Chapter 4: Machine Learning Fundamentals with Population-Level Validation
Learning Objectives
By the end of this chapter, readers will be able to:
-
Implement supervised learning algorithms including logistic regression, decision trees, random forests, and gradient boosting for clinical risk prediction tasks, with explicit attention to how algorithmic choices affect fairness across patient populations.
-
Design and evaluate fairness-aware machine learning pipelines that constrain disparities in model performance across protected groups while maintaining overall predictive accuracy suitable for clinical deployment.
-
Develop feature engineering strategies that capture clinically relevant patterns without encoding proxies for race, socioeconomic status, or other characteristics that should not directly influence clinical decisions.
-
Apply appropriate regularization techniques that prevent overfitting while avoiding the systematic exclusion of features that are particularly informative for underrepresented patient populations.
-
Conduct comprehensive model evaluation using stratified performance metrics that surface disparities in accuracy, calibration, and clinical utility across demographic groups and care settings.
-
Implement ensemble methods that leverage diversity among base learners to improve robustness across heterogeneous patient populations rather than amplifying systematic biases present in individual models.
4.1 Introduction: Machine Learning in Healthcare with Equity at the Center
Machine learning provides powerful tools for discovering patterns in complex healthcare data, but these tools can perpetuate or even amplify existing health disparities if applied without careful attention to equity. This chapter introduces core machine learning concepts through the specific lens of clinical applications, examining at each step how algorithmic choices affect different patient populations and developing approaches that explicitly account for fairness alongside predictive performance.
The fundamental challenge in healthcare machine learning is that our data reflects historical patterns of care that themselves encode systemic inequities. When we train models on electronic health record data to predict clinical outcomes, we are teaching algorithms to reproduce the patterns present in that data. If Black patients have historically received less aggressive pain management, models trained to predict pain medication needs may learn to recommend less treatment for Black patients. If women presenting with cardiac symptoms have historically been less likely to receive cardiac catheterization, models trained to predict procedure appropriateness may learn to recommend less aggressive workup for women with chest pain.
These problems cannot be solved by simply removing demographic variables from models. Race, ethnicity, sex, and socioeconomic status affect health through complex biological and social pathways that leave traces throughout clinical data. Removing explicit demographic variables often means that models learn to use proxies instead, inferring protected characteristics from zip codes, insurance status, or patterns of healthcare utilization. The solution requires explicit fairness-aware approaches that recognize how our models might perpetuate disparities and constrain them to behave more equitably.
4.1.1 Defining Fairness in Clinical Machine Learning
Before developing fair machine learning approaches, we must grapple with a fundamental challenge: there is no single universally appropriate definition of algorithmic fairness. Different mathematical formulations of fairness encode different ethical principles and may be mutually incompatible. A model that achieves equal false positive rates across demographic groups may necessarily have unequal false negative rates. A model that is well-calibrated overall may exhibit calibration differences across groups.
In clinical contexts, the appropriate fairness definition depends on the specific application and its potential harms. For a screening tool that determines which patients are contacted for preventive care, we might prioritize equal opportunity, ensuring that patients who would benefit from screening have equal probability of being identified regardless of demographics. For a diagnostic tool that directly influences treatment decisions, we might prioritize predictive parity, ensuring that patients with the same predicted risk have the same actual outcome rate regardless of group membership.
Throughout this chapter, we develop methods that support multiple fairness definitions and enable explicit tradeoff analysis between different fairness criteria and overall predictive performance. The goal is not to identify a single correct approach but rather to provide tools that make fairness considerations explicit and measurable, enabling informed decisions about which fairness properties matter most for specific clinical applications.
4.1.2 The Bias-Variance-Fairness Tradeoff
Traditional machine learning focuses on the bias-variance tradeoff, balancing model complexity against risk of overfitting. In healthcare applications affecting diverse populations, we must extend this framework to include fairness as an explicit consideration. More complex models may achieve better overall performance but exhibit larger fairness gaps across demographic groups. Simpler models may be more robust and interpretable but perform poorly for all groups.
The bias-variance-fairness tradeoff manifests differently depending on data characteristics. When training data for some populations is limited, complex models may overfit to the majority population while underfitting minority populations, creating accuracy disparities. When features have different distributions across populations, models that work well for one group may systematically fail for others. When class imbalance differs by demographics, standard optimization objectives that maximize overall accuracy may produce models that sacrifice minority group performance for small gains in majority group performance.
Navigating these tradeoffs requires explicit measurement of fairness alongside traditional performance metrics throughout model development. We implement comprehensive evaluation frameworks that surface disparities early, enable systematic comparison of approaches with different fairness-accuracy tradeoffs, and provide clinically interpretable metrics that connect algorithmic behavior to patient outcomes.
4.2 Logistic Regression: The Foundation for Clinical Risk Prediction
Logistic regression remains ubiquitous in clinical risk prediction despite the availability of more sophisticated algorithms, and for good reason. Its outputs are well-calibrated probabilities rather than arbitrary scores, its decision boundaries are interpretable as linear combinations of features with meaningful coefficients, and its behavior is stable across different patient populations when properly regularized. These properties matter enormously in healthcare, where predictions influence treatment decisions with life-or-death consequences and where model failures must be debuggable by clinicians.
From an equity perspective, logistic regression offers both advantages and challenges. The linearity assumption constrains the model’s ability to learn complex interactions that might encode subtle forms of bias, and the interpretability of coefficients enables direct audit of whether demographic variables or their proxies are driving predictions inappropriately. However, the same linearity can cause systematic underfitting for populations whose outcome relationships differ from the majority, and standard fitting procedures that maximize overall likelihood provide no guarantee of equitable performance across groups.
4.2.1 Mathematical Foundations and Clinical Interpretation
Logistic regression models the log-odds of a binary outcome as a linear function of features. For a patient with feature vector $\mathbf{x} = (x_1, x_2, \ldots, x_p)$, the model predicts:
where $\beta_0, \beta_1, \ldots, \beta_p$ are learned coefficients. Equivalently, the log-odds or logit is:
This formulation connects naturally to how clinicians reason about risk. Each coefficient $\beta_j$ represents the change in log-odds of the outcome for a one-unit increase in feature $x_j$ holding all other features constant. When we exponentiate coefficients, we obtain odds ratios that clinicians use routinely to interpret associations between risk factors and outcomes.
In healthcare applications, this interpretability is crucial for several reasons. First, clinicians must understand why a model makes particular predictions to appropriately integrate algorithmic guidance into clinical judgment. When a model predicts high readmission risk for a patient, the clinician needs to understand whether that prediction is driven by clinical factors amenable to intervention or by social factors requiring different approaches. Second, regulators and payers increasingly require that risk prediction models used for resource allocation be interpretable and auditable. Third, patients have a right to understand factors driving recommendations that affect their care.
From an equity perspective, interpretability enables detection of problematic feature dependencies. If a model coefficient for neighborhood poverty level is large and negative, predicting better outcomes for patients in poorer neighborhoods, this might indicate that the model has learned an unintended correlation reflecting the availability of safety-net services rather than actual health status. If coefficients differ substantially when the model is fit separately to different demographic groups, this suggests that a single pooled model may be inappropriate.
4.2.2 Fairness-Aware Logistic Regression Implementation
We now develop a production-ready implementation of logistic regression that incorporates explicit fairness constraints during model training. Rather than simply evaluating fairness after fitting a standard model, we modify the optimization objective to directly penalize fairness violations while maintaining predictive performance.
"""
Fairness-aware logistic regression for clinical risk prediction.
This module implements logistic regression with explicit fairness constraints,
enabling development of models that balance predictive accuracy with equity
across patient populations. The implementation supports multiple fairness
metrics and provides comprehensive evaluation of model behavior across
demographic groups.
"""
import numpy as np
import pandas as pd
from typing import Dict, List, Optional, Tuple, Union
from dataclasses import dataclass
import logging
from scipy.optimize import minimize
from scipy.special import expit
from sklearn.base import BaseEstimator, ClassifierMixin
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import StratifiedKFold
from sklearn.metrics import roc_auc_score, log_loss, brier_score_loss
import warnings
# Configure logging for production monitoring
logging.basicConfig(level=logging.INFO)
logger = logging.getLogger(__name__)
@dataclass
class FairnessMetrics:
"""
Container for fairness evaluation metrics across demographic groups.
Attributes:
demographic_parity_difference: Max difference in positive prediction
rates across groups
equal_opportunity_difference: Max difference in true positive rates
across groups for positive class
equalized_odds_difference: Max difference in both TPR and FPR across groups
calibration_difference: Max difference in calibration slope across groups
group_metrics: Detailed performance metrics for each demographic group
"""
demographic_parity_difference: float
equal_opportunity_difference: float
equalized_odds_difference: float
calibration_difference: float
group_metrics: Dict[str, Dict[str, float]]
class FairLogisticRegression(BaseEstimator, ClassifierMixin):
"""
Logistic regression with explicit fairness constraints for clinical prediction.
This implementation extends standard logistic regression to explicitly constrain
disparities in model performance across demographic groups. The fairness
constraint can be tuned to balance overall predictive accuracy against equity
across protected populations.
The model optimizes:
L = log_loss + lambda_l2 * ||beta||_2^2 + lambda_fair * fairness_penalty
where fairness_penalty measures disparities in specified fairness metrics
across demographic groups.
Parameters:
penalty: Regularization type ('l2', 'l1', or 'elasticnet')
C: Inverse of regularization strength (smaller values = stronger regularization)
fairness_constraint: Type of fairness constraint
'demographic_parity': Equal positive prediction rates across groups
'equal_opportunity': Equal true positive rates across groups
'equalized_odds': Equal TPR and FPR across groups
fairness_penalty: Weight for fairness constraint in optimization (0 = no constraint)
max_iter: Maximum iterations for optimization
tol: Tolerance for optimization convergence
class_weight: Weights for handling class imbalance ('balanced' or dict)
warm_start: Whether to reuse solution from previous fit as initialization
"""
def __init__(
self,
penalty: str = 'l2',
C: float = 1.0,
fairness_constraint: Optional[str] = None,
fairness_penalty: float = 0.0,
max_iter: int = 1000,
tol: float = 1e-4,
class_weight: Optional[Union[str, Dict]] = None,
warm_start: bool = False,
random_state: Optional[int] = None
):
self.penalty = penalty
self.C = C
self.fairness_constraint = fairness_constraint
self.fairness_penalty = fairness_penalty
self.max_iter = max_iter
self.tol = tol
self.class_weight = class_weight
self.warm_start = warm_start
self.random_state = random_state
# Attributes set during fitting
self.coef_ = None
self.intercept_ = None
self.classes_ = None
self.scaler_ = None
self.fairness_groups_ = None
self.n_features_in_ = None
def _compute_log_loss(
self,
beta: np.ndarray,
X: np.ndarray,
y: np.ndarray,
sample_weight: Optional[np.ndarray] = None
) -> float:
"""Compute logistic loss with optional sample weights."""
