def fit(self, X, p, treatment, y, verbose=True):
"""Fit the treatment effect and outcome models of the R learner.
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
p (np.ndarray or pd.Series or dict): an array of propensity scores of float (0,1) in the single-treatment
case; or, a dictionary of treatment groups that map to propensity vectors of float (0,1)
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
"""
X, treatment, y = convert_pd_to_np(X, treatment, y)
check_treatment_vector(treatment, self.control_name)
self.t_groups = np.unique(treatment[treatment != self.control_name])
self.t_groups.sort()
check_p_conditions(p, self.t_groups)
if isinstance(p, (np.ndarray, pd.Series)):
treatment_name = self.t_groups[0]
p = {treatment_name: convert_pd_to_np(p)}
elif isinstance(p, dict):
p = {
treatment_name: convert_pd_to_np(_p)
for treatment_name, _p in p.items()
}
self._classes = {group: i for i, group in enumerate(self.t_groups)}
self.models_tau = {
group: deepcopy(self.model_tau)
for group in self.t_groups
}
self.vars_c = {}
self.vars_t = {}
if verbose:
logger.info('generating out-of-fold CV outcome estimates')
yhat = cross_val_predict(self.model_mu,
X,
y,
cv=self.cv,
method='predict_proba',
n_jobs=-1)[:, 1]
for group in self.t_groups:
mask = (treatment == group) | (treatment == self.control_name)
treatment_filt = treatment[mask]
X_filt = X[mask]
y_filt = y[mask]
yhat_filt = yhat[mask]
p_filt = p[group][mask]
w = (treatment_filt == group).astype(int)
if verbose:
logger.info(
'training the treatment effect model for {} with R-loss'.
format(group))
self.models_tau[group].fit(X_filt,
(y_filt - yhat_filt) / (w - p_filt),
sample_weight=(w - p_filt)**2)
self.vars_c[group] = (y_filt[w == 0] - yhat_filt[w == 0]).var()
self.vars_t[group] = (y_filt[w == 1] - yhat_filt[w == 1]).var()
def fit_predict(self, X, treatment, y, p=None, return_ci=False,
n_bootstraps=1000, bootstrap_size=10000, verbose=True):
"""Fit the treatment effect and outcome models of the R learner and predict treatment effects.
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
p (np.ndarray or pd.Series or dict, optional): an array of propensity scores of float (0,1) in the
single-treatment case; or, a dictionary of treatment groups that map to propensity vectors of
float (0,1); if None will run ElasticNetPropensityModel() to generate the propensity scores.
return_ci (bool): whether to return confidence intervals
n_bootstraps (int): number of bootstrap iterations
bootstrap_size (int): number of samples per bootstrap
verbose (bool): whether to output progress logs
Returns:
(numpy.ndarray): Predictions of treatment effects. Output dim: [n_samples, n_treatment].
If return_ci, returns CATE [n_samples, n_treatment], LB [n_samples, n_treatment],
UB [n_samples, n_treatment]
"""
X, treatment, y = convert_pd_to_np(X, treatment, y)
self.fit(X, treatment, y, p, verbose=verbose)
te = self.predict(X)
if p is None:
p = self.propensity
else:
check_p_conditions(p, self.t_groups)
if isinstance(p, (np.ndarray, pd.Series)):
treatment_name = self.t_groups[0]
p = {treatment_name: convert_pd_to_np(p)}
elif isinstance(p, dict):
p = {treatment_name: convert_pd_to_np(_p) for treatment_name, _p in p.items()}
if not return_ci:
return te
else:
t_groups_global = self.t_groups
_classes_global = self._classes
model_mu_global = deepcopy(self.model_mu)
models_tau_global = deepcopy(self.models_tau)
te_bootstraps = np.zeros(shape=(X.shape[0], self.t_groups.shape[0], n_bootstraps))
logger.info('Bootstrap Confidence Intervals')
for i in tqdm(range(n_bootstraps)):
te_b = self.bootstrap(X, treatment, y, p, size=bootstrap_size)
te_bootstraps[:, :, i] = te_b
te_lower = np.percentile(te_bootstraps, (self.ate_alpha / 2) * 100, axis=2)
te_upper = np.percentile(te_bootstraps, (1 - self.ate_alpha / 2) * 100, axis=2)
# set member variables back to global (currently last bootstrapped outcome)
self.t_groups = t_groups_global
self._classes = _classes_global
self.model_mu = deepcopy(model_mu_global)
self.models_tau = deepcopy(models_tau_global)
return (te, te_lower, te_upper)
def predict(self,
X,
p,
treatment=None,
y=None,
return_components=False,
verbose=True):
"""Predict treatment effects.
Args:
X (np.matrix): a feature matrix
p (np.ndarray or dict): an array of propensity scores of float (0,1) in the single-treatment case
or, a dictionary of treatment groups that map to propensity vectors of float (0,1)
treatment (np.array, optional): a treatment vector
y (np.array, optional): an optional outcome vector
Returns:
(numpy.ndarray): Predictions of treatment effects.
"""
check_p_conditions(p, self.t_groups)
if isinstance(p, np.ndarray):
treatment_name = self.t_groups[0]
p = {treatment_name: p}
te = np.zeros((X.shape[0], self.t_groups.shape[0]))
dhat_cs = {}
dhat_ts = {}
for i, group in enumerate(self.t_groups):
model_tau_c = self.models_tau_c[group]
model_tau_t = self.models_tau_t[group]
dhat_cs[group] = model_tau_c.predict(X)
dhat_ts[group] = model_tau_t.predict(X)
_te = (p[group] * dhat_cs[group] +
(1 - p[group]) * dhat_ts[group]).reshape(-1, 1)
te[:, i] = np.ravel(_te)
if (y is not None) and (treatment is not None) and verbose:
mask = (treatment == group) | (treatment == self.control_name)
treatment_filt = treatment[mask]
X_filt = X[mask]
y_filt = y[mask]
w = (treatment_filt == group).astype(int)
yhat = np.zeros_like(y_filt, dtype=float)
yhat[w == 0] = self.models_mu_c[group].predict_proba(
X_filt[w == 0])[:, 1]
yhat[w == 1] = self.models_mu_t[group].predict_proba(
X_filt[w == 1])[:, 1]
logger.info('Error metrics for group {}'.format(group))
classification_metrics(y_filt, yhat, w)
if not return_components:
return te
else:
return te, dhat_cs, dhat_ts
def predict(self, X, p, treatment=None, y=None, return_components=False, verbose=True):
"""Predict treatment effects.
