5. DR-learner#

DR-learner is a two-stage doubly robust estimator for HTE estimation. Before Kennedy et al. 2020 [4], there are several related approaches trying to extend the doubly robust procedure to HTE estimation, such as [5, 6, 7]. Compared with the above three estimators, DR-learner is proved to be oracle efficient under some mild assumptions detailed in Theorem 2 of [4].

The basic steps of DR-learner is given below:

Step 1: Nuisance training:

(a) Using \(I_{1}^n\) to construct estimates \(\hat{\pi}\) for the propensity scores \(\pi\);

(b) Using \(I_{1}^n\) to construct estimates \(\hat\mu_a(s)\) for \(\mu_a(s):=\mathbb{E}[R|S=s,A=a]\);

Step 2: Pseudo-outcome regression:

Define \(\widehat{\phi}(Z)\) as the pseudo-outcome where

(31)#\[\begin{equation} \widehat{\phi}(Z)=\frac{A-\hat{\pi}(S)}{\hat{\pi}(S)\{1-\hat{\pi}(S)\}}\Big\{R-\hat{\mu}_A(S)\Big\}+\hat{\mu}_1(S)-\hat{\mu}_0(S), \end{equation}\]

and regress it on covariates \(S\) in the test sample \(I_2^n\), yielding

(32)#\[\begin{equation} \widehat{\tau}_{\text{DR-learner}}(s)=\widehat{\mathbb{E}}_n[\widehat{\phi}(Z)|S=s]. \end{equation}\]
# import related packages
import numpy as np
import pandas as pd
from matplotlib import pyplot as plt;
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.linear_model import LinearRegression
from sklearn.linear_model import LogisticRegression 
from causaldm.learners.CEL.Single_Stage import _env_getdata_CEL
from causaldm.learners.CEL.Single_Stage.DRlearner import DRlearner
import warnings
warnings.filterwarnings('ignore')

MovieLens Data#

# Get the MovieLens data
#import os
#os.chdir('/Users/alinaxu/Documents/CDM/CausalDM')
#MovieLens_CEL = pd.read_csv("./causaldm/data/MovieLens_CEL.csv")
MovieLens_CEL = _env_getdata_CEL.get_movielens_CEL()
MovieLens_CEL.pop(MovieLens_CEL.columns[0])
MovieLens_CEL = MovieLens_CEL[MovieLens_CEL.columns.drop(['Comedy','Action', 'Thriller'])]
MovieLens_CEL
user_id movie_id rating age Drama Sci-Fi gender_M occupation_academic/educator occupation_college/grad student occupation_executive/managerial occupation_other occupation_technician/engineer
0 48.0 1193.0 4.0 25.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0
1 48.0 919.0 4.0 25.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0
2 48.0 527.0 5.0 25.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0
3 48.0 1721.0 4.0 25.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0
4 48.0 150.0 4.0 25.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0
... ... ... ... ... ... ... ... ... ... ... ... ...
65637 5878.0 3300.0 2.0 25.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0
65638 5878.0 1391.0 1.0 25.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0
65639 5878.0 185.0 4.0 25.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0
65640 5878.0 2232.0 1.0 25.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0
65641 5878.0 426.0 3.0 25.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0

65642 rows × 12 columns

n = len(MovieLens_CEL)

# DR-learner for HTE estimation
np.random.seed(1)

outcome = 'rating'
treatment = 'Drama'
#controls = MovieLens_CEL.columns[userinfo_index]
controls = ['age', 'gender_M', 'occupation_academic/educator',
       'occupation_college/grad student', 'occupation_executive/managerial',
       'occupation_other', 'occupation_technician/engineer']
n_folds = 5
y_model = GradientBoostingRegressor(max_depth=2)
ps_model = LogisticRegression()
Rlearner_model = GradientBoostingRegressor(max_depth=2)

HTE_DR_learner = DRlearner(MovieLens_CEL, outcome, treatment, controls, y_model, ps_model)
HTE_DR_learner = HTE_DR_learner.to_numpy()

estimate with DR-learner

fold 1, testing r2 baselearner: 0.036, pslearner: 0.735

fold 2, testing r2 baselearner: 0.039, pslearner: 0.735

fold 3, testing r2 baselearner: 0.039, pslearner: 0.735

fold 4, testing r2 baselearner: 0.038, pslearner: 0.735

fold 5, testing r2 baselearner: 0.037, pslearner: 0.734

Let’s focus on the estimated HTEs for three randomly chosen users:

print("DR-learner:  ",HTE_DR_learner[np.array([0,1000,5000])])

DR-learner:   [ 1.05672212 -1.73726057  1.09360586]

ATE_DR_learner = np.sum(HTE_DR_learner)/n
print("Choosing Drama instead of Sci-Fi is expected to improve the rating of all users by",round(ATE_DR_learner,4), "out of 5 points.")

Choosing Drama instead of Sci-Fi is expected to improve the rating of all users by 0.3541 out of 5 points.

Conclusion: Choosing Drama instead of Sci-Fi is expected to improve the rating of all users by 0.3541 out of 5 points.

References#

  1. Xinkun Nie and Stefan Wager. Quasi-oracle estimation of heterogeneous treatment effects. Biometrika, 108(2):299–319, 2021.

  2. Peter M Robinson. Root-n-consistent semiparametric regression. Econometrica: Journal of the Econometric Society, pages 931–954, 1988.

  3. Edward H Kennedy. Optimal doubly robust estimation of heterogeneous causal effects. arXiv preprint arXiv:2004.14497, 2020

  4. M. J. van der Laan. Statistical inference for variable importance. The International Journal of Biostatistics, 2(1), 2006.

  5. S. Lee, R. Okui, and Y.-J. Whang. Doubly robust uniform confidence band for the conditional average treatment effect function. Journal of Applied Econometrics, 32(7):1207–1225, 2017.

  6. D. J. Foster and V. Syrgkanis. Orthogonal statistical learning. arXiv preprint arXiv:1901.09036, 2019.