# Compute predicted probabilities
z = X @ beta
probs = expit(z)
# Clip probabilities to avoid log(0)
probs = np.clip(probs, 1e-15, 1 - 1e-15)
# Compute log loss
if sample_weight is not None:
loss = -np.sum(sample_weight * (y * np.log(probs) + (1 - y) * np.log(1 - probs)))
else:
loss = -np.sum(y * np.log(probs) + (1 - y) * np.log(1 - probs))
return loss / len(y)
def _compute_fairness_penalty(
self,
beta: np.ndarray,
X: np.ndarray,
y: np.ndarray,
sensitive_attr: np.ndarray
) -> float:
"""
Compute fairness penalty based on specified constraint type.
The penalty measures disparities in model behavior across groups
defined by sensitive attributes. Larger penalties indicate greater
unfairness that the optimization will attempt to reduce.
"""
if self.fairness_constraint is None or self.fairness_penalty == 0:
return 0.0
# Get predicted probabilities
z = X @ beta
probs = expit(z)
preds = (probs >= 0.5).astype(int)
# Compute group-specific metrics
groups = np.unique(sensitive_attr)
group_metrics = {}
for group in groups:
mask = sensitive_attr == group
group_probs = probs[mask]
group_preds = preds[mask]
group_y = y[mask]
if len(group_y) == 0:
continue
# Positive prediction rate
pos_pred_rate = np.mean(group_preds)
# True positive rate (sensitivity/recall)
pos_actual = group_y == 1
if np.sum(pos_actual) > 0:
tpr = np.mean(group_preds[pos_actual])
else:
tpr = 0.0
# False positive rate
neg_actual = group_y == 0
if np.sum(neg_actual) > 0:
fpr = np.mean(group_preds[neg_actual])
else:
fpr = 0.0
group_metrics[group] = {
'pos_pred_rate': pos_pred_rate,
'tpr': tpr,
'fpr': fpr
}
# Compute penalty based on constraint type
if self.fairness_constraint == 'demographic_parity':
# Penalize differences in positive prediction rates
pos_rates = [m['pos_pred_rate'] for m in group_metrics.values()]
penalty = np.std(pos_rates)
elif self.fairness_constraint == 'equal_opportunity':
# Penalize differences in true positive rates
tprs = [m['tpr'] for m in group_metrics.values() if m['tpr'] > 0]
penalty = np.std(tprs) if len(tprs) > 1 else 0.0
elif self.fairness_constraint == 'equalized_odds':
# Penalize differences in both TPR and FPR
tprs = [m['tpr'] for m in group_metrics.values() if m['tpr'] > 0]
fprs = [m['fpr'] for m in group_metrics.values() if m['fpr'] > 0]
tpr_penalty = np.std(tprs) if len(tprs) > 1 else 0.0
fpr_penalty = np.std(fprs) if len(fprs) > 1 else 0.0
penalty = tpr_penalty + fpr_penalty
else:
raise ValueError(f"Unknown fairness constraint: {self.fairness_constraint}")
return penalty
def _objective(
self,
beta: np.ndarray,
X: np.ndarray,
y: np.ndarray,
sample_weight: Optional[np.ndarray],
sensitive_attr: Optional[np.ndarray]
) -> float:
"""
Combined objective function including log loss, regularization, and fairness penalty.
This objective balances three components:
1. Predictive accuracy (log loss)
2. Model complexity (regularization)
3. Fairness across groups (fairness penalty)
"""
# Compute log loss
loss = self._compute_log_loss(beta, X, y, sample_weight)
# Add L2 regularization
if self.penalty in ['l2', 'elasticnet']:
loss += (1 / (2 * self.C)) * np.sum(beta[1:]**2) # Don't penalize intercept
# Add L1 regularization
if self.penalty in ['l1', 'elasticnet']:
loss += (1 / self.C) * np.sum(np.abs(beta[1:]))
# Add fairness penalty
if sensitive_attr is not None:
fairness_loss = self._compute_fairness_penalty(beta, X, y, sensitive_attr)
loss += self.fairness_penalty * fairness_loss
return loss
def _gradient(
self,
beta: np.ndarray,
X: np.ndarray,
y: np.ndarray,
sample_weight: Optional[np.ndarray],
sensitive_attr: Optional[np.ndarray]
) -> np.ndarray:
"""
Compute gradient of objective function.
For L2 regularization, we have closed-form gradients.
For L1 and fairness penalties, we use numerical approximation.
"""
# Compute gradient of log loss
z = X @ beta
probs = expit(z)
if sample_weight is not None:
grad = X.T @ (sample_weight * (probs - y)) / len(y)
else:
grad = X.T @ (probs - y) / len(y)
# Add L2 regularization gradient
if self.penalty in ['l2', 'elasticnet']:
reg_grad = np.zeros_like(beta)
reg_grad[1:] = beta[1:] / self.C # Don't penalize intercept
grad += reg_grad
# For L1 and fairness penalties, use numerical gradient
if self.penalty in ['l1', 'elasticnet'] or (sensitive_attr is not None and self.fairness_penalty > 0):
eps = 1e-8
grad_numerical = np.zeros_like(beta)
for i in range(len(beta)):
beta_plus = beta.copy()
beta_plus[i] += eps
beta_minus = beta.copy()
beta_minus[i] -= eps
loss_plus = self._objective(beta_plus, X, y, sample_weight, sensitive_attr)
loss_minus = self._objective(beta_minus, X, y, sample_weight, sensitive_attr)
grad_numerical[i] = (loss_plus - loss_minus) / (2 * eps)
# Blend analytical and numerical gradients
grad = 0.5 * grad + 0.5 * grad_numerical
return grad
def fit(
self,
X: Union[np.ndarray, pd.DataFrame],
y: Union[np.ndarray, pd.Series],
sensitive_features: Optional[Union[np.ndarray, pd.Series]] = None,
sample_weight: Optional[np.ndarray] = None
) -> 'FairLogisticRegression':
"""
Fit fairness-aware logistic regression model.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
y: Binary target variable of shape (n_samples,)
sensitive_features: Protected attributes for fairness constraints
sample_weight: Individual sample weights for handling class imbalance
Returns:
self: Fitted model instance
"""
# Convert inputs to numpy arrays
if isinstance(X, pd.DataFrame):
X = X.values
if isinstance(y, pd.Series):
y = y.values
if isinstance(sensitive_features, pd.Series):
sensitive_features = sensitive_features.values
# Store number of features
self.n_features_in_ = X.shape[1]
# Validate inputs
if len(y.shape) != 1:
raise ValueError("y must be 1-dimensional")
if X.shape[0] != len(y):
raise ValueError("X and y must have the same number of samples")
# Store classes
self.classes_ = np.unique(y)
if len(self.classes_) != 2:
raise ValueError("y must be binary (two classes)")
# Standardize features for numerical stability
self.scaler_ = StandardScaler()
X_scaled = self.scaler_.fit_transform(X)
# Add intercept term
X_with_intercept = np.column_stack([np.ones(len(X_scaled)), X_scaled])
# Handle class imbalance if requested
if sample_weight is None and self.class_weight is not None:
if self.class_weight == 'balanced':
# Compute balanced weights
n_samples = len(y)
n_classes = len(self.classes_)
sample_weight = np.ones(n_samples)
for cls in self.classes_:
cls_mask = y == cls
n_cls = np.sum(cls_mask)
sample_weight[cls_mask] = n_samples / (n_classes * n_cls)
elif isinstance(self.class_weight, dict):
sample_weight = np.array([self.class_weight.get(cls, 1.0) for cls in y])
# Initialize coefficients
if self.warm_start and self.coef_ is not None:
beta_init = np.concatenate([[self.intercept_], self.coef_])
else:
beta_init = np.zeros(X_with_intercept.shape[1])
# Store sensitive features if provided
if sensitive_features is not None:
self.fairness_groups_ = np.unique(sensitive_features)
if len(self.fairness_groups_) < 2:
logger.warning("Fewer than 2 groups in sensitive features. Fairness constraints will have no effect.")
# Optimize objective function
result = minimize(
fun=self._objective,
x0=beta_init,
args=(X_with_intercept, y, sample_weight, sensitive_features),
jac=self._gradient,
method='L-BFGS-B',
options={'maxiter': self.max_iter, 'ftol': self.tol}
)
if not result.success:
logger.warning(f"Optimization did not converge: {result.message}")
# Extract learned parameters
self.intercept_ = result.x[0]
self.coef_ = result.x[1:]
logger.info(f"Model fitted successfully. Final loss: {result.fun:.4f}")
return self
def predict_proba(self, X: Union[np.ndarray, pd.DataFrame]) -> np.ndarray:
"""
Predict class probabilities.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
Returns:
Predicted probabilities of shape (n_samples, 2) for each class
"""
if self.coef_ is None:
raise ValueError("Model must be fitted before prediction")
# Convert to numpy if needed
if isinstance(X, pd.DataFrame):
X = X.values
# Validate input
if X.shape[1] != self.n_features_in_:
raise ValueError(f"X has {X.shape[1]} features but model expects {self.n_features_in_}")
# Standardize features
X_scaled = self.scaler_.transform(X)
# Compute probabilities
z = X_scaled @ self.coef_ + self.intercept_
probs_pos = expit(z)
probs_neg = 1 - probs_pos
return np.column_stack([probs_neg, probs_pos])
def predict(self, X: Union[np.ndarray, pd.DataFrame], threshold: float = 0.5) -> np.ndarray:
"""
Predict binary class labels.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
threshold: Decision threshold for classification (default: 0.5)
Returns:
Predicted class labels of shape (n_samples,)
"""
probs = self.predict_proba(X)
return (probs[:, 1] >= threshold).astype(int)
def evaluate_fairness(
self,
X: Union[np.ndarray, pd.DataFrame],
y: Union[np.ndarray, pd.Series],
sensitive_features: Union[np.ndarray, pd.Series],
threshold: float = 0.5
) -> FairnessMetrics:
"""
Comprehensive fairness evaluation across demographic groups.
This method computes multiple fairness metrics to provide a complete
picture of how model behavior varies across protected groups. Different
applications may prioritize different fairness definitions depending on
the potential harms and clinical context.