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
p (np.ndarray or pd.Series or dict): an array of propensity scores of float (0,1) in the single-treatment
case; or, a dictionary of treatment groups that map to propensity vectors of float (0,1)
treatment (np.array or pd.Series, optional): a treatment vector
y (np.array or pd.Series, optional): an outcome vector
return_components (bool, optional): whether to return outcome for treatment and control seperately
Returns:
(numpy.ndarray): Predictions of treatment effects.
"""
X, treatment, y = convert_pd_to_np(X, treatment, y)
check_p_conditions(p, self.t_groups)
if isinstance(p, (np.ndarray, pd.Series)):
treatment_name = self.t_groups[0]
p = {treatment_name: convert_pd_to_np(p)}
elif isinstance(p, dict):
p = {treatment_name: convert_pd_to_np(_p) for treatment_name, _p in p.items()}
te = np.zeros((X.shape[0], self.t_groups.shape[0]))
dhat_cs = {}
dhat_ts = {}
for i, group in enumerate(self.t_groups):
model_tau_c = self.models_tau_c[group]
model_tau_t = self.models_tau_t[group]
dhat_cs[group] = model_tau_c.predict(X)
dhat_ts[group] = model_tau_t.predict(X)
_te = (p[group] * dhat_cs[group] + (1 - p[group]) * dhat_ts[group]).reshape(-1, 1)
te[:, i] = np.ravel(_te)
if (y is not None) and (treatment is not None) and verbose:
mask = (treatment == group) | (treatment == self.control_name)
treatment_filt = treatment[mask]
X_filt = X[mask]
y_filt = y[mask]
w = (treatment_filt == group).astype(int)
yhat = np.zeros_like(y_filt, dtype=float)
yhat[w == 0] = self.models_mu_c[group].predict(X_filt[w == 0])
yhat[w == 1] = self.models_mu_t[group].predict(X_filt[w == 1])
logger.info('Error metrics for group {}'.format(group))
regression_metrics(y_filt, yhat, w)
if not return_components:
return te
else:
return te, dhat_cs, dhat_ts
def _format_p(p, t_groups):
"""Format propensity scores into a dictionary of {treatment group: propensity scores}.
Args:
p (np.ndarray, pd.Series, or dict): propensity scores
t_groups (list): treatment group names.
Returns:
dict of {treatment group: propensity scores}
"""
check_p_conditions(p, t_groups)
if isinstance(p, (np.ndarray, pd.Series)):
treatment_name = t_groups[0]
p = {treatment_name: convert_pd_to_np(p)}
elif isinstance(p, dict):
p = {
treatment_name: convert_pd_to_np(_p) for treatment_name, _p in p.items()
}
return p
def fit(self, X, treatment, y, p=None):
"""Fit the inference model.
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
p (np.ndarray or pd.Series or dict, optional): an array of propensity scores of float (0,1) in the
single-treatment case; or, a dictionary of treatment groups that map to propensity vectors of
float (0,1); if None will run ElasticNetPropensityModel() to generate the propensity scores.
"""
X, treatment, y = convert_pd_to_np(X, treatment, y)
check_treatment_vector(treatment, self.control_name)
self.t_groups = np.unique(treatment[treatment != self.control_name])
self.t_groups.sort()
if p is None:
logger.info('Generating propensity score')
p = dict()
p_model = dict()
for group in self.t_groups:
mask = (treatment == group) | (treatment == self.control_name)
treatment_filt = treatment[mask]
X_filt = X[mask]
w_filt = (treatment_filt == group).astype(int)
w = (treatment == group).astype(int)
p[group], p_model[group] = compute_propensity_score(
X=X_filt, treatment=w_filt, X_pred=X, treatment_pred=w)
self.propensity_model = p_model
self.propensity = p
else:
check_p_conditions(p, self.t_groups)
if isinstance(p, (np.ndarray, pd.Series)):
treatment_name = self.t_groups[0]
p = {treatment_name: convert_pd_to_np(p)}
elif isinstance(p, dict):
p = {
treatment_name: convert_pd_to_np(_p)
for treatment_name, _p in p.items()
}
self._classes = {group: i for i, group in enumerate(self.t_groups)}
self.models_mu_c = {
group: deepcopy(self.model_mu_c)
for group in self.t_groups
}
self.models_mu_t = {
group: deepcopy(self.model_mu_t)
for group in self.t_groups
}
self.models_tau_c = {
group: deepcopy(self.model_tau_c)
for group in self.t_groups
}
self.models_tau_t = {
group: deepcopy(self.model_tau_t)
for group in self.t_groups
}
self.vars_c = {}
self.vars_t = {}
for group in self.t_groups:
mask = (treatment == group) | (treatment == self.control_name)
treatment_filt = treatment[mask]
X_filt = X[mask]
y_filt = y[mask]
w = (treatment_filt == group).astype(int)
# Train outcome models
self.models_mu_c[group].fit(X_filt[w == 0], y_filt[w == 0])
self.models_mu_t[group].fit(X_filt[w == 1], y_filt[w == 1])
# Calculate variances and treatment effects
var_c = (y_filt[w == 0] -
self.models_mu_c[group].predict(X_filt[w == 0])).var()
self.vars_c[group] = var_c
var_t = (y_filt[w == 1] -
self.models_mu_t[group].predict(X_filt[w == 1])).var()
self.vars_t[group] = var_t
# Train treatment models
d_c = self.models_mu_t[group].predict(
X_filt[w == 0]) - y_filt[w == 0]
d_t = y_filt[w == 1] - self.models_mu_c[group].predict(
X_filt[w == 1])
self.models_tau_c[group].fit(X_filt[w == 0], d_c)
self.models_tau_t[group].fit(X_filt[w == 1], d_t)
def estimate_ate(self,
X,
treatment,
y,
p=None,
bootstrap_ci=False,
n_bootstraps=1000,
bootstrap_size=10000):
"""Estimate the Average Treatment Effect (ATE).
