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Download Jupyter notebook:
https://docs.doubleml.org/stable/examples/py_double_ml_cate_plr.ipynb.
Python: Conditional Average Treatment Effects (CATEs) for PLR models#
In this simple example, we illustrate how the DoubleML package can be used to estimate conditional average treatment effects with B-splines for one or two-dimensional effects in the DoubleMLPLR model.
[1]:
import numpy as np
import pandas as pd
import doubleml as dml
from doubleml.datasets import make_heterogeneous_data
Data#
We define a data generating process to create synthetic data to compare the estimates to the true effect. The data generating process is based on the Monte Carlo simulation from Oprescu et al. (2019).
The documentation of the data generating process can be found here.
One-dimensional Example#
We start with an one-dimensional effect and create our training data. In this example the true effect depends only the first covariate \(X_0\) and takes the following form
The generated dictionary also contains a callable with key treatment_effect to calculate the true treatment effect for new observations.
[2]:
np.random.seed(42)
data_dict = make_heterogeneous_data(
n_obs=2000,
p=10,
support_size=5,
n_x=1,
)
treatment_effect = data_dict['treatment_effect']
data = data_dict['data']
print(data.head())
y d X_0 X_1 X_2 X_3 X_4 \
0 1.564451 0.241064 0.259828 0.886086 0.895690 0.297287 0.229994
1 1.114570 0.040912 0.824350 0.396992 0.156317 0.737951 0.360475
2 8.901013 1.392623 0.988421 0.977280 0.793818 0.659423 0.577807
3 -1.315155 -0.551317 0.427486 0.330285 0.564232 0.850575 0.201528
4 1.314625 0.683487 0.016200 0.818380 0.040139 0.889913 0.991963
X_5 X_6 X_7 X_8 X_9
0 0.411304 0.240532 0.672384 0.826065 0.673092
1 0.671271 0.270644 0.081230 0.992582 0.156202
2 0.866102 0.289440 0.467681 0.619390 0.411190
3 0.934433 0.689088 0.823273 0.556191 0.779517
4 0.294067 0.210319 0.765363 0.253026 0.865562
First, define the DoubleMLData object.
[3]:
data_dml_base = dml.DoubleMLData(
data,
y_col='y',
d_cols='d'
)
Next, define the learners for the nuisance functions and fit the PLR Model. Remark that linear learners would usually be optimal due to the data generating process.
[4]:
# First stage estimation
from sklearn.ensemble import RandomForestClassifier, RandomForestRegressor
ml_l = RandomForestRegressor(n_estimators=500)
ml_m = RandomForestRegressor(n_estimators=500)
np.random.seed(42)
dml_plr = dml.DoubleMLPLR(data_dml_base,
ml_l=ml_l,
ml_m=ml_m,
n_folds=5)
print("Training PLR Model")
dml_plr.fit()
print(dml_plr.summary)
Training PLR Model
coef std err t P>|t| 2.5 % 97.5 %
d 4.377669 0.043998 99.497422 0.0 4.291434 4.463903
To estimate the CATE, we rely on the best-linear-predictor of the linear score as in Semenova et al. (2021) To approximate the target function \(\theta_0(x)\) with a linear form, we have to define a data frame of basis functions. Here, we rely on patsy to construct a suitable basis of B-splines.
[5]:
import patsy
design_matrix = patsy.dmatrix("bs(x, df=5, degree=2)", {"x": data["X_0"]})
spline_basis = pd.DataFrame(design_matrix)
To estimate the parameters to calculate the CATE estimate call the cate() method and supply the dataframe of basis elements.