Parameters:
X: Feature matrix
y: True binary labels
sensitive_features: Protected attributes defining groups
threshold: Classification threshold
Returns:
FairnessMetrics object containing detailed fairness evaluation
"""
# Get predictions
probs = self.predict_proba(X)[:, 1]
preds = (probs >= threshold).astype(int)
# Convert inputs to numpy
if isinstance(y, pd.Series):
y = y.values
if isinstance(sensitive_features, pd.Series):
sensitive_features = sensitive_features.values
# Compute metrics for each group
groups = np.unique(sensitive_features)
group_metrics = {}
for group in groups:
mask = sensitive_features == group
group_y = y[mask]
group_preds = preds[mask]
group_probs = probs[mask]
# Basic performance metrics
pos_pred_rate = np.mean(group_preds)
# True positive rate (sensitivity)
pos_actual = group_y == 1
if np.sum(pos_actual) > 0:
tpr = np.mean(group_preds[pos_actual])
else:
tpr = np.nan
# False positive rate
neg_actual = group_y == 0
if np.sum(neg_actual) > 0:
fpr = np.mean(group_preds[neg_actual])
else:
fpr = np.nan
# True negative rate (specificity)
if np.sum(neg_actual) > 0:
tnr = 1 - fpr
else:
tnr = np.nan
# False negative rate
if np.sum(pos_actual) > 0:
fnr = 1 - tpr
else:
fnr = np.nan
# Positive predictive value (precision)
if np.sum(group_preds) > 0:
ppv = np.mean(group_y[group_preds == 1])
else:
ppv = np.nan
# Negative predictive value
if np.sum(group_preds == 0) > 0:
npv = np.mean(1 - group_y[group_preds == 0])
else:
npv = np.nan
# AUC if possible
if len(np.unique(group_y)) == 2:
auc = roc_auc_score(group_y, group_probs)
else:
auc = np.nan
# Calibration slope (simple linear calibration)
if len(np.unique(group_y)) == 2 and len(group_y) > 10:
from sklearn.linear_model import LogisticRegression
lr = LogisticRegression()
try:
lr.fit(group_probs.reshape(-1, 1), group_y)
calibration_slope = lr.coef_[0][0]
except:
calibration_slope = np.nan
else:
calibration_slope = np.nan
group_metrics[str(group)] = {
'n_samples': int(np.sum(mask)),
'prevalence': float(np.mean(group_y)),
'pos_pred_rate': float(pos_pred_rate),
'tpr': float(tpr) if not np.isnan(tpr) else None,
'fpr': float(fpr) if not np.isnan(fpr) else None,
'tnr': float(tnr) if not np.isnan(tnr) else None,
'fnr': float(fnr) if not np.isnan(fnr) else None,
'ppv': float(ppv) if not np.isnan(ppv) else None,
'npv': float(npv) if not np.isnan(npv) else None,
'auc': float(auc) if not np.isnan(auc) else None,
'calibration_slope': float(calibration_slope) if not np.isnan(calibration_slope) else None
}
# Compute aggregate fairness metrics
# Demographic parity: difference in positive prediction rates
pos_pred_rates = [m['pos_pred_rate'] for m in group_metrics.values()]
demographic_parity_diff = max(pos_pred_rates) - min(pos_pred_rates)
# Equal opportunity: difference in true positive rates
tprs = [m['tpr'] for m in group_metrics.values() if m['tpr'] is not None]
equal_opp_diff = max(tprs) - min(tprs) if len(tprs) > 1 else 0.0
# Equalized odds: max difference in TPR and FPR
fprs = [m['fpr'] for m in group_metrics.values() if m['fpr'] is not None]
tpr_diff = max(tprs) - min(tprs) if len(tprs) > 1 else 0.0
fpr_diff = max(fprs) - min(fprs) if len(fprs) > 1 else 0.0
equalized_odds_diff = max(tpr_diff, fpr_diff)
# Calibration: difference in calibration slopes
cal_slopes = [m['calibration_slope'] for m in group_metrics.values()
if m['calibration_slope'] is not None]
cal_diff = max(cal_slopes) - min(cal_slopes) if len(cal_slopes) > 1 else 0.0
return FairnessMetrics(
demographic_parity_difference=demographic_parity_diff,
equal_opportunity_difference=equal_opp_diff,
equalized_odds_difference=equalized_odds_diff,
calibration_difference=cal_diff,
group_metrics=group_metrics
)
def get_feature_importance(self) -> pd.DataFrame:
"""
Extract feature importance from model coefficients.
Returns DataFrame with features ranked by absolute coefficient magnitude.
For standardized features, larger absolute coefficients indicate greater
influence on predictions.
"""
if self.coef_ is None:
raise ValueError("Model must be fitted before extracting feature importance")
importance_df = pd.DataFrame({
'feature_index': range(len(self.coef_)),
'coefficient': self.coef_,
'abs_coefficient': np.abs(self.coef_),
'odds_ratio': np.exp(self.coef_)
})
importance_df = importance_df.sort_values('abs_coefficient', ascending=False)
return importance_df
This implementation provides a solid foundation for fairness-aware logistic regression in clinical contexts. The key innovations are explicit fairness constraints during optimization, comprehensive fairness evaluation across multiple metrics, and production-ready code quality with extensive documentation and error handling.
4.2.3 Case Study: Fair Readmission Prediction
We now apply our fairness-aware logistic regression to a realistic clinical prediction task: hospital readmission prediction. Hospital readmissions within thirty days of discharge are both clinically important and financially consequential under current payment models. However, standard readmission prediction models have documented fairness issues, with disparities in prediction accuracy across racial and ethnic groups.
"""
Case study: Fair hospital readmission prediction.
This example demonstrates how to develop and evaluate a fairness-aware
readmission prediction model using realistic clinical data. We show how
standard approaches can produce biased predictions and how fairness-aware
methods can mitigate these disparities while maintaining clinical utility.
"""
import numpy as np
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
import matplotlib.pyplot as plt
import seaborn as sns
from typing import Dict, Tuple
def load_readmission_data() -> Tuple[pd.DataFrame, pd.Series, pd.Series]:
"""
Load and preprocess hospital readmission data.
In production, this would load from a clinical data warehouse.
For this example, we simulate realistic clinical data with known
disparities in readmission rates and documentation completeness.
Returns:
X: Feature DataFrame
y: Binary readmission outcome
race: Protected attribute for fairness evaluation
"""
np.random.seed(42)
n_samples = 10000
# Simulate demographic characteristics
# Ensure realistic population proportions
race = np.random.choice(
['White', 'Black', 'Hispanic', 'Asian', 'Other'],
size=n_samples,
p=[0.60, 0.20, 0.12, 0.05, 0.03]
)
# Simulate clinical features with realistic correlations
# Features are chosen based on established readmission risk factors
# Age (higher for some groups due to population demographics)
age = np.random.normal(65, 15, n_samples)
age[race == 'White'] += 3 # Older average age
age = np.clip(age, 18, 100)
# Comorbidity burden (Charlson Comorbidity Index)
# Simulate higher burden in disadvantaged groups
charlson_score = np.random.poisson(2.5, n_samples)
charlson_score[race == 'Black'] += np.random.poisson(0.5, np.sum(race == 'Black'))
charlson_score = np.clip(charlson_score, 0, 15)
# Length of stay (days)
los = np.random.lognormal(1.5, 0.8, n_samples)
los = np.clip(los, 1, 30)
# Number of ED visits in past year
# Higher for groups with less access to primary care
ed_visits = np.random.poisson(1.5, n_samples)
ed_visits[race == 'Black'] += np.random.poisson(0.8, np.sum(race == 'Black'))
ed_visits[race == 'Hispanic'] += np.random.poisson(0.6, np.sum(race == 'Hispanic'))
ed_visits = np.clip(ed_visits, 0, 20)
# Insurance status (strong proxy for socioeconomic status)
# Medicare: 0, Medicaid: 1, Commercial: 2, Uninsured: 3
insurance = np.zeros(n_samples, dtype=int)
# Age-based Medicare eligibility
insurance[age >= 65] = 0
# For younger patients, assign based on demographics
young_mask = age < 65
insurance[young_mask & (race == 'Black')] = np.random.choice(
[1, 2, 3], size=np.sum(young_mask & (race == 'Black')), p=[0.35, 0.50, 0.15]
)
insurance[young_mask & (race == 'Hispanic')] = np.random.choice(
[1, 2, 3], size=np.sum(young_mask & (race == 'Hispanic')), p=[0.30, 0.50, 0.20]
)
insurance[young_mask & (race == 'White')] = np.random.choice(
[1, 2], size=np.sum(young_mask & (race == 'White')), p=[0.15, 0.85]
)
# Social determinants (Area Deprivation Index - higher = more deprived)
# Simulated based on demographic patterns
adi = np.random.uniform(1, 10, n_samples)
adi[race == 'Black'] += np.random.uniform(0, 3, np.sum(race == 'Black'))
adi[race == 'Hispanic'] += np.random.uniform(0, 2.5, np.sum(race == 'Hispanic'))
adi = np.clip(adi, 1, 10)
# Discharge disposition (0: home, 1: SNF/rehab)
# Correlated with both severity and insurance
discharge_to_facility = np.random.binomial(1, 0.25, n_samples)
discharge_to_facility[charlson_score > 5] = np.random.binomial(
1, 0.45, np.sum(charlson_score > 5)
)
# Number of medications at discharge
n_medications = np.random.poisson(8, n_samples)
n_medications += charlson_score # More meds with more comorbidities
n_medications = np.clip(n_medications, 0, 30)
# Follow-up appointment scheduled (proxy for care coordination)
# Systematically lower for disadvantaged groups
followup_scheduled = np.random.binomial(1, 0.70, n_samples)
followup_scheduled[insurance == 3] = np.random.binomial( # Uninsured
1, 0.45, np.sum(insurance == 3)
)
followup_scheduled[insurance == 1] = np.random.binomial( # Medicaid
1, 0.55, np.sum(insurance == 1)
)
# Simulate readmission outcome
# True readmission risk is driven by clinical and social factors
# Clinical risk score
clinical_risk = (
0.02 * (age - 65) +
0.15 * charlson_score +
0.05 * los +
0.10 * ed_visits +
0.08 * n_medications +
-0.30 * followup_scheduled
)
# Social determinants contribution
social_risk = (
0.20 * adi +
0.25 * (insurance == 3) + # Uninsured penalty
0.15 * (insurance == 1) # Medicaid penalty
)
# Total risk (logit scale)
total_risk_logit = -2.5 + clinical_risk + social_risk
# Add random noise
total_risk_logit += np.random.normal(0, 0.5, n_samples)
# Convert to probability and generate outcomes
readmission_prob = 1 / (1 + np.exp(-total_risk_logit))
y = np.random.binomial(1, readmission_prob)
# Create feature DataFrame
X = pd.DataFrame({
'age': age,
'charlson_score': charlson_score,
'length_of_stay': los,
'ed_visits_past_year': ed_visits,
'n_medications': n_medications,
'discharge_to_facility': discharge_to_facility,
'followup_scheduled': followup_scheduled,
'insurance_medicaid': (insurance == 1).astype(int),
'insurance_commercial': (insurance == 2).astype(int),
'insurance_uninsured': (insurance == 3).astype(int),
'area_deprivation_index': adi
})
return X, pd.Series(y), pd.Series(race)
def compare_fairness_approaches(
X: pd.DataFrame,
y: pd.Series,
sensitive_feature: pd.Series
) -> Dict[str, Tuple[FairLogisticRegression, FairnessMetrics]]:
"""
Compare standard and fairness-aware approaches for readmission prediction.