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
p (np.ndarray or pd.Series or dict, optional): an array of propensity scores of float (0,1) in the
single-treatment case; or, a dictionary of treatment groups that map to propensity vectors of
float (0,1); if None will run ElasticNetPropensityModel() to generate the propensity scores.
bootstrap_ci (bool): whether run bootstrap for confidence intervals
n_bootstraps (int): number of bootstrap iterations
bootstrap_size (int): number of samples per bootstrap
Returns:
The mean and confidence interval (LB, UB) of the ATE estimate.
"""
te, dhat_cs, dhat_ts = self.fit_predict(X,
treatment,
y,
p,
return_components=True)
X, treatment, y = convert_pd_to_np(X, treatment, y)
if p is None:
p = self.propensity
else:
check_p_conditions(p, self.t_groups)
if isinstance(p, np.ndarray):
treatment_name = self.t_groups[0]
p = {treatment_name: convert_pd_to_np(p)}
elif isinstance(p, dict):
p = {
treatment_name: convert_pd_to_np(_p)
for treatment_name, _p in p.items()
}
ate = np.zeros(self.t_groups.shape[0])
ate_lb = np.zeros(self.t_groups.shape[0])
ate_ub = np.zeros(self.t_groups.shape[0])
for i, group in enumerate(self.t_groups):
_ate = te[:, i].mean()
mask = (treatment == group) | (treatment == self.control_name)
treatment_filt = treatment[mask]
w = (treatment_filt == group).astype(int)
prob_treatment = float(sum(w)) / w.shape[0]
dhat_c = dhat_cs[group][mask]
dhat_t = dhat_ts[group][mask]
p_filt = p[group][mask]
# SE formula is based on the lower bound formula (7) from Imbens, Guido W., and Jeffrey M. Wooldridge. 2009.
# "Recent Developments in the Econometrics of Program Evaluation." Journal of Economic Literature
se = np.sqrt(
(self.vars_t[group] / prob_treatment + self.vars_c[group] /
(1 - prob_treatment) +
(p_filt * dhat_c + (1 - p_filt) * dhat_t).var()) / w.shape[0])
_ate_lb = _ate - se * norm.ppf(1 - self.ate_alpha / 2)
_ate_ub = _ate + se * norm.ppf(1 - self.ate_alpha / 2)
ate[i] = _ate
ate_lb[i] = _ate_lb
ate_ub[i] = _ate_ub
if not bootstrap_ci:
return ate, ate_lb, ate_ub
else:
t_groups_global = self.t_groups
_classes_global = self._classes
models_mu_c_global = deepcopy(self.models_mu_c)
models_mu_t_global = deepcopy(self.models_mu_t)
models_tau_c_global = deepcopy(self.models_tau_c)
models_tau_t_global = deepcopy(self.models_tau_t)
logger.info('Bootstrap Confidence Intervals for ATE')
ate_bootstraps = np.zeros(shape=(self.t_groups.shape[0],
n_bootstraps))
for n in tqdm(range(n_bootstraps)):
cate_b = self.bootstrap(X,
treatment,
y,
p,
size=bootstrap_size)
ate_bootstraps[:, n] = cate_b.mean()
ate_lower = np.percentile(ate_bootstraps,
(self.ate_alpha / 2) * 100,
axis=1)
ate_upper = np.percentile(ate_bootstraps,
(1 - self.ate_alpha / 2) * 100,
axis=1)
# set member variables back to global (currently last bootstrapped outcome)
self.t_groups = t_groups_global
self._classes = _classes_global
self.models_mu_c = deepcopy(models_mu_c_global)
self.models_mu_t = deepcopy(models_mu_t_global)
self.models_tau_c = deepcopy(models_tau_c_global)
self.models_tau_t = deepcopy(models_tau_t_global)
return ate, ate_lower, ate_upper
def fit(self, X, p, treatment, y, verbose=True):
"""Fit the treatment effect and outcome models of the R learner.
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
p (np.ndarray or pd.Series or dict): an array of propensity scores of float (0,1) in the single-treatment
case; or, a dictionary of treatment groups that map to propensity vectors of float (0,1)
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
"""
check_treatment_vector(treatment, self.control_name)
X, treatment, y = convert_pd_to_np(X, treatment, y)
self.t_groups = np.unique(treatment[treatment != self.control_name])
self.t_groups.sort()
check_p_conditions(p, self.t_groups)
if isinstance(p, np.ndarray):
treatment_name = self.t_groups[0]
p = {treatment_name: convert_pd_to_np(p)}
elif isinstance(p, dict):
p = {treatment_name: convert_pd_to_np(_p) for treatment_name, _p in p.items()}
self._classes = {group: i for i, group in enumerate(self.t_groups)}
self.models_tau = {group: deepcopy(self.model_tau) for group in self.t_groups}
self.vars_c = {}
self.vars_t = {}
if verbose:
logger.info('generating out-of-fold CV outcome estimates')
yhat = cross_val_predict(self.model_mu, X, y, cv=self.cv, n_jobs=-1)
for group in self.t_groups:
treatment_mask = (treatment == group) | (treatment == self.control_name)
treatment_filt = treatment[treatment_mask]
w = (treatment_filt == group).astype(int)
X_filt = X[treatment_mask]
y_filt = y[treatment_mask]
yhat_filt = yhat[treatment_mask]
p_filt = p[group][treatment_mask]
if verbose:
logger.info('training the treatment effect model for {} with R-loss'.format(group))
if self.early_stopping:
X_train_filt, X_test_filt, y_train_filt, y_test_filt, yhat_train_filt, yhat_test_filt, \
w_train, w_test, p_train_filt, p_test_filt = train_test_split(
X_filt, y_filt, yhat_filt, w, p_filt,
test_size=self.test_size, random_state=self.random_state
)
self.models_tau[group].fit(X=X_train_filt,
y=(y_train_filt - yhat_train_filt) / (w_train - p_train_filt),
sample_weight=(w_train - p_train_filt) ** 2,
eval_set=[(X_test_filt,
(y_test_filt - yhat_test_filt) / (w_test - p_test_filt))],
sample_weight_eval_set=[(w_test - p_test_filt) ** 2],
eval_metric=self.effect_learner_eval_metric,
early_stopping_rounds=self.early_stopping_rounds,
verbose=verbose)
else:
self.models_tau[group].fit(X_filt, (y_filt - yhat_filt) / (w - p_filt),
sample_weight=(w - p_filt) ** 2,
eval_metric=self.effect_learner_eval_metric)
self.vars_c[group] = (y_filt[w == 0] - yhat_filt[w == 0]).var()
self.vars_t[group] = (y_filt[w == 1] - yhat_filt[w == 1]).var()
def estimate_ate(self, X, p, treatment, y, bootstrap_ci=False, n_bootstraps=1000, bootstrap_size=10000):
"""Estimate the Average Treatment Effect (ATE).
