Additional estimands

Dual bounds can apply beyond the settings described in the previous sections.

Variance of the CATE

Dual bounds can also be used to lower-bound the variance of the conditional average treatment effect \(\theta = \text{Var}(E[Y(1) - Y(0) \mid X])\), as shown below.

[1]:

# Import packages
import sys; sys.path.insert(0, "../../../")
import numpy as np
import dualbounds as db
from dualbounds.generic import DualBounds
# Generate synthetic data
data = db.gen_data.gen_regression_data(n=500, p=30)

[2]:

vdb = db.varcate.CalibratedVarCATEDualBounds(
    outcome=data['y'],
    treatment=data['W'],
    covariates=data['X'],
    propensities=data['pis'],
    outcome_model='elasticnet',
)
vdb.fit()
print(vdb.results().to_markdown())

Cross-fitting the outcome model.

Fitting cluster bootstrap to aggregate results.

|            |    Lower |   Upper |
|:-----------|---------:|--------:|
| Estimate   | 8.48493  |     nan |
| SE         | 0.748497 |     nan |
| Conf. Int. | 6.98069  |     nan |

We broadly recommend using the class CalibratedVarCATEDualBounds instead of VarCATEDualBounds. (Both have the same API, but the former will yield more powerful results.)

Variance of the ITE

Dual bounds can also be used to upper and lower bound the variance of the individual treatment effect \(\theta = \text{Var}(Y(1) - Y(0))\), as shown below.

[3]:

vdb = db.varite.VarITEDualBounds(
    outcome=data['y'],
    treatment=data['W'],
    covariates=data['X'],
    propensities=data['pis'],
    outcome_model='elasticnet',
)
vdb.fit()
print(vdb.results().to_markdown())

Cross-fitting the outcome model.

Estimating optimal dual variables.

|            |    Lower |     Upper |
|:-----------|---------:|----------:|
| Estimate   | 8.37082  | 12.8302   |
| SE         | 0.746638 |  0.818289 |
| Conf. Int. | 6.90744  | 14.434    |

Lee Bounds under monotonicity

Lee bounds are a method to bound the average treatment effect in the face of post-treatment nonrandom sample selection, named in honor of Lee (2009). Precisely, we assume we observe the following data:

  • Pre-treatment covariates \(X_i \in \mathcal{X}\)

  • A binary treatment \(W_i \in \{0,1\}\)

  • A post-treatment selection indicator \(S_i \in \{0,1\}\).

  • An outcome \(Y_i \in \mathbb{R}\).

Note that both \(Y_i\) and \(S_i\) have potential outcomes \((Y_i(0), Y_i(1))\) and \((S_i(0), S_i(1))\) since both potentially depend on the treatment.

A classic example is a setting where \(W_i\) denotes enrollment in a job training program, \(S_i\) denotes whether a subject entered the labor market, and the outcome \(Y_i\) denotes wages. A natural estimand in these settings is the average treatment effect for subjects who would have entered the labor market no matter their treatment status; e.g.,

\[E[Y(1) - Y(0) \mid S(1) = S(0) = 1].\]

Dual bounds can be used to bound this partially identified estimand under the monotonicity assumption that \(S(1) \ge S(0)\) a.s., as shown below.

[4]:

# create data
lee_data = db.gen_data.gen_lee_bound_data(n=900, p=30, sample_seed=123)

# fit lee bounds
ldb = db.lee.LeeDualBounds(
    # data
    selections=lee_data['S'],
    covariates=lee_data['X'],
    treatment=lee_data['W'],
    propensities=lee_data['pis'],
    outcome=lee_data['y'],
    # Model specifications
    outcome_model='ridge',
    selection_model='monotone_logistic',
)
ldb.fit().summary()

Cross-fitting the selection model.

Cross-fitting the outcome model.

Estimating optimal dual variables.

___________________Inference_____________________
               Lower     Upper
Estimate    2.220620  3.155182
SE          0.191205  0.193274
Conf. Int.  1.845866  3.533992

________________Selection model__________________
                            Model  No covariates
Out-of-sample R^2        0.172740       0.000000
Accuracy                 0.703333       0.594444
Likelihood (geom. mean)  0.557705       0.508488

_________________Outcome model___________________
                      Model  No covariates
Out-of-sample R^2  0.919644       0.000000
RMSE               1.057244       3.729639
MAE                0.846142       2.954590

________________Treatment model__________________
                            Model  No covariates
Out-of-sample R^2        0.001111       0.000000
Accuracy                 0.500000       0.516667
Likelihood (geom. mean)  0.500000       0.499721

It is also possible to bound this estimand without the monotonicity assumption using the generic DualBounds class, although we caution that without the monotonicity assumption, the bounds might be too wide to be useful. Please see Ji et al. (2023), Section 2.5 for details.