We train three models:
1. Standard logistic regression (no fairness constraint)
2. Demographic parity constrained model
3. Equal opportunity constrained model
Returns:
Dictionary mapping approach name to (fitted model, fairness metrics)
"""
# Split data
X_train, X_test, y_train, y_test, race_train, race_test = train_test_split(
X, y, sensitive_feature, test_size=0.3, random_state=42, stratify=sensitive_feature
)
results = {}
# 1. Standard logistic regression (baseline)
print("\n" + "="*80)
print("Training standard logistic regression (no fairness constraint)...")
print("="*80)
model_standard = FairLogisticRegression(
C=1.0,
penalty='l2',
class_weight='balanced',
random_state=42
)
model_standard.fit(X_train, y_train)
fairness_standard = model_standard.evaluate_fairness(
X_test, y_test, race_test
)
print("\nStandard Model Performance:")
print(f"Demographic Parity Difference: {fairness_standard.demographic_parity_difference:.4f}")
print(f"Equal Opportunity Difference: {fairness_standard.equal_opportunity_difference:.4f}")
print(f"Equalized Odds Difference: {fairness_standard.equalized_odds_difference:.4f}")
results['standard'] = (model_standard, fairness_standard)
# 2. Demographic parity constrained
print("\n" + "="*80)
print("Training with demographic parity constraint...")
print("="*80)
model_dp = FairLogisticRegression(
C=1.0,
penalty='l2',
fairness_constraint='demographic_parity',
fairness_penalty=1.0,
class_weight='balanced',
random_state=42
)
model_dp.fit(X_train, y_train, sensitive_features=race_train)
fairness_dp = model_dp.evaluate_fairness(
X_test, y_test, race_test
)
print("\nDemographic Parity Model Performance:")
print(f"Demographic Parity Difference: {fairness_dp.demographic_parity_difference:.4f}")
print(f"Equal Opportunity Difference: {fairness_dp.equal_opportunity_difference:.4f}")
print(f"Equalized Odds Difference: {fairness_dp.equalized_odds_difference:.4f}")
results['demographic_parity'] = (model_dp, fairness_dp)
# 3. Equal opportunity constrained
print("\n" + "="*80)
print("Training with equal opportunity constraint...")
print("="*80)
model_eo = FairLogisticRegression(
C=1.0,
penalty='l2',
fairness_constraint='equal_opportunity',
fairness_penalty=1.5,
class_weight='balanced',
random_state=42
)
model_eo.fit(X_train, y_train, sensitive_features=race_train)
fairness_eo = model_eo.evaluate_fairness(
X_test, y_test, race_test
)
print("\nEqual Opportunity Model Performance:")
print(f"Demographic Parity Difference: {fairness_eo.demographic_parity_difference:.4f}")
print(f"Equal Opportunity Difference: {fairness_eo.equal_opportunity_difference:.4f}")
print(f"Equalized Odds Difference: {fairness_eo.equalized_odds_difference:.4f}")
results['equal_opportunity'] = (model_eo, fairness_eo)
return results
def visualize_fairness_comparison(results: Dict) -> None:
"""
Create visualizations comparing fairness across modeling approaches.
Generates bar charts showing key fairness metrics and group-specific
performance for each modeling approach.
"""
# Extract data for visualization
approaches = list(results.keys())
# Aggregate fairness metrics
dp_diffs = [results[app][1].demographic_parity_difference for app in approaches]
eo_diffs = [results[app][1].equal_opportunity_difference for app in approaches]
eod_diffs = [results[app][1].equalized_odds_difference for app in approaches]
# Create figure
fig, axes = plt.subplots(1, 3, figsize=(15, 5))
# Demographic parity
axes[0].bar(approaches, dp_diffs, color=['#FF6B6B', '#4ECDC4', '#45B7D1'])
axes[0].set_ylabel('Difference')
axes[0].set_title('Demographic Parity Difference\n(Lower is More Fair)')
axes[0].axhline(y=0.1, color='r', linestyle='--', label='Threshold')
axes[0].legend()
axes[0].set_ylim([0, max(dp_diffs) * 1.2])
# Equal opportunity
axes[1].bar(approaches, eo_diffs, color=['#FF6B6B', '#4ECDC4', '#45B7D1'])
axes[1].set_ylabel('Difference')
axes[1].set_title('Equal Opportunity Difference\n(Lower is More Fair)')
axes[1].axhline(y=0.1, color='r', linestyle='--', label='Threshold')
axes[1].legend()
axes[1].set_ylim([0, max(eo_diffs) * 1.2])
# Equalized odds
axes[2].bar(approaches, eod_diffs, color=['#FF6B6B', '#4ECDC4', '#45B7D1'])
axes[2].set_ylabel('Difference')
axes[2].set_title('Equalized Odds Difference\n(Lower is More Fair)')
axes[2].axhline(y=0.1, color='r', linestyle='--', label='Threshold')
axes[2].legend()
axes[2].set_ylim([0, max(eod_diffs) * 1.2])
plt.tight_layout()
plt.savefig('/mnt/user-data/outputs/readmission_fairness_comparison.png', dpi=300, bbox_inches='tight')
print("\nFairness comparison visualization saved.")
# Group-specific performance
fig, axes = plt.subplots(len(approaches), 1, figsize=(12, 4*len(approaches)))
for idx, approach in enumerate(approaches):
_, fairness_metrics = results[approach]
groups = list(fairness_metrics.group_metrics.keys())
aucs = [fairness_metrics.group_metrics[g]['auc'] for g in groups]
tprs = [fairness_metrics.group_metrics[g]['tpr'] for g in groups]
x = np.arange(len(groups))
width = 0.35
axes[idx].bar(x - width/2, aucs, width, label='AUC', color='#4ECDC4')
axes[idx].bar(x + width/2, tprs, width, label='TPR', color='#FF6B6B')
axes[idx].set_ylabel('Performance')
axes[idx].set_title(f'{approach.replace("_", " ").title()} - Group Performance')
axes[idx].set_xticks(x)
axes[idx].set_xticklabels(groups)
axes[idx].legend()
axes[idx].set_ylim([0, 1.0])
plt.tight_layout()
plt.savefig('/mnt/user-data/outputs/readmission_group_performance.png', dpi=300, bbox_inches='tight')
print("Group performance visualization saved.")
# Run the case study
if __name__ == "__main__":
print("Loading readmission data...")
X, y, race = load_readmission_data()
print(f"\nDataset statistics:")
print(f"Total samples: {len(y)}")
print(f"Readmission rate: {y.mean():.1%}")
print(f"\nRacial distribution:")
print(race.value_counts())
print(f"\nReadmission rates by race:")
for r in race.unique():
rate = y[race == r].mean()
print(f" {r}: {rate:.1%}")
print("\n" + "="*80)
print("COMPARING FAIRNESS APPROACHES")
print("="*80)
results = compare_fairness_approaches(X, y, race)
visualize_fairness_comparison(results)
print("\n" + "="*80)
print("ANALYSIS COMPLETE")
print("="*80)
This case study demonstrates several key principles of fairness-aware clinical machine learning. First, we see that standard approaches often produce disparate performance across demographic groups even when demographic variables are not explicitly used as features. This occurs because other features serve as proxies for protected characteristics. Second, fairness-aware constraints can meaningfully reduce these disparities, though with tradeoffs depending on which fairness definition we prioritize. Third, the appropriate choice of fairness constraint depends on the clinical application and its potential harms.
For readmission prediction used to target care management resources, we might prioritize equal opportunity, ensuring that patients who will readmit have equal probability of being identified for intervention regardless of demographics. For prediction used to adjust risk scores in payment models, we might prioritize calibration fairness, ensuring that predicted risks accurately reflect true risks across all groups. The key is making these choices explicitly and evaluating their consequences comprehensively.
4.3 Decision Trees: Interpretable Nonlinear Risk Models
Decision trees offer an appealing alternative to logistic regression when relationships between features and outcomes are fundamentally nonlinear or involve complex interactions. A decision tree recursively partitions the feature space based on threshold rules, creating a hierarchical structure that naturally handles nonlinearities and interactions without requiring manual feature engineering. The resulting model is highly interpretable as a sequence of if-then rules that clinicians can directly inspect and validate against domain knowledge.
From an equity perspective, decision trees present both opportunities and challenges. Their nonparametric nature enables them to capture heterogeneous outcome relationships that might differ across patient populations, potentially improving performance for underrepresented groups compared to simpler linear models. Their interpretability allows direct inspection of whether the tree is making clinically appropriate splits or learning inappropriate shortcuts. However, their greedy splitting algorithm can exacerbate bias when some populations are underrepresented in training data, as the algorithm preferentially splits to improve accuracy for the majority while ignoring minority group performance.
4.3.1 Tree Construction and Splitting Criteria
A decision tree is constructed through recursive binary splitting of the feature space. Starting with the full dataset at the root node, the algorithm selects a feature and threshold that best separates the data according to some splitting criterion. The most common criteria are:
Gini impurity measures the probability of misclassifying a randomly chosen element if it were randomly labeled according to the distribution of labels in the node. For a node with class proportions $p_1, p_2, \ldots, p_K$, the Gini impurity is:
The algorithm selects splits that maximize the reduction in weighted average Gini impurity across child nodes. This criterion favors splits that create pure nodes where most samples belong to a single class.
Entropy or information gain measures the reduction in uncertainty about class labels. For a node with class proportions $p_1, p_2, \ldots, p_K$, the entropy is:
Splits are chosen to maximize information gain, the difference between parent node entropy and weighted average child node entropy. Entropy and Gini impurity typically produce similar trees, though entropy is more computationally expensive due to the logarithm computation.
Mean squared error can be used for regression trees, measuring the variance in outcomes within each node. For classification with probability estimates, we might instead use log loss or Brier score.
The greedy nature of tree construction has important implications for fairness. At each split, the algorithm considers only the immediate reduction in impurity without lookahead to future splits. If one demographic group is underrepresented, splits that would benefit that group may not provide sufficient immediate impurity reduction to be selected, especially if they would harm majority group performance. This means that tree structure can systematically disadvantage minority populations even when no demographic information is explicitly used.
4.3.2 Fairness-Aware Tree Construction
We now develop methods for constructing decision trees that explicitly account for fairness during the splitting process. Rather than selecting splits based solely on overall impurity reduction, we modify the splitting criterion to penalize splits that create disparate outcomes across demographic groups.
"""
Fairness-aware decision tree for clinical risk prediction.
This implementation extends standard decision tree construction to explicitly
account for fairness during splitting decisions. Rather than pure greedy
optimization of overall impurity, we incorporate penalties for splits that
create disparate outcomes across protected groups.
"""
import numpy as np
import pandas as pd
from typing import Dict, List, Optional, Tuple, Union
from dataclasses import dataclass
from collections import Counter
import logging
logging.basicConfig(level=logging.INFO)
logger = logging.getLogger(__name__)
@dataclass
class TreeNode:
"""
Node in a decision tree.
Internal nodes contain a split rule (feature and threshold).
Leaf nodes contain class predictions and probabilities.
"""
# For internal nodes
feature: Optional[int] = None
threshold: Optional[float] = None
left: Optional['TreeNode'] = None
right: Optional['TreeNode'] = None
# For leaf nodes
value: Optional[np.ndarray] = None # Class probabilities
n_samples: Optional[int] = None
impurity: Optional[float] = None
# Fairness tracking
group_distributions: Optional[Dict[str, np.ndarray]] = None
def is_leaf(self) -> bool:
"""Check if this is a leaf node."""
return self.value is not None
class FairDecisionTreeClassifier:
"""
Decision tree classifier with fairness constraints during construction.
This implementation modifies the splitting criterion to explicitly penalize
splits that create disparate outcomes across demographic groups. The fairness
penalty is incorporated directly into split selection, ensuring that fairness
is optimized jointly with predictive accuracy rather than added post-hoc.
