Bias and fairness

import numpy as np
import pandas as pd

rng = np.random.default_rng(7)
n = 20000
group = rng.choice(["A", "B"], n, p=[0.7, 0.3])
need = rng.gamma(2.0, 1.0, n)                   # true health need (same distribution in both groups)
access = np.where(group == "A", 1.0, 0.6)       # group B spends less for the same need
spending = need * access * rng.lognormal(0, 0.3, n)
df = pd.DataFrame({"group": group, "need": need, "spending": spending})

selected = df.spending >= df.spending.quantile(0.90)
high_need = df.need >= df.need.quantile(0.90)
summary = pd.DataFrame({
    "share of population": df.group.value_counts(normalize=True),
    "share of high-need patients": df[high_need].group.value_counts(normalize=True),
    "share of patients selected": df[selected].group.value_counts(normalize=True),
}).round(2)
print(summary.to_string())
print(f"\nHigh-need patients missed: group A {1 - selected[high_need & (df.group == 'A')].mean():.0%}, "
      f"group B {1 - selected[high_need & (df.group == 'B')].mean():.0%}")

#        share of population  share of high-need patients  share of patients selected
# group
# A                      0.7                         0.68                        0.91
# B                      0.3                         0.32                        0.09
#
# High-need patients missed: group A 24%, group B 75%

import numpy as np
import pandas as pd
from sklearn.linear_model import LogisticRegression

rng = np.random.default_rng(8)
n = 20000
group = rng.choice(["A", "B"], n, p=[0.8, 0.2])
x = rng.normal(size=(n, 3))
baseline = np.where(group == "A", -2.0, -1.0)          # outcome more common in group B
y = rng.binomial(1, 1 / (1 + np.exp(-(baseline + x @ np.array([1.0, 0.7, 0.4])))))

def subgroup_table(p, threshold=0.2):
    flag = p >= threshold
    rows = []
    for g in ["A", "B"]:
        m = group == g
        tp, fn = np.sum(flag & (y == 1) & m), np.sum(~flag & (y == 1) & m)
        fp = np.sum(flag & (y == 0) & m)
        rows.append({"group": g, "outcome rate": y[m].mean(), "mean predicted": p[m].mean(),
                     "sensitivity": tp / (tp + fn), "PPV": tp / (tp + fp)})
    return pd.DataFrame(rows).round(2).to_string(index=False)

p = LogisticRegression().fit(x, y).predict_proba(x)[:, 1]   # group not used
print(subgroup_table(p))

# group  outcome rate  mean predicted  sensitivity  PPV
#     A          0.18             0.2         0.75 0.34
#     B          0.31             0.2         0.67 0.55

X_with_group = np.column_stack([x, group == "B"])
p2 = LogisticRegression().fit(X_with_group, y).predict_proba(X_with_group)[:, 1]
print(subgroup_table(p2))

# group  outcome rate  mean predicted  sensitivity  PPV
#     A          0.18            0.18         0.67 0.37
#     B          0.31            0.31         0.87 0.45