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
p (np.ndarray or pd.Series or dict): an array of propensity scores of float (0,1) in the single-treatment
case; or, a dictionary of treatment groups that map to propensity vectors of float (0,1)
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
bootstrap_ci (bool): whether run bootstrap for confidence intervals
n_bootstraps (int): number of bootstrap iterations
bootstrap_size (int): number of samples per bootstrap
verbose (str): whether to output progress logs
Returns:
The mean and confidence interval (LB, UB) of the ATE estimate.
"""
X, treatment, y = convert_pd_to_np(X, treatment, y)
te = self.fit_predict(X, p, treatment, y)
check_p_conditions(p, self.t_groups)
if isinstance(p, np.ndarray):
treatment_name = self.t_groups[0]
p = {treatment_name: convert_pd_to_np(p)}
elif isinstance(p, dict):
p = {treatment_name: convert_pd_to_np(_p) for treatment_name, _p in p.items()}
ate = np.zeros(self.t_groups.shape[0])
ate_lb = np.zeros(self.t_groups.shape[0])
ate_ub = np.zeros(self.t_groups.shape[0])
for i, group in enumerate(self.t_groups):
w = (treatment == group).astype(int)
prob_treatment = float(sum(w)) / X.shape[0]
_ate = te[:, i].mean()
se = (np.sqrt((self.vars_t[group] / prob_treatment)
+ (self.vars_c[group] / (1 - prob_treatment))
+ te[:, i].var())
/ X.shape[0])
_ate_lb = _ate - se * norm.ppf(1 - self.ate_alpha / 2)
_ate_ub = _ate + se * norm.ppf(1 - self.ate_alpha / 2)
ate[i] = _ate
ate_lb[i] = _ate_lb
ate_ub[i] = _ate_ub
if not bootstrap_ci:
return ate, ate_lb, ate_ub
else:
t_groups_global = self.t_groups
_classes_global = self._classes
model_mu_global = deepcopy(self.model_mu)
models_tau_global = deepcopy(self.models_tau)
logger.info('Bootstrap Confidence Intervals for ATE')
ate_bootstraps = np.zeros(shape=(self.t_groups.shape[0], n_bootstraps))
for n in tqdm(range(n_bootstraps)):
cate_b = self.bootstrap(X, p, treatment, y, size=bootstrap_size)
ate_bootstraps[:, n] = cate_b.mean()
ate_lower = np.percentile(ate_bootstraps, (self.ate_alpha / 2) * 100, axis=1)
ate_upper = np.percentile(ate_bootstraps, (1 - self.ate_alpha / 2) * 100, axis=1)
# set member variables back to global (currently last bootstrapped outcome)
self.t_groups = t_groups_global
self._classes = _classes_global
self.model_mu = deepcopy(model_mu_global)
self.models_tau = deepcopy(models_tau_global)
return ate, ate_lower, ate_upper
def fit(self, X, treatment, y, p=None, verbose=True):
"""Fit the treatment effect and outcome models of the R learner.
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
p (np.ndarray or pd.Series or dict, optional): an array of propensity scores of float (0,1) in the
single-treatment case; or, a dictionary of treatment groups that map to propensity vectors of
float (0,1); if None will run ElasticNetPropensityModel() to generate the propensity scores.
verbose (bool, optional): whether to output progress logs
"""
X, treatment, y = convert_pd_to_np(X, treatment, y)
check_treatment_vector(treatment, self.control_name)
self.t_groups = np.unique(treatment[treatment != self.control_name])
self.t_groups.sort()
if p is None:
logger.info('Generating propensity score')
p = dict()
p_model = dict()
for group in self.t_groups:
mask = (treatment == group) | (treatment == self.control_name)
treatment_filt = treatment[mask]
X_filt = X[mask]
w_filt = (treatment_filt == group).astype(int)
w = (treatment == group).astype(int)
p[group], p_model[group] = compute_propensity_score(
X=X_filt,
treatment=w_filt,
X_pred=X,
treatment_pred=w,
cv=self.cv)
self.propensity_model = p_model
self.propensity = p
else:
check_p_conditions(p, self.t_groups)
if isinstance(p, (np.ndarray, pd.Series)):
treatment_name = self.t_groups[0]
p = {treatment_name: convert_pd_to_np(p)}
elif isinstance(p, dict):
p = {
treatment_name: convert_pd_to_np(_p)
for treatment_name, _p in p.items()
}
self._classes = {group: i for i, group in enumerate(self.t_groups)}
self.models_tau = {
group: deepcopy(self.model_tau)
for group in self.t_groups
}
self.vars_c = {}
self.vars_t = {}
if verbose:
logger.info('generating out-of-fold CV outcome estimates')
yhat = cross_val_predict(self.model_mu, X, y, cv=self.cv, n_jobs=-1)
for group in self.t_groups:
mask = (treatment == group) | (treatment == self.control_name)
treatment_filt = treatment[mask]
X_filt = X[mask]
y_filt = y[mask]
yhat_filt = yhat[mask]
p_filt = p[group][mask]
w = (treatment_filt == group).astype(int)
if verbose:
logger.info(
'training the treatment effect model for {} with R-loss'.
format(group))
self.models_tau[group].fit(X_filt,
(y_filt - yhat_filt) / (w - p_filt),
sample_weight=(w - p_filt)**2)
self.vars_c[group] = (y_filt[w == 0] - yhat_filt[w == 0]).var()
self.vars_t[group] = (y_filt[w == 1] - yhat_filt[w == 1]).var()
def fit(self, X, treatment, y, p=None, seed=None):
"""Fit the inference model.