Parameters:
max_depth: Maximum depth of the tree (None = unlimited)
min_samples_split: Minimum samples required to split a node
min_samples_leaf: Minimum samples required in each leaf
min_impurity_decrease: Minimum impurity decrease to make a split
fairness_penalty: Weight for fairness constraint (0 = no constraint)
fairness_metric: Type of fairness to optimize
'demographic_parity': Balance positive rates across groups
'equal_opportunity': Balance true positive rates across groups
criterion: Impurity measure ('gini' or 'entropy')
max_features: Number of features to consider per split
random_state: Random seed for reproducibility
"""
def __init__(
self,
max_depth: Optional[int] = None,
min_samples_split: int = 2,
min_samples_leaf: int = 1,
min_impurity_decrease: float = 0.0,
fairness_penalty: float = 0.0,
fairness_metric: str = 'demographic_parity',
criterion: str = 'gini',
max_features: Optional[Union[int, str]] = None,
random_state: Optional[int] = None
):
self.max_depth = max_depth
self.min_samples_split = min_samples_split
self.min_samples_leaf = min_samples_leaf
self.min_impurity_decrease = min_impurity_decrease
self.fairness_penalty = fairness_penalty
self.fairness_metric = fairness_metric
self.criterion = criterion
self.max_features = max_features
self.random_state = random_state
self.tree_ = None
self.classes_ = None
self.n_features_ = None
if random_state is not None:
np.random.seed(random_state)
def _gini(self, y: np.ndarray) -> float:
"""Compute Gini impurity."""
n = len(y)
if n == 0:
return 0.0
counts = np.bincount(y)
probs = counts / n
return 1.0 - np.sum(probs**2)
def _entropy(self, y: np.ndarray) -> float:
"""Compute entropy."""
n = len(y)
if n == 0:
return 0.0
counts = np.bincount(y)
probs = counts / n
probs = probs[probs > 0] # Avoid log(0)
return -np.sum(probs * np.log2(probs))
def _impurity(self, y: np.ndarray) -> float:
"""Compute impurity based on selected criterion."""
if self.criterion == 'gini':
return self._gini(y)
elif self.criterion == 'entropy':
return self._entropy(y)
else:
raise ValueError(f"Unknown criterion: {self.criterion}")
def _compute_fairness_cost(
self,
y_left: np.ndarray,
y_right: np.ndarray,
sensitive_left: np.ndarray,
sensitive_right: np.ndarray
) -> float:
"""
Compute fairness cost for a potential split.
The cost measures how much the split creates disparate outcomes
across demographic groups. Higher costs indicate greater unfairness.
"""
if self.fairness_penalty == 0:
return 0.0
# Combine both sides to evaluate overall fairness
y_combined = np.concatenate([y_left, y_right])
sensitive_combined = np.concatenate([sensitive_left, sensitive_right])
preds_combined = np.concatenate([
np.ones(len(y_left)), # Left side gets positive prediction
np.zeros(len(y_right)) # Right side gets negative prediction
])
# Compute group-specific metrics
groups = np.unique(sensitive_combined)
if len(groups) < 2:
return 0.0 # No fairness cost if only one group
group_metrics = {}
for group in groups:
mask = sensitive_combined == group
group_y = y_combined[mask]
group_preds = preds_combined[mask]
if len(group_y) == 0:
continue
if self.fairness_metric == 'demographic_parity':
# Positive prediction rate
metric = np.mean(group_preds)
elif self.fairness_metric == 'equal_opportunity':
# True positive rate for positive class
pos_mask = group_y == 1
if np.sum(pos_mask) > 0:
metric = np.mean(group_preds[pos_mask])
else:
metric = 0.0
else:
raise ValueError(f"Unknown fairness metric: {self.fairness_metric}")
group_metrics[group] = metric
# Fairness cost is variance across groups
if len(group_metrics) > 1:
return np.var(list(group_metrics.values()))
else:
return 0.0
def _find_best_split(
self,
X: np.ndarray,
y: np.ndarray,
sensitive_features: Optional[np.ndarray],
feature_indices: np.ndarray
) -> Tuple[Optional[int], Optional[float], float]:
"""
Find the best split for a node.
Returns:
best_feature: Index of feature to split on (None if no valid split)
best_threshold: Threshold value for split
best_gain: Information gain from split
"""
n_samples = len(y)
parent_impurity = self._impurity(y)
best_gain = -np.inf
best_feature = None
best_threshold = None
# Try each candidate feature
for feature_idx in feature_indices:
# Get unique values for this feature
thresholds = np.unique(X[:, feature_idx])
# Skip if only one unique value
if len(thresholds) == 1:
continue
# Try each potential threshold
# For efficiency, only consider midpoints between sorted unique values
for i in range(len(thresholds) - 1):
threshold = (thresholds[i] + thresholds[i+1]) / 2
# Split samples
left_mask = X[:, feature_idx] <= threshold
right_mask = ~left_mask
n_left = np.sum(left_mask)
n_right = np.sum(right_mask)
# Check minimum samples constraint
if n_left < self.min_samples_leaf or n_right < self.min_samples_leaf:
continue
# Compute weighted impurity after split
y_left = y[left_mask]
y_right = y[right_mask]
impurity_left = self._impurity(y_left)
impurity_right = self._impurity(y_right)
weighted_impurity = (n_left * impurity_left + n_right * impurity_right) / n_samples
# Information gain
gain = parent_impurity - weighted_impurity
# Add fairness penalty if sensitive features provided
if sensitive_features is not None and self.fairness_penalty > 0:
fairness_cost = self._compute_fairness_cost(
y_left, y_right,
sensitive_features[left_mask],
sensitive_features[right_mask]
)
# Subtract fairness cost from gain (higher cost = less desirable split)
gain -= self.fairness_penalty * fairness_cost
# Track best split
if gain > best_gain:
best_gain = gain
best_feature = feature_idx
best_threshold = threshold
# Check minimum impurity decrease
if best_gain < self.min_impurity_decrease:
return None, None, 0.0
return best_feature, best_threshold, best_gain
def _build_tree(
self,
X: np.ndarray,
y: np.ndarray,
sensitive_features: Optional[np.ndarray],
depth: int = 0
) -> TreeNode:
"""
Recursively build decision tree.
This method implements the core tree construction algorithm with
fairness-aware splitting decisions.
"""
n_samples = len(y)
n_classes = len(self.classes_)
# Compute class probabilities for this node
class_counts = np.bincount(y, minlength=n_classes)
probabilities = class_counts / n_samples
# Compute group distributions if sensitive features provided
group_distributions = None
if sensitive_features is not None:
groups = np.unique(sensitive_features)
group_distributions = {}
for group in groups:
mask = sensitive_features == group
group_y = y[mask]
if len(group_y) > 0:
group_counts = np.bincount(group_y, minlength=n_classes)
group_probs = group_counts / len(group_y)
group_distributions[str(group)] = group_probs
# Create leaf node if stopping criteria met
if (self.max_depth is not None and depth >= self.max_depth) or \
n_samples < self.min_samples_split or \
len(np.unique(y)) == 1: # Pure node
return TreeNode(
value=probabilities,
n_samples=n_samples,
impurity=self._impurity(y),
group_distributions=group_distributions
)
# Select features to consider for splitting
if self.max_features is None:
feature_indices = np.arange(self.n_features_)
elif isinstance(self.max_features, int):
feature_indices = np.random.choice(
self.n_features_,
size=min(self.max_features, self.n_features_),
replace=False
)
elif self.max_features == 'sqrt':
n_features_to_try = int(np.sqrt(self.n_features_))
feature_indices = np.random.choice(
self.n_features_,
size=n_features_to_try,
replace=False
)
elif self.max_features == 'log2':
n_features_to_try = int(np.log2(self.n_features_))
feature_indices = np.random.choice(
self.n_features_,
size=n_features_to_try,
replace=False
)
else:
raise ValueError(f"Invalid max_features: {self.max_features}")
# Find best split
best_feature, best_threshold, best_gain = self._find_best_split(
X, y, sensitive_features, feature_indices
)
# Create leaf if no valid split found
if best_feature is None:
return TreeNode(
value=probabilities,
n_samples=n_samples,
impurity=self._impurity(y),
group_distributions=group_distributions
)
# Split data and recursively build subtrees
left_mask = X[:, best_feature] <= best_threshold
right_mask = ~left_mask
left_subtree = self._build_tree(
X[left_mask],
y[left_mask],
sensitive_features[left_mask] if sensitive_features is not None else None,
depth + 1
)
right_subtree = self._build_tree(
X[right_mask],
y[right_mask],
sensitive_features[right_mask] if sensitive_features is not None else None,
depth + 1
)
# Create internal node
return TreeNode(
feature=best_feature,
threshold=best_threshold,
left=left_subtree,
right=right_subtree,
n_samples=n_samples,
impurity=self._impurity(y),
group_distributions=group_distributions
)
def fit(
self,
X: Union[np.ndarray, pd.DataFrame],
y: Union[np.ndarray, pd.Series],
sensitive_features: Optional[Union[np.ndarray, pd.Series]] = None
) -> 'FairDecisionTreeClassifier':
"""
Build a decision tree from training data.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
y: Binary target variable of shape (n_samples,)
sensitive_features: Protected attributes for fairness constraints
Returns:
self: Fitted tree instance
"""
# Convert to numpy arrays
if isinstance(X, pd.DataFrame):
X = X.values
if isinstance(y, pd.Series):
y = y.values
if isinstance(sensitive_features, pd.Series):
sensitive_features = sensitive_features.values
# Store classes and number of features
self.classes_ = np.unique(y)
self.n_features_ = X.shape[1]
if len(self.classes_) != 2:
raise ValueError("Only binary classification is supported")
# Build tree
self.tree_ = self._build_tree(X, y, sensitive_features)
logger.info("Decision tree fitted successfully")
return self
def _traverse(self, x: np.ndarray, node: TreeNode) -> np.ndarray:
"""Traverse tree to make prediction for a single sample."""
if node.is_leaf():
return node.value
if x[node.feature] <= node.threshold:
return self._traverse(x, node.left)
else:
return self._traverse(x, node.right)
def predict_proba(self, X: Union[np.ndarray, pd.DataFrame]) -> np.ndarray:
"""
Predict class probabilities.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
Returns:
Predicted probabilities of shape (n_samples, 2)
"""
if self.tree_ is None:
raise ValueError("Tree must be fitted before prediction")
if isinstance(X, pd.DataFrame):
X = X.values
probabilities = np.array([self._traverse(x, self.tree_) for x in X])
return probabilities
def predict(self, X: Union[np.ndarray, pd.DataFrame]) -> np.ndarray:
"""
Predict class labels.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
Returns:
Predicted class labels of shape (n_samples,)
"""
probabilities = self.predict_proba(X)
return (probabilities[:, 1] >= 0.5).astype(int)
def _get_tree_structure(self, node: TreeNode, feature_names: Optional[List[str]] = None, depth: int = 0) -> str:
"""
Generate string representation of tree structure.
This creates a human-readable representation showing the splitting
rules and leaf predictions, useful for interpretation and debugging.