[6]:
cate = dml_plr.cate(spline_basis)
print(cate)
================== DoubleMLBLP Object ==================
------------------ Fit summary ------------------
coef std err t P>|t| [0.025 0.975]
0 1.239313 0.141729 8.744228 4.675733e-18 0.961360 1.517266
1 1.581849 0.237292 6.666259 3.385877e-11 1.116483 2.047215
2 4.178218 0.150136 27.829619 2.742758e-144 3.883778 4.472657
3 4.040919 0.182692 22.118721 3.950131e-97 3.682631 4.399207
4 3.272408 0.186795 17.518682 5.025783e-64 2.906073 3.638742
5 3.796384 0.194232 19.545602 5.614185e-78 3.415465 4.177304
To obtain the confidence intervals for the CATE, we have to call the confint() method and a supply a dataframe of basis elements. This could be the same basis as for fitting the CATE model or a new basis to e.g. evaluate the CATE model on a grid. Here, we will evaluate the CATE on a grid from 0.1 to 0.9 to plot the final results. Further, we construct uniform confidence intervals by setting the option joint and providing a number of bootstrap repetitions n_rep_boot.
[7]:
new_data = {"x": np.linspace(0.1, 0.9, 100)}
spline_grid = pd.DataFrame(patsy.build_design_matrices([design_matrix.design_info], new_data)[0])
df_cate = cate.confint(spline_grid, level=0.95, joint=True, n_rep_boot=2000)
print(df_cate)
2.5 % effect 97.5 %
0 2.199412 2.429057 2.658702
1 2.289357 2.521611 2.753866
2 2.376806 2.613622 2.850439
3 2.462567 2.705090 2.947613
4 2.547324 2.796014 3.044704
.. ... ... ...
95 4.494089 4.734770 4.975450
96 4.502901 4.738065 4.973229
97 4.514173 4.743341 4.972509
98 4.527452 4.750597 4.973741
99 4.542159 4.759833 4.977507
[100 rows x 3 columns]
Finally, we can plot our results and compare them with the true effect.
[8]:
from matplotlib import pyplot as plt
plt.rcParams['figure.figsize'] = 10., 7.5
df_cate['x'] = new_data['x']
df_cate['true_effect'] = treatment_effect(new_data["x"].reshape(-1, 1))
fig, ax = plt.subplots()
ax.plot(df_cate['x'],df_cate['effect'], label='Estimated Effect')
ax.plot(df_cate['x'],df_cate['true_effect'], color="green", label='True Effect')
ax.fill_between(df_cate['x'], df_cate['2.5 %'], df_cate['97.5 %'], color='b', alpha=.3, label='Confidence Interval')
plt.legend()
plt.title('CATE')
plt.xlabel('x')
_ = plt.ylabel('Effect and 95%-CI')
If the effect is not one-dimensional, the estimate still corresponds to the projection of the true effect on the basis functions.
Two-Dimensional Example#
It is also possible to estimate multi-dimensional conditional effects. We will use a similar data generating process but now the effect depends on the first two covariates \(X_0\) and \(X_1\) and takes the following form
With the argument n_x=2 we can specify set the effect to be two-dimensional.
[9]:
np.random.seed(42)
data_dict = make_heterogeneous_data(
n_obs=5000,
p=10,
support_size=5,
n_x=2,
)
treatment_effect = data_dict['treatment_effect']
data = data_dict['data']
print(data.head())
y d X_0 X_1 X_2 X_3 X_4 \
0 -0.359307 -0.479722 0.014080 0.006958 0.240127 0.100807 0.260211
1 0.578557 -0.587135 0.152148 0.912230 0.892796 0.653901 0.672234
2 1.479882 0.172083 0.344787 0.893649 0.291517 0.562712 0.099731
3 4.468072 0.480579 0.619351 0.232134 0.000943 0.757151 0.985207
4 5.949866 0.974213 0.477130 0.447624 0.775191 0.526769 0.316717
X_5 X_6 X_7 X_8 X_9
0 0.177043 0.028520 0.909304 0.008223 0.736082
1 0.005339 0.984872 0.877833 0.895106 0.659245
2 0.921956 0.140770 0.224897 0.558134 0.764093
3 0.809913 0.460207 0.903767 0.409848 0.524934
4 0.258158 0.037747 0.583195 0.229961 0.148134
As univariate example estimate the PLR Model.