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
p (np.ndarray or pd.Series or dict, optional): an array of propensity scores of float (0,1) in the
single-treatment case; or, a dictionary of treatment groups that map to propensity vectors of
float (0,1); if None will run ElasticNetPropensityModel() to generate the propensity scores.
seed (int): random seed for cross-fitting
"""
X, treatment, y = convert_pd_to_np(X, treatment, y)
check_treatment_vector(treatment, self.control_name)
self.t_groups = np.unique(treatment[treatment != self.control_name])
self.t_groups.sort()
self._classes = {group: i for i, group in enumerate(self.t_groups)}
# The estimator splits the data into 3 partitions for cross-fit on the propensity score estimation,
# the outcome regression, and the treatment regression on the doubly robust estimates. The use of
# the partitions is rotated so we do not lose on the sample size.
cv = KFold(n_splits=3, shuffle=True, random_state=seed)
split_indices = [index for _, index in cv.split(y)]
self.models_mu_c = [
deepcopy(self.model_mu_c),
deepcopy(self.model_mu_c),
deepcopy(self.model_mu_c),
]
self.models_mu_t = {
group: [
deepcopy(self.model_mu_t),
deepcopy(self.model_mu_t),
deepcopy(self.model_mu_t),
]
for group in self.t_groups
}
self.models_tau = {
group: [
deepcopy(self.model_tau),
deepcopy(self.model_tau),
deepcopy(self.model_tau),
]
for group in self.t_groups
}
if p is None:
self.propensity = {
group: np.zeros(y.shape[0])
for group in self.t_groups
}
for ifold in range(3):
treatment_idx = split_indices[ifold]
outcome_idx = split_indices[(ifold + 1) % 3]
tau_idx = split_indices[(ifold + 2) % 3]
treatment_treat, treatment_out, treatment_tau = (
treatment[treatment_idx],
treatment[outcome_idx],
treatment[tau_idx],
)
y_out, y_tau = y[outcome_idx], y[tau_idx]
X_treat, X_out, X_tau = X[treatment_idx], X[outcome_idx], X[
tau_idx]
if p is None:
logger.info("Generating propensity score")
cur_p = dict()
for group in self.t_groups:
mask = (treatment_treat == group) | (treatment_treat
== self.control_name)
treatment_filt = treatment_treat[mask]
X_filt = X_treat[mask]
w_filt = (treatment_filt == group).astype(int)
w = (treatment_tau == group).astype(int)
cur_p[group], _ = compute_propensity_score(
X=X_filt,
treatment=w_filt,
X_pred=X_tau,
treatment_pred=w)
self.propensity[group][tau_idx] = cur_p[group]
else:
cur_p = dict()
if isinstance(p, (np.ndarray, pd.Series)):
cur_p = {self.t_groups[0]: convert_pd_to_np(p[tau_idx])}
else:
cur_p = {g: prop[tau_idx] for g, prop in p.items()}
check_p_conditions(cur_p, self.t_groups)
logger.info("Generate outcome regressions")
self.models_mu_c[ifold].fit(
X_out[treatment_out == self.control_name],
y_out[treatment_out == self.control_name],
)
for group in self.t_groups:
self.models_mu_t[group][ifold].fit(
X_out[treatment_out == group],
y_out[treatment_out == group])
logger.info("Fit pseudo outcomes from the DR formula")
for group in self.t_groups:
mask = (treatment_tau == group) | (treatment_tau
== self.control_name)
treatment_filt = treatment_tau[mask]
X_filt = X_tau[mask]
y_filt = y_tau[mask]
w_filt = (treatment_filt == group).astype(int)
p_filt = cur_p[group][mask]
mu_t = self.models_mu_t[group][ifold].predict(X_filt)
mu_c = self.models_mu_c[ifold].predict(X_filt)
dr = ((w_filt - p_filt) / p_filt / (1 - p_filt) *
(y_filt - mu_t * w_filt - mu_c *
(1 - w_filt)) + mu_t - mu_c)
self.models_tau[group][ifold].fit(X_filt, dr)
def fit(
self, X, assignment, treatment, y, p=None, pZ=None, seed=None, calibrate=True
):
"""Fit the inference model.
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
assignment (np.array or pd.Series): a (0,1)-valued assignment vector. The assignment is the
instrumental variable that does not depend on unknown confounders. The assignment status
influences treatment in a monotonic way, i.e. one can only be more likely to take the
treatment if assigned.
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
p (2-tuple of np.ndarray or pd.Series or dict, optional): The first (second) element corresponds to
unassigned (assigned) units. Each is an array of propensity scores of float (0,1) in the single-treatment
case; or, a dictionary of treatment groups that map to propensity vectors of float (0,1). If None will run
ElasticNetPropensityModel() to generate the propensity scores.
pZ (np.array or pd.Series, optional): an array of assignment probability of float (0,1); if None
will run ElasticNetPropensityModel() to generate the assignment probability score.
seed (int): random seed for cross-fitting
"""
X, treatment, assignment, y = convert_pd_to_np(X, treatment, assignment, y)
check_treatment_vector(treatment, self.control_name)
self.t_groups = np.unique(treatment[treatment != self.control_name])
self.t_groups.sort()