"""
indent = " " * depth
if node.is_leaf():
class_probs = node.value
pred_class = np.argmax(class_probs)
return f"{indent}Leaf: class={pred_class}, probs={class_probs}, n_samples={node.n_samples}\n"
feature_name = f"X[{node.feature}]" if feature_names is None else feature_names[node.feature]
structure = f"{indent}Node: {feature_name} <= {node.threshold:.3f} (n_samples={node.n_samples})\n"
structure += self._get_tree_structure(node.left, feature_names, depth + 1)
structure += f"{indent}Node: {feature_name} > {node.threshold:.3f}\n"
structure += self._get_tree_structure(node.right, feature_names, depth + 1)
return structure
def print_tree(self, feature_names: Optional[List[str]] = None) -> None:
"""Print human-readable tree structure."""
if self.tree_ is None:
print("Tree not fitted yet")
return
print("Decision Tree Structure:")
print("=" * 80)
print(self._get_tree_structure(self.tree_, feature_names))
This implementation of fairness-aware decision trees directly incorporates equity considerations into the splitting process. By adding a fairness penalty to the splitting criterion, we can construct trees that balance overall predictive performance with fairness across demographic groups. The key advantage over post-processing approaches is that fairness is optimized during tree construction, potentially enabling better tradeoffs between accuracy and equity.
4.3.3 Tree Interpretation and Clinical Validation
One of the most valuable properties of decision trees for healthcare applications is their interpretability. Unlike complex ensemble methods or neural networks, a single decision tree can be directly examined by clinicians to validate whether splits are clinically sensible and whether the model is making appropriate use of available features.
"""
Tools for interpreting and validating decision trees in clinical contexts.
This module provides utilities for extracting decision rules, identifying
potentially problematic splits, and generating clinical validation reports
that domain experts can review.
"""
from typing import List, Dict, Tuple
import pandas as pd
import numpy as np
class TreeInterpreter:
"""
Extract and analyze decision rules from fitted trees.
This class provides methods for understanding what a tree has learned,
including extracting decision paths, identifying important splits, and
flagging potentially problematic rules that might encode bias.
"""
def __init__(self, tree: FairDecisionTreeClassifier, feature_names: List[str]):
"""
Initialize interpreter with fitted tree and feature names.
Parameters:
tree: Fitted FairDecisionTreeClassifier
feature_names: Names corresponding to feature indices
"""
self.tree = tree
self.feature_names = feature_names
def extract_rules(self, node: Optional[TreeNode] = None, path: List[str] = None) -> List[Tuple[List[str], np.ndarray]]:
"""
Extract all decision rules from root to leaves.
Returns list of (rule_path, leaf_probabilities) tuples where rule_path
is a list of string conditions that must be satisfied to reach the leaf.
"""
if path is None:
path = []
if node is None:
node = self.tree.tree_
if node.is_leaf():
return [(path.copy(), node.value)]
rules = []
# Left subtree (condition satisfied)
feature_name = self.feature_names[node.feature]
left_condition = f"{feature_name} <= {node.threshold:.3f}"
rules.extend(self.extract_rules(node.left, path + [left_condition]))
# Right subtree (condition not satisfied)
right_condition = f"{feature_name} > {node.threshold:.3f}"
rules.extend(self.extract_rules(node.right, path + [right_condition]))
return rules
def get_decision_path(self, x: np.ndarray) -> Tuple[List[str], np.ndarray]:
"""
Get the decision path for a specific sample.
Returns the sequence of conditions that led to the prediction for this
sample along with the final predicted probabilities.
"""
node = self.tree.tree_
path = []
while not node.is_leaf():
feature_name = self.feature_names[node.feature]
threshold = node.threshold
value = x[node.feature]
if value <= threshold:
path.append(f"{feature_name} <= {threshold:.3f} (value: {value:.3f})")
node = node.left
else:
path.append(f"{feature_name} > {threshold:.3f} (value: {value:.3f})")
node = node.right
return path, node.value
def identify_sensitive_splits(self, sensitive_feature_names: List[str]) -> List[Dict]:
"""
Identify splits that directly use sensitive features.
Such splits may be appropriate (e.g., sex-specific risk models) or
problematic (e.g., race-based rules without clinical justification).
Clinical review is essential.
Returns list of dictionaries describing each sensitive split.
"""
sensitive_splits = []
def traverse(node: TreeNode, depth: int = 0):
if node.is_leaf():
return
feature_name = self.feature_names[node.feature]
if feature_name in sensitive_feature_names:
sensitive_splits.append({
'depth': depth,
'feature': feature_name,
'threshold': node.threshold,
'n_samples': node.n_samples
})
traverse(node.left, depth + 1)
traverse(node.right, depth + 1)
traverse(self.tree.tree_)
return sensitive_splits
def get_feature_importance(self) -> pd.DataFrame:
"""
Compute feature importance based on impurity reduction.
Features that appear higher in the tree and create larger impurity
reductions are more important. This metric helps identify which
features the model relies on most heavily.
"""
importance = np.zeros(len(self.feature_names))
def traverse(node: TreeNode):
if node.is_leaf():
return
# Impurity reduction from this split
n_samples = node.n_samples
parent_impurity = node.impurity
left_impurity = node.left.impurity
right_impurity = node.right.impurity
n_left = node.left.n_samples
n_right = node.right.n_samples
weighted_child_impurity = (n_left * left_impurity + n_right * right_impurity) / n_samples
impurity_reduction = n_samples * (parent_impurity - weighted_child_impurity)
importance[node.feature] += impurity_reduction
traverse(node.left)
traverse(node.right)
traverse(self.tree.tree_)
# Normalize
if importance.sum() > 0:
importance = importance / importance.sum()
importance_df = pd.DataFrame({
'feature': self.feature_names,
'importance': importance
})
importance_df = importance_df.sort_values('importance', ascending=False)
return importance_df
def generate_clinical_validation_report(self) -> str:
"""
Generate report for clinical validation of tree structure.
This report extracts key information that clinical experts need to
validate whether the tree has learned clinically appropriate rules.
"""
lines = []
lines.append("=" * 80)
lines.append("CLINICAL VALIDATION REPORT FOR DECISION TREE")
lines.append("=" * 80)
lines.append("")
# Tree statistics
def count_nodes(node):
if node.is_leaf():
return 1, 1, 0
left_total, left_leaves, left_internal = count_nodes(node.left)
right_total, right_leaves, right_internal = count_nodes(node.right)
return left_total + right_total + 1, left_leaves + right_leaves, left_internal + right_internal + 1
n_total, n_leaves, n_internal = count_nodes(self.tree.tree_)
lines.append("TREE STRUCTURE:")
lines.append(f" Total nodes: {n_total}")
lines.append(f" Internal nodes: {n_internal}")
lines.append(f" Leaf nodes: {n_leaves}")
lines.append("")
# Feature importance
lines.append("FEATURE IMPORTANCE:")
importance_df = self.get_feature_importance()
for _, row in importance_df.head(10).iterrows():
lines.append(f" {row['feature']}: {row['importance']:.4f}")
lines.append("")
# All decision rules
lines.append("COMPLETE DECISION RULES:")
lines.append("(Clinical review required to validate appropriateness)")
lines.append("")
rules = self.extract_rules()
for idx, (path, probs) in enumerate(rules, 1):
pred_class = np.argmax(probs)
pred_prob = probs[pred_class]
lines.append(f"Rule {idx}:")
lines.append(f" Conditions:")
for condition in path:
lines.append(f" {condition}")
lines.append(f" Prediction: Class {pred_class} (probability: {pred_prob:.3f})")
lines.append(f" All class probabilities: {probs}")
lines.append("")
lines.append("=" * 80)
lines.append("END OF CLINICAL VALIDATION REPORT")
lines.append("=" * 80)
return "\n".join(lines)
This interpretation framework enables clinicians to systematically review what the tree has learned and validate whether the splits make clinical sense. For instance, if a tree splits heavily on neighborhood deprivation index but not on established clinical risk factors, this might indicate that the tree is learning shortcuts based on access to care rather than true health risk. Such patterns require careful clinical review before deployment.
4.4 Random Forests: Ensemble Learning for Robustness
Random forests extend single decision trees by training an ensemble of trees on bootstrap samples of the data and averaging their predictions. This ensemble approach dramatically improves predictive performance compared to single trees while maintaining many of their desirable properties. The averaging across trees reduces variance, making random forests more robust to outliers and less prone to overfitting than individual trees. The random feature selection at each split introduces diversity among trees, further improving generalization.
From an equity perspective, random forests offer important advantages but also new challenges. The ensemble averaging can improve robustness for underrepresented populations by reducing the impact of individual trees that may have learned biased patterns from unrepresentative training subsets. The feature subsampling reduces reliance on any single feature, potentially mitigating the impact of features that serve as problematic proxies for protected characteristics. However, if all trees in the forest learn similar biases from the training data, ensemble averaging amplifies rather than corrects these biases. Careful attention to fairness during both tree construction and ensemble aggregation remains essential.
4.4.1 Random Forest Algorithm and Equity Considerations
A random forest consists of $T$ decision trees, each trained on a bootstrap sample of the training data with random feature subsampling at each split. For prediction, the forest aggregates predictions across all trees, typically using majority voting for classification or averaging for probability estimates:
where $\hat{p}_t(y=1 \mid \mathbf{x})$ is the predicted probability from tree $t$.
The bootstrap sampling creates training subsets that differ across trees, introducing diversity that improves generalization. However, if demographic groups are unevenly represented in the full training set, bootstrap samples will maintain or exacerbate this imbalance. Trees trained on samples with very few examples from minority groups may learn poor models for those groups, and aggregating many such trees still produces poor overall performance for underrepresented populations.
The random feature subsampling at each split considers only a subset of features when determining the best split. Typically $\sqrt{p}$ features are considered at each split for $p$ total features, though this hyperparameter can be tuned. This randomness introduces additional diversity and reduces overfitting, but also means that features particularly informative for minority groups may be missed if they don’t appear in the random subset at critical splits.
4.4.2 Fairness-Aware Random Forest Implementation
We now develop a random forest implementation that extends our fairness-aware decision trees to the ensemble setting. Beyond simply training multiple fair trees, we implement stratified bootstrap sampling to ensure adequate representation across demographic groups in each tree’s training set, and we monitor fairness metrics across the ensemble to detect when individual trees might be introducing bias.
"""
Fairness-aware random forest for clinical risk prediction.
This implementation extends random forests to explicitly account for equity
across patient populations through stratified bootstrap sampling and fairness-
constrained tree construction.
"""
import numpy as np
import pandas as pd
from typing import List, Optional, Union, Dict
import logging
from concurrent.futures import ProcessPoolExecutor, as_completed
from sklearn.model_selection import StratifiedKFold
logging.basicConfig(level=logging.INFO)
logger = logging.getLogger(__name__)
class FairRandomForestClassifier:
"""
Random forest with fairness constraints for clinical prediction.
This implementation trains an ensemble of fairness-aware decision trees
with stratified bootstrap sampling to ensure adequate representation of
all demographic groups in each tree's training data.
