2. T-learner#

The second learner is called T-learner, which denotes ``two learners”. Instead of fitting a single model to estimate the potential outcomes under both treatment and control groups, T-learner aims to learn different models for \(\mathbb{E}[R(1)|S]\) and \(\mathbb{E}[R(0)|S]\) separately, and finally combines them to obtain a final HTE estimator.

Define the control response function as \(\mu_0(s)=\mathbb{E}[R(0)|S=s]\), and the treatment response function as \(\mu_1(s)=\mathbb{E}[R(1)|S=s]\). The algorithm of T-learner is summarized below:

Step 1: Estimate \(\mu_0(s)\) and \(\mu_1(s)\) separately with any regression algorithms or supervised machine learning methods;

Step 2: Estimate HTE by

\[\begin{equation*} \hat{\tau}_{\text{T-learner}}(s)=\hat\mu_1(s)-\hat\mu_0(s). \end{equation*}\]
# 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 causaldm.learners.CEL.Single_Stage import _env_getdata_CEL

MovieLens Data#

# Get the MovieLens data
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)
userinfo_index = np.array([3,6,7,8,9,10,11])
SandA = MovieLens_CEL.iloc[:, np.array([3,5,6,7,8,9,10,11])]

mu0 = GradientBoostingRegressor(max_depth=3)
mu1 = GradientBoostingRegressor(max_depth=3)

mu0.fit(MovieLens_CEL.iloc[np.where(MovieLens_CEL['Drama']==0)[0],userinfo_index],MovieLens_CEL.iloc[np.where(MovieLens_CEL['Drama']==0)[0],2] )
mu1.fit(MovieLens_CEL.iloc[np.where(MovieLens_CEL['Drama']==1)[0],userinfo_index],MovieLens_CEL.iloc[np.where(MovieLens_CEL['Drama']==1)[0],2] )


# estimate the HTE by T-learner
HTE_T_learner = mu1.predict(MovieLens_CEL.iloc[:,userinfo_index]) - mu0.predict(MovieLens_CEL.iloc[:,userinfo_index])

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

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

T-learner:   [0.3598282  0.34648075 0.35533324]

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

Choosing Drama instead of Sci-Fi is expected to improve the rating of all users by 0.3571 out of 5 points.

Conclusion: Same as the estimation result provided by S-learner, people are more inclined to give higher ratings to drama than science fictions. The expected causal effect estiamted by T-learner is larger than S-learner. In some cases when the treatment effect is relatively complex, it’s likely to yield better performance by fitting two models separately.

However, in an extreme case when both \(\mu_0(s)\) and \(\mu_1(s)\) are nonlinear complicated function of state \(s\) while their difference is just a constant, T-learner will overfit each model very easily, yielding a nonlinear treatment effect estimator. In this case, other estimators are often preferred.

References#

  1. Kunzel, S. R., Sekhon, J. S., Bickel, P. J., and Yu, B. (2019). Metalearners for estimating heterogeneous treatment effects using machine learning. Proceedings of the national academy of sciences 116, 4156–4165.