self._classes = {group: i for i, group in enumerate(self.t_groups)}
# The estimator splits the data into 3 partitions for cross-fit on the propensity score estimation,
# the outcome regression, and the treatment regression on the doubly robust estimates. The use of
# the partitions is rotated so we do not lose on the sample size. We do not cross-fit the assignment
# score estimation as the assignment process is usually simple.
cv = KFold(n_splits=3, shuffle=True, random_state=seed)
split_indices = [index for _, index in cv.split(y)]
self.models_mu_c = {
group: [
deepcopy(self.model_mu_c),
deepcopy(self.model_mu_c),
deepcopy(self.model_mu_c),
]
for group in self.t_groups
}
self.models_mu_t = {
group: [
deepcopy(self.model_mu_t),
deepcopy(self.model_mu_t),
deepcopy(self.model_mu_t),
]
for group in self.t_groups
}
self.models_tau = {
group: [
deepcopy(self.model_tau),
deepcopy(self.model_tau),
deepcopy(self.model_tau),
]
for group in self.t_groups
}
if p is None:
self.propensity_1 = {
group: np.zeros(y.shape[0]) for group in self.t_groups
} # propensity scores for those assigned
self.propensity_0 = {
group: np.zeros(y.shape[0]) for group in self.t_groups
} # propensity scores for those not assigned
if pZ is None:
self.propensity_assign, _ = compute_propensity_score(
X=X,
treatment=assignment,
X_pred=X,
treatment_pred=assignment,
calibrate_p=calibrate,
)
else:
self.propensity_assign = pZ
for ifold in range(3):
treatment_idx = split_indices[ifold]
outcome_idx = split_indices[(ifold + 1) % 3]
tau_idx = split_indices[(ifold + 2) % 3]
treatment_treat, treatment_out, treatment_tau = (
treatment[treatment_idx],
treatment[outcome_idx],
treatment[tau_idx],
)
assignment_treat, assignment_out, assignment_tau = (
assignment[treatment_idx],
assignment[outcome_idx],
assignment[tau_idx],
)
y_out, y_tau = y[outcome_idx], y[tau_idx]
X_treat, X_out, X_tau = X[treatment_idx], X[outcome_idx], X[tau_idx]
pZ_tau = self.propensity_assign[tau_idx]
if p is None:
logger.info("Generating propensity score")
cur_p_1 = dict()
cur_p_0 = dict()
for group in self.t_groups:
mask = (treatment_treat == group) | (
treatment_treat == self.control_name
)
mask_1, mask_0 = mask & (assignment_treat == 1), mask & (
assignment_treat == 0
)
cur_p_1[group], _ = compute_propensity_score(
X=X_treat[mask_1],
treatment=(treatment_treat[mask_1] == group).astype(int),
X_pred=X_tau,
treatment_pred=(treatment_tau == group).astype(int),
)
if (treatment_treat[mask_0] == group).sum() == 0:
cur_p_0[group] = np.zeros(X_tau.shape[0])
else:
cur_p_0[group], _ = compute_propensity_score(
X=X_treat[mask_0],
treatment=(treatment_treat[mask_0] == group).astype(int),
X_pred=X_tau,
treatment_pred=(treatment_tau == group).astype(int),
)
self.propensity_1[group][tau_idx] = cur_p_1[group]
self.propensity_0[group][tau_idx] = cur_p_0[group]
else:
cur_p_1 = dict()
cur_p_0 = dict()
if isinstance(p[0], (np.ndarray, pd.Series)):
cur_p_0 = {self.t_groups[0]: convert_pd_to_np(p[0][tau_idx])}
else:
cur_p_0 = {g: prop[tau_idx] for g, prop in p[0].items()}
check_p_conditions(cur_p_0, self.t_groups)
if isinstance(p[1], (np.ndarray, pd.Series)):
cur_p_1 = {self.t_groups[0]: convert_pd_to_np(p[1][tau_idx])}
else:
cur_p_1 = {g: prop[tau_idx] for g, prop in p[1].items()}
check_p_conditions(cur_p_1, self.t_groups)
logger.info("Generate outcome regressions")
for group in self.t_groups:
mask = (treatment_out == group) | (treatment_out == self.control_name)
mask_1, mask_0 = mask & (assignment_out == 1), mask & (
assignment_out == 0
)
self.models_mu_c[group][ifold].fit(X_out[mask_0], y_out[mask_0])
self.models_mu_t[group][ifold].fit(X_out[mask_1], y_out[mask_1])
logger.info("Fit pseudo outcomes from the DR formula")
for group in self.t_groups:
mask = (treatment_tau == group) | (treatment_tau == self.control_name)
treatment_filt = treatment_tau[mask]
X_filt = X_tau[mask]
y_filt = y_tau[mask]
w_filt = (treatment_filt == group).astype(int)
p_1_filt = cur_p_1[group][mask]
p_0_filt = cur_p_0[group][mask]
z_filt = assignment_tau[mask]
pZ_filt = pZ_tau[mask]
mu_t = self.models_mu_t[group][ifold].predict(X_filt)
mu_c = self.models_mu_c[group][ifold].predict(X_filt)
dr = (
z_filt * (y_filt - mu_t) / pZ_filt
- (1 - z_filt) * (y_filt - mu_c) / (1 - pZ_filt)
+ mu_t
- mu_c
)
weight = (
z_filt * (w_filt - p_1_filt) / pZ_filt
- (1 - z_filt) * (w_filt - p_0_filt) / (1 - pZ_filt)
+ p_1_filt
- p_0_filt
)
dr /= weight
self.models_tau[group][ifold].fit(X_filt, dr, sample_weight=weight**2)
def estimate_ate(
self,
X,
assignment,
treatment,
y,
p=None,
pZ=None,
bootstrap_ci=False,
n_bootstraps=1000,
bootstrap_size=10000,
seed=None,
calibrate=True,
):
"""Estimate the Average Treatment Effect (ATE) for compliers.
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
assignment (np.array or pd.Series): an assignment vector. The assignment is the
instrumental variable that does not depend on unknown confounders. The assignment status
influences treatment in a monotonic way, i.e. one can only be more likely to take the
treatment if assigned.
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
p (2-tuple of np.ndarray or pd.Series or dict, optional): The first (second) element corresponds to
unassigned (assigned) units. Each is an array of propensity scores of float (0,1) in the single-treatment
case; or, a dictionary of treatment groups that map to propensity vectors of float (0,1). If None will run
ElasticNetPropensityModel() to generate the propensity scores.
pZ (np.array or pd.Series, optional): an array of assignment probability of float (0,1); if None
will run ElasticNetPropensityModel() to generate the assignment probability score.
bootstrap_ci (bool): whether run bootstrap for confidence intervals
n_bootstraps (int): number of bootstrap iterations
bootstrap_size (int): number of samples per bootstrap
seed (int): random seed for cross-fitting
Returns:
The mean and confidence interval (LB, UB) of the ATE estimate.
