forest, X_tr)$predictions
cates_hon <- predict(forest, X_hon)$predictions

groupings <- build_aggtree(Y_tr, D_tr, X_tr, # Training sample.
                           Y_hon, D_hon, X_hon, # Honest sample.
                           cates_tr, cates_hon) # Predicted CATEs.

## We have compatibility with generic S3-methods.
summary(groupings)
print(groupings)
plot(groupings) # Try also setting 'sequence = TRUE'.

## To predict, do the following.
tree <- subtree(groupings$tree, cv = TRUE) # Select by cross-validation.
head(predict(tree, data.frame(X_hon)))

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

summary(results$model) # Coefficient of leafk is GATE in k-th leaf.

results$gates_diff_pairs$gates_diff # GATEs differences.
results$gates_diff_pairs$holm_pvalues # leaves 1-2 not statistically different.

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

Run the code above in your browser using DataLab