[10]:
data_dml_base = dml.DoubleMLData(
data,
y_col='y',
d_cols='d'
)
[11]:
# First stage estimation
from sklearn.ensemble import RandomForestClassifier, RandomForestRegressor
ml_l = RandomForestRegressor(n_estimators=500)
ml_m = RandomForestRegressor(n_estimators=500)
np.random.seed(42)
dml_plr = dml.DoubleMLPLR(data_dml_base,
ml_l=ml_l,
ml_m=ml_m,
n_folds=5)
print("Training PLR Model")
dml_plr.fit()
print(dml_plr.summary)
Training PLR Model
coef std err t P>|t| 2.5 % 97.5 %
d 4.469895 0.04973 89.883485 0.0 4.372427 4.567364
As above, we will rely on the patsy package to construct the basis elements. In the two-dimensional case, we will construct a tensor product of B-splines (for more information see here).
[12]:
design_matrix = patsy.dmatrix("te(bs(x_0, df=7, degree=3), bs(x_1, df=7, degree=3))", {"x_0": data["X_0"], "x_1": data["X_1"]})
spline_basis = pd.DataFrame(design_matrix)
cate = dml_plr.cate(spline_basis)
print(cate)
================== DoubleMLBLP Object ==================
------------------ Fit summary ------------------
coef std err t P>|t| [0.025 0.975]
0 2.785038 0.103186 26.990377 8.380432e-150 2.582747 2.987329
1 -3.326148 0.865540 -3.842859 1.231330e-04 -5.022991 -1.629306
2 3.068073 0.886989 3.458976 5.468051e-04 1.329181 4.806964
3 1.733047 0.768331 2.255598 2.413933e-02 0.226776 3.239317
4 2.016011 0.772253 2.610559 9.066689e-03 0.502054 3.529969
5 -3.886041 0.929598 -4.180348 2.960875e-05 -5.708465 -2.063618
6 -5.448842 1.011511 -5.386834 7.502843e-08 -7.431852 -3.465832
7 -7.348700 0.916806 -8.015548 1.356886e-15 -9.146046 -5.551355
8 -0.629549 0.909141 -0.692465 4.886777e-01 -2.411869 1.152772
9 -0.213070 0.963051 -0.221245 8.249109e-01 -2.101076 1.674936
10 1.905042 0.826391 2.305255 2.119348e-02 0.284949 3.525135
11 0.572991 0.826215 0.693513 4.880202e-01 -1.046757 2.192739
12 -0.871923 1.044062 -0.835125 4.036874e-01 -2.918747 1.174901
13 -1.231153 1.127831 -1.091611 2.750571e-01 -3.442202 0.979896
14 -2.161049 0.955926 -2.260687 2.382188e-02 -4.035088 -0.287011
15 0.084156 0.788400 0.106743 9.149973e-01 -1.461458 1.629771
16 2.889326 0.795558 3.631821 2.842770e-04 1.329679 4.448973
17 1.899716 0.682353 2.784066 5.388593e-03 0.562001 3.237430
18 2.681246 0.666912 4.020387 5.896681e-05 1.373802 3.988690
19 -1.993201 0.845059 -2.358653 1.838006e-02 -3.649891 -0.336510
20 -1.965341 0.842901 -2.331640 1.975957e-02 -3.617800 -0.312882
21 -4.128651 0.735054 -5.616797 2.051578e-08 -5.569684 -2.687619
22 0.161288 0.780120 0.206748 8.362155e-01 -1.368092 1.690668
23 4.430595 0.796220 5.564537 2.766850e-08 2.869651 5.991539
24 2.308568 0.692199 3.335121 8.588992e-04 0.951550 3.665585
25 3.346269 0.656526 5.096934 3.580477e-07 2.059187 4.633350
26 0.029022 0.860261 0.033737 9.730884e-01 -1.657470 1.715515
27 -1.186237 0.951115 -1.247207 2.123806e-01 -3.050843 0.678369
28 -1.137213 0.892331 -1.274430 2.025708e-01 -2.886577 0.612151