"""
te, yhat_cs, yhat_ts = self.fit_predict(
X,
assignment,
treatment,
y,
p,
return_components=True,
seed=seed,
calibrate=calibrate,
)
X, assignment, treatment, y = convert_pd_to_np(X, assignment, treatment, y)
if p is None:
p = (self.propensity_0, self.propensity_1)
else:
check_p_conditions(p[0], self.t_groups)
check_p_conditions(p[1], self.t_groups)
if isinstance(p[0], (np.ndarray, pd.Series)):
treatment_name = self.t_groups[0]
p = (
{treatment_name: convert_pd_to_np(p[0])},
{treatment_name: convert_pd_to_np(p[1])},
)
elif isinstance(p[0], dict):
p = (
{
treatment_name: convert_pd_to_np(_p)
for treatment_name, _p in p[0].items()
},
{
treatment_name: convert_pd_to_np(_p)
for treatment_name, _p in p[1].items()
},
)
ate = np.zeros(self.t_groups.shape[0])
ate_lb = np.zeros(self.t_groups.shape[0])
ate_ub = np.zeros(self.t_groups.shape[0])
for i, group in enumerate(self.t_groups):
_ate = te[:, i].mean()
mask = (treatment == group) | (treatment == self.control_name)
mask_1, mask_0 = mask & (assignment == 1), mask & (assignment == 0)
Gamma = (treatment[mask_1] == group).mean() - (
treatment[mask_0] == group
).mean()
y_filt_1, y_filt_0 = y[mask_1], y[mask_0]
yhat_0 = yhat_cs[group][mask_0]
yhat_1 = yhat_ts[group][mask_1]
treatment_filt_1, treatment_filt_0 = treatment[mask_1], treatment[mask_0]
prob_treatment_1, prob_treatment_0 = (
p[1][group][mask_1],
p[0][group][mask_0],
)
w = (assignment[mask]).mean()
part_1 = (
(y_filt_1 - yhat_1).var()
+ _ate**2 * (treatment_filt_1 - prob_treatment_1).var()
- 2
* _ate
* (y_filt_1 * treatment_filt_1 - yhat_1 * prob_treatment_1).mean()
)
part_0 = (
(y_filt_0 - yhat_0).var()
+ _ate**2 * (treatment_filt_0 - prob_treatment_0).var()
- 2
* _ate
* (y_filt_0 * treatment_filt_0 - yhat_0 * prob_treatment_0).mean()
)
part_2 = np.mean(
(
yhat_ts[group][mask]
- yhat_cs[group][mask]
- _ate * (p[1][group][mask] - p[0][group][mask])
)
** 2
)
# SE formula is based on the lower bound formula (9) from Frölich, Markus. 2006.
# "Nonparametric IV estimation of local average treatment effects wth covariates."
# Journal of Econometrics.
se = np.sqrt((part_1 / w + part_2 / (1 - w)) + part_2) / Gamma
_ate_lb = _ate - se * norm.ppf(1 - self.ate_alpha / 2)
_ate_ub = _ate + se * norm.ppf(1 - self.ate_alpha / 2)
ate[i] = _ate
ate_lb[i] = _ate_lb
ate_ub[i] = _ate_ub
if not bootstrap_ci:
return ate, ate_lb, ate_ub
else:
t_groups_global = self.t_groups
_classes_global = self._classes
models_mu_c_global = deepcopy(self.models_mu_c)
models_mu_t_global = deepcopy(self.models_mu_t)
models_tau_global = deepcopy(self.models_tau)
logger.info("Bootstrap Confidence Intervals for ATE")
ate_bootstraps = np.zeros(shape=(self.t_groups.shape[0], n_bootstraps))
for n in tqdm(range(n_bootstraps)):
cate_b = self.bootstrap(
X, assignment, treatment, y, p, pZ, size=bootstrap_size, seed=seed
)
ate_bootstraps[:, n] = cate_b.mean()
ate_lower = np.percentile(
ate_bootstraps, (self.ate_alpha / 2) * 100, axis=1
)
ate_upper = np.percentile(
ate_bootstraps, (1 - self.ate_alpha / 2) * 100, axis=1
)
# set member variables back to global (currently last bootstrapped outcome)
self.t_groups = t_groups_global
self._classes = _classes_global
self.models_mu_c = deepcopy(models_mu_c_global)
self.models_mu_t = deepcopy(models_mu_t_global)
self.models_tau = deepcopy(models_tau_global)
return ate, ate_lower, ate_upper
def fit_predict(
self,
X,
assignment,
treatment,
y,
p=None,
pZ=None,
return_ci=False,
n_bootstraps=1000,
bootstrap_size=10000,
return_components=False,
verbose=True,
seed=None,
calibrate=True,
):
"""Fit the treatment effect and outcome models of the R learner and predict treatment effects.
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
assignment (np.array or pd.Series): a (0,1)-valued assignment vector. The assignment is the
instrumental variable that does not depend on unknown confounders. The assignment status
influences treatment in a monotonic way, i.e. one can only be more likely to take the
treatment if assigned.
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
p (2-tuple of np.ndarray or pd.Series or dict, optional): The first (second) element corresponds to
unassigned (assigned) units. Each is an array of propensity scores of float (0,1) in the single-treatment
case; or, a dictionary of treatment groups that map to propensity vectors of float (0,1). If None will run
ElasticNetPropensityModel() to generate the propensity scores.
pZ (np.array or pd.Series, optional): an array of assignment probability of float (0,1); if None
will run ElasticNetPropensityModel() to generate the assignment probability score.