Parameters:
n_estimators: Number of trees in the forest
max_depth: Maximum depth of each tree
min_samples_split: Minimum samples to split a node
min_samples_leaf: Minimum samples in each leaf
fairness_penalty: Weight for fairness constraint in each tree
fairness_metric: Type of fairness to optimize
criterion: Impurity measure for splitting
max_features: Features to consider per split ('sqrt', 'log2', or int)
bootstrap: Whether to use bootstrap sampling
stratify_groups: Whether to use stratified bootstrap by sensitive groups
max_samples: If bootstrap, fraction of samples to draw
n_jobs: Number of parallel jobs (-1 for all CPUs)
random_state: Random seed for reproducibility
verbose: Verbosity level
"""
def __init__(
self,
n_estimators: int = 100,
max_depth: Optional[int] = None,
min_samples_split: int = 2,
min_samples_leaf: int = 1,
fairness_penalty: float = 0.0,
fairness_metric: str = 'demographic_parity',
criterion: str = 'gini',
max_features: Union[str, int] = 'sqrt',
bootstrap: bool = True,
stratify_groups: bool = True,
max_samples: Optional[float] = None,
n_jobs: int = -1,
random_state: Optional[int] = None,
verbose: int = 0
):
self.n_estimators = n_estimators
self.max_depth = max_depth
self.min_samples_split = min_samples_split
self.min_samples_leaf = min_samples_leaf
self.fairness_penalty = fairness_penalty
self.fairness_metric = fairness_metric
self.criterion = criterion
self.max_features = max_features
self.bootstrap = bootstrap
self.stratify_groups = stratify_groups
self.max_samples = max_samples
self.n_jobs = n_jobs
self.random_state = random_state
self.verbose = verbose
self.estimators_ = []
self.classes_ = None
self.n_features_ = None
if random_state is not None:
np.random.seed(random_state)
def _get_bootstrap_sample(
self,
X: np.ndarray,
y: np.ndarray,
sensitive_features: Optional[np.ndarray],
random_state: int
) -> Tuple[np.ndarray, np.ndarray, Optional[np.ndarray], np.ndarray]:
"""
Generate bootstrap sample with optional stratification.
If stratify_groups is True, we ensure that each bootstrap sample
maintains approximately the same proportion of each demographic group
as the full dataset. This prevents individual trees from being trained
on highly unrepresentative samples.
"""
np.random.seed(random_state)
n_samples = len(y)
if self.max_samples is not None:
n_samples = int(self.max_samples * n_samples)
if not self.stratify_groups or sensitive_features is None:
# Standard bootstrap sampling
indices = np.random.choice(len(y), size=n_samples, replace=self.bootstrap)
else:
# Stratified bootstrap by sensitive features
groups = np.unique(sensitive_features)
indices = []
for group in groups:
group_indices = np.where(sensitive_features == group)[0]
n_group = len(group_indices)
# Sample proportionally from each group
n_group_samples = int(n_samples * n_group / len(y))
if n_group_samples > 0:
group_sample = np.random.choice(
group_indices,
size=n_group_samples,
replace=self.bootstrap
)
indices.extend(group_sample)
indices = np.array(indices)
np.random.shuffle(indices)
X_boot = X[indices]
y_boot = y[indices]
sensitive_boot = sensitive_features[indices] if sensitive_features is not None else None
return X_boot, y_boot, sensitive_boot, indices
def _train_single_tree(
self,
X: np.ndarray,
y: np.ndarray,
sensitive_features: Optional[np.ndarray],
tree_idx: int
) -> FairDecisionTreeClassifier:
"""
Train a single tree on a bootstrap sample.
This method is designed to be called in parallel across multiple trees.
"""
# Generate bootstrap sample with tree-specific random state
random_state = self.random_state + tree_idx if self.random_state is not None else tree_idx
X_boot, y_boot, sensitive_boot, _ = self._get_bootstrap_sample(
X, y, sensitive_features, random_state
)
# Train tree
tree = FairDecisionTreeClassifier(
max_depth=self.max_depth,
min_samples_split=self.min_samples_split,
min_samples_leaf=self.min_samples_leaf,
fairness_penalty=self.fairness_penalty,
fairness_metric=self.fairness_metric,
criterion=self.criterion,
max_features=self.max_features,
random_state=random_state
)
tree.fit(X_boot, y_boot, sensitive_boot)
if self.verbose > 0:
logger.info(f"Trained tree {tree_idx + 1}/{self.n_estimators}")
return tree
def fit(
self,
X: Union[np.ndarray, pd.DataFrame],
y: Union[np.ndarray, pd.Series],
sensitive_features: Optional[Union[np.ndarray, pd.Series]] = None
) -> 'FairRandomForestClassifier':
"""
Build a random forest from training data.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
y: Binary target variable of shape (n_samples,)
sensitive_features: Protected attributes for fairness constraints
Returns:
self: Fitted forest instance
"""
# Convert to numpy arrays
if isinstance(X, pd.DataFrame):
X = X.values
if isinstance(y, pd.Series):
y = y.values
if isinstance(sensitive_features, pd.Series):
sensitive_features = sensitive_features.values
# Store classes and number of features
self.classes_ = np.unique(y)
self.n_features_ = X.shape[1]
if len(self.classes_) != 2:
raise ValueError("Only binary classification is supported")
logger.info(f"Training random forest with {self.n_estimators} trees...")
# Train trees in parallel if n_jobs != 1
if self.n_jobs == 1:
# Sequential training
self.estimators_ = []
for tree_idx in range(self.n_estimators):
tree = self._train_single_tree(X, y, sensitive_features, tree_idx)
self.estimators_.append(tree)
else:
# Parallel training
max_workers = None if self.n_jobs == -1 else self.n_jobs
with ProcessPoolExecutor(max_workers=max_workers) as executor:
futures = [
executor.submit(self._train_single_tree, X, y, sensitive_features, i)
for i in range(self.n_estimators)
]
self.estimators_ = []
for future in as_completed(futures):
tree = future.result()
self.estimators_.append(tree)
logger.info("Random forest training complete")
return self
def predict_proba(self, X: Union[np.ndarray, pd.DataFrame]) -> np.ndarray:
"""
Predict class probabilities by averaging across trees.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
Returns:
Predicted probabilities of shape (n_samples, 2)
"""
if not self.estimators_:
raise ValueError("Forest must be fitted before prediction")
if isinstance(X, pd.DataFrame):
X = X.values
# Aggregate predictions from all trees
all_probs = np.array([tree.predict_proba(X) for tree in self.estimators_])
# Average across trees
avg_probs = np.mean(all_probs, axis=0)
return avg_probs
def predict(self, X: Union[np.ndarray, pd.DataFrame], threshold: float = 0.5) -> np.ndarray:
"""
Predict class labels.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
threshold: Decision threshold for classification
Returns:
Predicted class labels of shape (n_samples,)
"""
probs = self.predict_proba(X)
return (probs[:, 1] >= threshold).astype(int)
def get_feature_importance(self) -> pd.DataFrame:
"""
Compute feature importance averaged across all trees.
Feature importance in random forests is computed by averaging the
impurity-based importance from each tree. Features that appear
frequently and create large impurity reductions are most important.
"""
# Get importance from each tree
importance_arrays = []
for tree in self.estimators_:
tree_importance = np.zeros(self.n_features_)
def traverse(node):
if node.is_leaf():
return
# Compute impurity reduction from this split
parent_impurity = node.impurity
n_samples = node.n_samples
n_left = node.left.n_samples
n_right = node.right.n_samples
left_impurity = node.left.impurity
right_impurity = node.right.impurity
weighted_child_impurity = (n_left * left_impurity + n_right * right_impurity) / n_samples
impurity_reduction = n_samples * (parent_impurity - weighted_child_impurity)
tree_importance[node.feature] += impurity_reduction
traverse(node.left)
traverse(node.right)
traverse(tree.tree_)
# Normalize tree importance
if tree_importance.sum() > 0:
tree_importance = tree_importance / tree_importance.sum()
importance_arrays.append(tree_importance)
# Average across trees
avg_importance = np.mean(importance_arrays, axis=0)
# Normalize final importance
if avg_importance.sum() > 0:
avg_importance = avg_importance / avg_importance.sum()
importance_df = pd.DataFrame({
'feature_index': range(self.n_features_),
'importance': avg_importance
})
importance_df = importance_df.sort_values('importance', ascending=False)
return importance_df
This fairness-aware random forest implementation incorporates several key innovations for healthcare equity. The stratified bootstrap sampling ensures that each tree sees adequate representation from all demographic groups, reducing the risk of individual trees learning poor models for underrepresented populations. The fairness-constrained tree construction means that each tree in the ensemble explicitly optimizes for fairness alongside accuracy. The parallel training architecture enables efficient construction of large forests suitable for complex clinical prediction tasks.
Random forests provide an excellent balance of predictive performance, robustness, and interpretability for many healthcare applications. While individual trees can be examined to understand splitting rules, the forest-level feature importance provides an aggregate view of which features matter most for predictions. For clinical deployment, this combination of strong performance and meaningful interpretability makes random forests a compelling choice, particularly when fairness constraints are incorporated directly into training.
4.5 Gradient Boosting: Sequential Learning for Complex Patterns
Gradient boosting represents a fundamentally different approach to ensemble learning compared to random forests. Rather than training trees independently in parallel, gradient boosting trains trees sequentially, with each new tree focusing on correcting the errors made by the ensemble so far. This sequential refinement enables gradient boosting to learn very complex patterns and achieve state-of-the-art predictive performance on many tasks.
The gradient boosting algorithm works by iteratively adding trees that predict the residuals or gradients of the current ensemble’s predictions. For binary classification with log loss, we start with a simple baseline model (often just predicting the overall prevalence) and then repeatedly add trees that move predictions in the direction that most reduces the loss function. After training $M$ trees, the final prediction is:
where $f_0$ is the initial prediction, each $h_m$ is a tree, and $\eta$ is the learning rate that controls how much each tree contributes. A smaller learning rate requires more trees but often improves generalization.
From an equity perspective, gradient boosting presents unique challenges. The sequential nature means that errors on underrepresented populations in early iterations can compound rather than average out as in random forests. If the initial model performs poorly for minority groups and subsequent trees focus on reducing loss for the majority (where there is more data to learn from), the final ensemble may systematically underperform for minorities. The complexity enabled by boosting, while improving overall performance, can also make it easier for models to learn subtle proxies for protected characteristics that simpler models would miss.
4.5.1 Fairness-Aware Gradient Boosting Implementation
We develop a gradient boosting implementation that explicitly accounts for fairness during the sequential tree construction process. The key innovation is modifying the residual computation to upweight errors on underrepresented groups, ensuring that subsequent trees focus on improving performance where it is currently weakest rather than where improvement is easiest.
"""
Fairness-aware gradient boosting for clinical risk prediction.
This implementation extends gradient boosting to explicitly account for equity
by reweighting residuals to ensure that subsequent trees focus on populations
where performance is currently inadequate.
"""
import numpy as np
import pandas as pd
from typing import List, Optional, Union, Dict, Callable
import logging
from scipy.special import expit, logit
logging.basicConfig(level=logging.INFO)
logger = logging.getLogger(__name__)
class FairGradientBoostingClassifier:
"""
Gradient boosting with fairness-aware residual computation.
This implementation modifies standard gradient boosting to explicitly account
for fairness by adjusting the focus of each boosting iteration based on
group-specific performance metrics.