29 5.555137 0.996454 5.574904 2.608003e-08 3.601645 7.508630
30 1.414533 0.972732 1.454185 1.459584e-01 -0.492454 3.321520
31 7.397155 0.858579 8.615574 9.234812e-18 5.713959 9.080351
32 2.910895 0.815574 3.569135 3.615573e-04 1.312008 4.509782
33 2.228630 1.040141 2.142624 3.219196e-02 0.189493 4.267767
34 0.202846 1.133343 0.178980 8.579605e-01 -2.019008 2.424700
35 1.050538 1.048326 1.002110 3.163393e-01 -1.004645 3.105722
36 6.569540 1.028001 6.390599 1.804081e-10 4.554203 8.584877
37 5.115636 1.050399 4.870185 1.149671e-06 3.056389 7.174884
38 7.074617 0.836515 8.457252 3.561183e-17 5.434677 8.714557
39 5.933322 0.907198 6.540270 6.760494e-11 4.154811 7.711834
40 4.316863 1.114591 3.873048 1.088696e-04 2.131771 6.501954
41 1.626633 1.254551 1.296585 1.948344e-01 -0.832844 4.086109
42 -0.761429 1.095835 -0.694839 4.871887e-01 -2.909752 1.386894
43 8.384443 0.996946 8.410124 5.297779e-17 6.429986 10.338900
44 7.251480 1.161088 6.245416 4.580231e-10 4.975232 9.527728
45 8.031820 0.825824 9.725820 3.690796e-22 6.412838 9.650802
46 7.031156 0.869136 8.089825 7.452484e-16 5.327265 8.735048
47 4.639580 1.041112 4.456370 8.519888e-06 2.598539 6.680620
48 2.492656 0.956110 2.607080 9.159202e-03 0.618256 4.367056
49 3.651127 0.821870 4.442462 9.087491e-06 2.039897 5.262357
Finally, we create a new grid to evaluate and plot the effects.
[13]:
grid_size = 100
x_0 = np.linspace(0.1, 0.9, grid_size)
x_1 = np.linspace(0.1, 0.9, grid_size)
x_0, x_1 = np.meshgrid(x_0, x_1)
new_data = {"x_0": x_0.ravel(), "x_1": x_1.ravel()}
[14]:
spline_grid = pd.DataFrame(patsy.build_design_matrices([design_matrix.design_info], new_data)[0])
df_cate = cate.confint(spline_grid, joint=True, n_rep_boot=2000)
print(df_cate)
2.5 % effect 97.5 %
0 1.333704 2.035264 2.736823
1 1.349638 2.035441 2.721245
2 1.376760 2.042249 2.707738
3 1.412726 2.055171 2.697616
4 1.455091 2.073694 2.692297
... ... ... ...
9995 3.543691 4.251412 4.959132
9996 3.582146 4.332502 5.082858
9997 3.619128 4.416132 5.213135
9998 3.659339 4.502595 5.345852
9999 3.707125 4.592186 5.477247
[10000 rows x 3 columns]
[15]:
import plotly.graph_objects as go
grid_array = np.array(list(zip(x_0.ravel(), x_1.ravel())))
true_effect = treatment_effect(grid_array).reshape(x_0.shape)
effect = np.asarray(df_cate['effect']).reshape(x_0.shape)
lower_bound = np.asarray(df_cate['2.5 %']).reshape(x_0.shape)
upper_bound = np.asarray(df_cate['97.5 %']).reshape(x_0.shape)
fig = go.Figure(data=[
go.Surface(x=x_0,
y=x_1,
z=true_effect),
go.Surface(x=x_0,
y=x_1,
z=upper_bound, showscale=False, opacity=0.4,colorscale='purp'),
go.Surface(x=x_0,
y=x_1,
z=lower_bound, showscale=False, opacity=0.4,colorscale='purp'),
])
fig.update_traces(contours_z=dict(show=True, usecolormap=True,
highlightcolor="limegreen", project_z=True))
fig.update_layout(scene = dict(
xaxis_title='X_0',
yaxis_title='X_1',
zaxis_title='Effect'),
width=700,
margin=dict(r=20, b=10, l=10, t=10))
fig.show()