return_ci (bool): whether to return confidence intervals
n_bootstraps (int): number of bootstrap iterations
bootstrap_size (int): number of samples per bootstrap
return_components (bool, optional): whether to return outcome for treatment and control seperately
verbose (str): whether to output progress logs
seed (int): random seed for cross-fitting
Returns:
(numpy.ndarray): Predictions of treatment effects for compliers, , i.e. those individuals
who take the treatment only if they are assigned. Output dim: [n_samples, n_treatment]
If return_ci, returns CATE [n_samples, n_treatment], LB [n_samples, n_treatment],
UB [n_samples, n_treatment]
"""
X, assignment, treatment, y = convert_pd_to_np(X, assignment, treatment, y)
self.fit(X, assignment, treatment, y, p, seed, calibrate)
if p is None:
p = (self.propensity_0, self.propensity_1)
else:
check_p_conditions(p[0], self.t_groups)
check_p_conditions(p[1], self.t_groups)
if isinstance(p[0], (np.ndarray, pd.Series)):
treatment_name = self.t_groups[0]
p = (
{treatment_name: convert_pd_to_np(p[0])},
{treatment_name: convert_pd_to_np(p[1])},
)
elif isinstance(p[0], dict):
p = (
{
treatment_name: convert_pd_to_np(_p)
for treatment_name, _p in p[0].items()
},
{
treatment_name: convert_pd_to_np(_p)
for treatment_name, _p in p[1].items()
},
)
if pZ is None:
pZ = self.propensity_assign
te = self.predict(
X, treatment=treatment, y=y, return_components=return_components
)
if not return_ci:
return te
else:
t_groups_global = self.t_groups
_classes_global = self._classes
models_mu_c_global = deepcopy(self.models_mu_c)
models_mu_t_global = deepcopy(self.models_mu_t)
models_tau_global = deepcopy(self.models_tau)
te_bootstraps = np.zeros(
shape=(X.shape[0], self.t_groups.shape[0], n_bootstraps)
)
logger.info("Bootstrap Confidence Intervals")
for i in tqdm(range(n_bootstraps)):
te_b = self.bootstrap(
X, assignment, treatment, y, p, pZ, size=bootstrap_size, seed=seed
)
te_bootstraps[:, :, i] = te_b
te_lower = np.percentile(te_bootstraps, (self.ate_alpha / 2) * 100, axis=2)
te_upper = np.percentile(
te_bootstraps, (1 - self.ate_alpha / 2) * 100, axis=2
)
# set member variables back to global (currently last bootstrapped outcome)
self.t_groups = t_groups_global
self._classes = _classes_global
self.models_mu_c = deepcopy(models_mu_c_global)
self.models_mu_t = deepcopy(models_mu_t_global)
self.models_tau = deepcopy(models_tau_global)
return (te, te_lower, te_upper)
def estimate_ate(self, X, treatment, y, p, segment=None, return_ci=False):
"""Estimate the Average Treatment Effect (ATE).
Args:
X (np.matrix or np.array or pd.Dataframe): a feature matrix
treatment (np.array or pd.Series): a treatment vector
y (np.array or pd.Series): an outcome vector
p (np.ndarray or pd.Series or dict): an array of propensity scores of float (0,1) in the single-treatment
case; or, a dictionary of treatment groups that map to propensity vectors of float (0,1)
segment (np.array, optional): An optional segment vector of int. If given, the ATE and its CI will be
estimated for each segment.
return_ci (bool, optional): Whether to return confidence intervals
Returns:
(tuple): The ATE and its confidence interval (LB, UB) for each treatment, t and segment, s
"""
X, treatment, y = convert_pd_to_np(X, treatment, y)
check_treatment_vector(treatment, self.control_name)
self.t_groups = np.unique(treatment[treatment != self.control_name])
self.t_groups.sort()
check_p_conditions(p, self.t_groups)
if isinstance(p, (np.ndarray, pd.Series)):
treatment_name = self.t_groups[0]
p = {treatment_name: convert_pd_to_np(p)}
elif isinstance(p, dict):
p = {
treatment_name: convert_pd_to_np(_p)
for treatment_name, _p in p.items()
}
ate = []
ate_lb = []
ate_ub = []
for i, group in enumerate(self.t_groups):
logger.info("Estimating ATE for group {}.".format(group))
w_group = (treatment == group).astype(int)
p_group = p[group]
if self.calibrate_propensity:
logger.info("Calibrating propensity scores.")
p_group = calibrate(p_group, w_group)
yhat_c = np.zeros_like(y, dtype=float)
yhat_t = np.zeros_like(y, dtype=float)
if self.cv:
for i_fold, (i_trn,
i_val) in enumerate(self.cv.split(X, y), 1):
logger.info(
"Training an outcome model for CV #{}".format(i_fold))
self.model_tau.fit(
np.hstack((X[i_trn], w_group[i_trn].reshape(-1, 1))),
y[i_trn])
yhat_c[i_val] = self.model_tau.predict(
np.hstack((X[i_val], np.zeros((len(i_val), 1)))))
yhat_t[i_val] = self.model_tau.predict(
np.hstack((X[i_val], np.ones((len(i_val), 1)))))
else:
self.model_tau.fit(np.hstack((X, w_group.reshape(-1, 1))), y)
yhat_c = self.model_tau.predict(
np.hstack((X, np.zeros((len(y), 1)))))
yhat_t = self.model_tau.predict(
np.hstack((X, np.ones((len(y), 1)))))
if segment is None:
logger.info("Training the TMLE learner.")
_ate, se = simple_tmle(y, w_group, yhat_c, yhat_t, p_group)
_ate_lb = _ate - se * norm.ppf(1 - self.ate_alpha / 2)
_ate_ub = _ate + se * norm.ppf(1 - self.ate_alpha / 2)
else:
assert (segment.shape[0] == X.shape[0] and segment.ndim
== 1), "Segment must be the 1-d np.array of int."
segments = np.unique(segment)
_ate = []
_ate_lb = []
_ate_ub = []
for s in sorted(segments):
logger.info(
"Training the TMLE learner for segment {}.".format(s))
filt = (segment
== s) & (yhat_c < np.quantile(yhat_c, q=0.99))
_ate_s, se = simple_tmle(
y[filt],
w_group[filt],
yhat_c[filt],
yhat_t[filt],
p_group[filt],
)
_ate_lb_s = _ate_s - se * norm.ppf(1 - self.ate_alpha / 2)
_ate_ub_s = _ate_s + se * norm.ppf(1 - self.ate_alpha / 2)
_ate.append(_ate_s)
_ate_lb.append(_ate_lb_s)
_ate_ub.append(_ate_ub_s)
ate.append(_ate)
ate_lb.append(_ate_lb)
ate_ub.append(_ate_ub)
return np.array(ate), np.array(ate_lb), np.array(ate_ub)