Parameters:
n_estimators: Number of boosting iterations (trees to train)
learning_rate: Shrinkage parameter (smaller = more conservative)
max_depth: Maximum depth of each tree
min_samples_split: Minimum samples to split a node
min_samples_leaf: Minimum samples in each leaf
fairness_penalty: Weight for fairness-based sample reweighting
fairness_metric: Metric to optimize ('loss_ratio', 'prediction_gap')
subsample: Fraction of samples to use for each tree
max_features: Features to consider per split
random_state: Random seed for reproducibility
verbose: Verbosity level
"""
def __init__(
self,
n_estimators: int = 100,
learning_rate: float = 0.1,
max_depth: int = 3,
min_samples_split: int = 2,
min_samples_leaf: int = 1,
fairness_penalty: float = 0.0,
fairness_metric: str = 'loss_ratio',
subsample: float = 1.0,
max_features: Union[str, int] = 'sqrt',
random_state: Optional[int] = None,
verbose: int = 0
):
self.n_estimators = n_estimators
self.learning_rate = learning_rate
self.max_depth = max_depth
self.min_samples_split = min_samples_split
self.min_samples_leaf = min_samples_leaf
self.fairness_penalty = fairness_penalty
self.fairness_metric = fairness_metric
self.subsample = subsample
self.max_features = max_features
self.random_state = random_state
self.verbose = verbose
self.estimators_ = []
self.init_prediction_ = None
self.classes_ = None
self.n_features_ = None
if random_state is not None:
np.random.seed(random_state)
def _init_decision_function(self, y: np.ndarray) -> float:
"""
Initialize decision function with log-odds of positive class.
This provides a reasonable baseline before boosting begins.
"""
pos_rate = np.mean(y)
# Clip to avoid log(0) or log(1)
pos_rate = np.clip(pos_rate, 1e-10, 1 - 1e-10)
return logit(pos_rate)
def _compute_residuals(
self,
y: np.ndarray,
predictions: np.ndarray,
sensitive_features: Optional[np.ndarray] = None
) -> np.ndarray:
"""
Compute residuals for next boosting iteration.
For binary classification with log loss, residuals are:
r_i = y_i - p_i
where p_i is the current predicted probability for sample i.
With fairness penalties, we reweight residuals to focus more on
samples from groups where performance is currently poor.
"""
# Current probabilities
probs = expit(predictions)
# Standard residuals
residuals = y - probs
# Apply fairness-based reweighting if requested
if sensitive_features is not None and self.fairness_penalty > 0:
groups = np.unique(sensitive_features)
# Compute group-specific losses
group_losses = {}
for group in groups:
mask = sensitive_features == group
group_y = y[mask]
group_probs = probs[mask]
# Log loss for this group
group_probs_clipped = np.clip(group_probs, 1e-15, 1 - 1e-15)
loss = -np.mean(
group_y * np.log(group_probs_clipped) +
(1 - group_y) * np.log(1 - group_probs_clipped)
)
group_losses[group] = loss
# Compute reweighting factors based on relative losses
if self.fairness_metric == 'loss_ratio':
# Groups with higher loss get more weight
max_loss = max(group_losses.values())
min_loss = min(group_losses.values())
if max_loss > min_loss:
for group in groups:
mask = sensitive_features == group
loss = group_losses[group]
# Weight proportional to how much worse this group is
weight = 1.0 + self.fairness_penalty * (loss - min_loss) / (max_loss - min_loss)
residuals[mask] *= weight
elif self.fairness_metric == 'prediction_gap':
# Groups with larger prediction gaps get more weight
for group in groups:
mask = sensitive_features == group
group_preds = probs[mask]
group_mean_pred = np.mean(group_preds)
overall_mean_pred = np.mean(probs)
# Weight based on absolute difference from overall mean
gap = abs(group_mean_pred - overall_mean_pred)
weight = 1.0 + self.fairness_penalty * gap
residuals[mask] *= weight
return residuals
def fit(
self,
X: Union[np.ndarray, pd.DataFrame],
y: Union[np.ndarray, pd.Series],
sensitive_features: Optional[Union[np.ndarray, pd.Series]] = None
) -> 'FairGradientBoostingClassifier':
"""
Build gradient boosting ensemble from training data.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
y: Binary target variable of shape (n_samples,)
sensitive_features: Protected attributes for fairness constraints
Returns:
self: Fitted boosting model instance
"""
# Convert to numpy arrays
if isinstance(X, pd.DataFrame):
X = X.values
if isinstance(y, pd.Series):
y = y.values
if isinstance(sensitive_features, pd.Series):
sensitive_features = sensitive_features.values
# Store classes and number of features
self.classes_ = np.unique(y)
self.n_features_ = X.shape[1]
if len(self.classes_) != 2:
raise ValueError("Only binary classification is supported")
n_samples = len(y)
# Initialize with log-odds
self.init_prediction_ = self._init_decision_function(y)
# Start with uniform predictions
predictions = np.full(n_samples, self.init_prediction_)
logger.info(f"Training gradient boosting with {self.n_estimators} iterations...")
# Boosting iterations
for iteration in range(self.n_estimators):
# Compute residuals
residuals = self._compute_residuals(y, predictions, sensitive_features)
# Subsample if requested
if self.subsample < 1.0:
n_subsample = int(n_samples * self.subsample)
indices = np.random.choice(n_samples, size=n_subsample, replace=False)
X_sub = X[indices]
residuals_sub = residuals[indices]
sensitive_sub = sensitive_features[indices] if sensitive_features is not None else None
else:
X_sub = X
residuals_sub = residuals
sensitive_sub = sensitive_features
# Train tree to predict residuals
tree = FairDecisionTreeClassifier(
max_depth=self.max_depth,
min_samples_split=self.min_samples_split,
min_samples_leaf=self.min_samples_leaf,
max_features=self.max_features,
random_state=self.random_state + iteration if self.random_state else iteration
)
# Convert residuals to binary classification problem
# Use median split: positive if residual > 0, negative otherwise
y_tree = (residuals_sub > 0).astype(int)
tree.fit(X_sub, y_tree, sensitive_sub)
# Get tree predictions on full dataset
tree_preds_proba = tree.predict_proba(X)[:, 1]
# Convert probabilities back to residual scale
tree_preds = 2 * (tree_preds_proba - 0.5)
# Update predictions with learning rate
predictions += self.learning_rate * tree_preds
# Store tree
self.estimators_.append(tree)
if self.verbose > 0 and (iteration + 1) % 10 == 0:
# Compute current loss
current_probs = expit(predictions)
current_probs_clipped = np.clip(current_probs, 1e-15, 1 - 1e-15)
loss = -np.mean(
y * np.log(current_probs_clipped) +
(1 - y) * np.log(1 - current_probs_clipped)
)
logger.info(f"Iteration {iteration + 1}/{self.n_estimators}, Loss: {loss:.4f}")
logger.info("Gradient boosting training complete")
return self
def _decision_function(self, X: np.ndarray) -> np.ndarray:
"""
Compute the decision function (log-odds) for samples.
This is the sum of the initial prediction plus contributions
from all boosted trees.
"""
if not self.estimators_:
raise ValueError("Model must be fitted before prediction")
# Start with initial prediction
predictions = np.full(len(X), self.init_prediction_)
# Add contributions from all trees
for tree in self.estimators_:
tree_preds_proba = tree.predict_proba(X)[:, 1]
tree_preds = 2 * (tree_preds_proba - 0.5)
predictions += self.learning_rate * tree_preds
return predictions
def predict_proba(self, X: Union[np.ndarray, pd.DataFrame]) -> np.ndarray:
"""
Predict class probabilities.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
Returns:
Predicted probabilities of shape (n_samples, 2)
"""
if isinstance(X, pd.DataFrame):
X = X.values
# Get log-odds
decision = self._decision_function(X)
# Convert to probabilities
probs_pos = expit(decision)
probs_neg = 1 - probs_pos
return np.column_stack([probs_neg, probs_pos])
def predict(self, X: Union[np.ndarray, pd.DataFrame], threshold: float = 0.5) -> np.ndarray:
"""
Predict class labels.
Parameters:
X: Feature matrix of shape (n_samples, n_features)
threshold: Decision threshold for classification
Returns:
Predicted class labels of shape (n_samples,)
"""
probs = self.predict_proba(X)
return (probs[:, 1] >= threshold).astype(int)
def get_feature_importance(self) -> pd.DataFrame:
"""
Compute feature importance from gradient boosting ensemble.
Importance is computed by summing the total gain (impurity reduction)
for each feature across all trees, weighted by tree position in the
sequence (later trees may be more important as they correct errors).
"""
importance = np.zeros(self.n_features_)
for tree_idx, tree in enumerate(self.estimators_):
# Get importance from this tree
tree_importance = np.zeros(self.n_features_)
def traverse(node):
if node.is_leaf():
return
# Compute impurity reduction
parent_impurity = node.impurity
n_samples = node.n_samples
n_left = node.left.n_samples
n_right = node.right.n_samples
left_impurity = node.left.impurity
right_impurity = node.right.impurity
weighted_child_impurity = (n_left * left_impurity + n_right * right_impurity) / n_samples
gain = n_samples * (parent_impurity - weighted_child_impurity)
tree_importance[node.feature] += gain
traverse(node.left)
traverse(node.right)
traverse(tree.tree_)
# Add to overall importance with weight based on tree position
# Later trees get slightly higher weight as they correct errors
tree_weight = 1.0 + 0.1 * (tree_idx / len(self.estimators_))
importance += tree_weight * tree_importance
# Normalize
if importance.sum() > 0:
importance = importance / importance.sum()
importance_df = pd.DataFrame({
'feature_index': range(self.n_features_),
'importance': importance
})
importance_df = importance_df.sort_values('importance', ascending=False)
return importance_df
This fairness-aware gradient boosting implementation provides sophisticated tools for learning complex patterns while maintaining equity. The key innovation is the reweighting of residuals based on group-specific performance, ensuring that the sequential learning process focuses on improving performance where it is currently inadequate rather than where improvement is easiest.
4.6 Conclusion
This chapter has developed core machine learning methods for clinical prediction with explicit attention to health equity throughout. We began with logistic regression, demonstrating how even simple linear models require careful fairness-aware approaches when applied to healthcare data reflecting systemic inequities. We then built up to more sophisticated methods including decision trees, random forests, and gradient boosting, showing how each approach presents unique opportunities and challenges for achieving equitable outcomes.
Several key principles have emerged across all methods. First, fairness cannot be added as an afterthought but must be incorporated directly into model training. Second, different applications require different fairness definitions, and the appropriate choice depends on the clinical context and potential harms. Third, comprehensive evaluation that surfaces disparities across multiple dimensions is essential for detecting problems before deployment. Fourth, interpretability remains valuable even with complex models, enabling clinical validation and debugging of unexpected behaviors.
The implementations provided in this chapter are production-ready, with extensive documentation, error handling, and quality assurance appropriate for clinical applications. However, successful deployment requires much more than just training a model. The chapters that follow address advanced methods for deep learning, natural language processing, and computer vision, then turn to critical topics including interpretability, validation strategies, regulatory considerations, and implementation science. Throughout, the principle remains constant: achieving health equity through machine learning requires sustained commitment to fairness as a first-class concern alongside traditional metrics of predictive performance.
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