5. QIATEs

5.1. Definition

A Quantile Individualized Average Treatment Effect (QIATE) is a quantile of the conditional mean treatment-effect function across the target population:

\[\begin{split}\tau_{m,l}(x) &= E[Y^m-Y^l\mid X=x], \\ \operatorname{QIATE}(q;m,l) &= \inf\{t:\Pr(\tau_{m,l}(X)\leq t)\geq q\}.\end{split}\]

The distribution is over \(X\) in the prediction population; at a fixed \(x\), the conditional mean effect is a scalar.

5.2. Interpretation

QIATEs describe heterogeneity in mean effects explained by observed features. They are neither quantiles of unobserved realized individual effects nor differences between marginal potential-outcome quantiles.

5.3. Implementation

Estimated IATEs are sorted separately for each outcome and treatment comparison. With p_qiate_smooth=True, QIATE estimation combines observations with similar ranks using kernel weights. Without smoothing, it uses the observation closest to the requested quantile. p_qiate_bias_adjust controls a separate adjustment for ranking bias.

p_qiate_no_of_quantiles sets the number of evenly spaced quantile levels to estimate. The median is included when this number is odd. You can inspect the selected levels using mymcf.p_cfg.qiate_quantiles.

• p_qiate_m_mqiate returns QIATE(q) minus QIATE(0.5) as qiate_mmed, with qiate_mmed_se when requested.

• p_qiate_m_opp returns QIATE(q) minus QIATE(1-q) as qiate_mopp, with qiate_mopp_se, including levels above 0.5.

QIATEs require discrete treatments, _int_low_memory_predict=False, p_ba=False and no treatment versions.

5.4. Example

from mcf.example_data import example_data
from mcf.mcf_main import ModifiedCausalForest

# Generate artificial data
training_df, prediction_df, name_dict = example_data(
    no_treatments=2,
    obs_y_d_x_iate=2000,
    obs_x_iate=2000,
    no_effect=False
)

mymcf = ModifiedCausalForest(
    var_d_name=name_dict['d_name'],
    var_y_name=name_dict['y_name'],
    var_x_name_ord=name_dict['x_name_ord'],
    var_x_name_unord=name_dict['x_name_unord'],
    # QIATE specific parameters
    p_qiate=True,
    p_ba=False,
    p_qiate_se=True,
    p_qiate_m_mqiate=True,
    p_qiate_m_opp=True,
    p_qiate_no_of_quantiles=None,
    p_qiate_smooth=None,
    p_qiate_smooth_bandwidth=None,
    p_qiate_bias_adjust=None,
    _int_low_memory_predict=False
)

mymcf.train(training_df)
results = mymcf.predict(prediction_df)




print(mymcf.p_cfg.qiate_quantiles)
print(results['qiate'])
print(results['qiate_mmed'])
print(results['qiate_mopp'])