build_aggtree {aggTrees}R Documentation

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). 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,
  D,
  X,
  honest_frac = 0.5,
  method = "aipw",
  scores = NULL,
  cates = NULL,
  is_honest = NULL,
  ...
)

inference_aggtree(object, n_groups, boot_ci = FALSE, boot_R = 2000)

Arguments

y

Outcome vector.

D

Treatment vector.

X

Covariate matrix (no intercept).

honest_frac

Fraction of observations to be allocated to honest sample.

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".

cates

Optional, estimated CATEs. If not provided by the user, CATEs are estimated internally via a causal_forest.

is_honest

Logical vector denoting which observations belong to the honest sample. Required only if the cates argument is used.

...

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.

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 argument. If so, the user needs to specify a logical vector to denote which observations belong to the honest sample. If honesty is not desired, is_honest must be a vector of FALSEs. If no vector of CATEs is provided, these are estimated internally via a causal_forest.

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 because the user either sets honest_frac = 0 or passes a vector of FALSEs as is_honest when calling build_aggtree), the same data used to construct the tree are used to estimate the above quantities. This is fine for prediction but invalidates inference.

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.

Author(s)

Riccardo Di Francesco

References

See Also

plot.aggTrees print.aggTrees.inference

Examples

## 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)

## Construct sequence of groupings. CATEs estimated internally.
groupings <- build_aggtree(y, D, X, method = "aipw")

## Alternatively, we can estimate the CATEs and pass them.
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, ]

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

groupings <- build_aggtree(y, D, X, method = "aipw", cates = cates,
                           is_honest = 1:length(y) %in% honest_idx)

## 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)))

## 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")
[Package aggTrees version 2.0.2 Index]