First Course Ding: Chapter 2

Potential outcomes and treatment selection

Chapter 2 is the basic potential-outcomes setup: treatment effects are individual-level objects, while observed outcomes depend on both the potential outcomes and the treatment rule.

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

np.set_printoptions(precision=4, suppress=True)

1 Build A Population With Heterogeneous Effects

rng = np.random.default_rng(2)
n = 1000
y0 = rng.normal(loc=0.0, scale=1.0, size=n)
tau = -0.4 + 0.9 * y0
y1 = y0 + tau

ate = tau.mean()
att_if_treat_positive = tau[tau >= 0.0].mean()

pd.DataFrame(
    {
        "estimand": ["ATE", "Share with positive treatment effect", "ATT if only gainers are treated"],
        "value": [ate, np.mean(tau >= 0.0), att_if_treat_positive],
    }
)

	estimand	value
0	ATE	-0.420168
1	Share with positive treatment effect	0.311000
2	ATT if only gainers are treated	0.623081

fig, ax = plt.subplots(figsize=(6, 4))
ax.hist(tau, bins=40, color="tab:blue", alpha=0.75)
ax.axvline(0.0, color="black", linestyle="--")
ax.set_xlabel("Individual treatment effect")
ax.set_ylabel("Count")
ax.set_title("Treatment effects are heterogeneous")
fig.tight_layout()

2 A Perfect Doctor And A Coin-Flip Doctor

The perfect doctor treats exactly when the individual treatment effect is nonnegative. The coin-flip doctor ignores the potential outcomes entirely.

z_perfect = (tau >= 0.0).astype(float)
z_coin = rng.binomial(1, 0.5, size=n).astype(float)

y_perfect = z_perfect * y1 + (1.0 - z_perfect) * y0
y_coin = z_coin * y1 + (1.0 - z_coin) * y0

def observed_difference(y_obs, z_obs):
    treated = z_obs == 1.0
    control = ~treated
    return y_obs[treated].mean() - y_obs[control].mean()

selection_table = pd.DataFrame(
    {
        "rule": ["Perfect doctor", "Coin-flip doctor"],
        "observed_difference": [
            observed_difference(y_perfect, z_perfect),
            observed_difference(y_coin, z_coin),
        ],
        "treated_share": [z_perfect.mean(), z_coin.mean()],
        "average_effect_among_treated": [
            tau[z_perfect == 1.0].mean(),
            tau[z_coin == 1.0].mean(),
        ],
    }
)
selection_table

	rule	observed_difference	treated_share	average_effect_among_treated
0	Perfect doctor	2.305471	0.311	0.623081
1	Coin-flip doctor	-0.356470	0.503	-0.400482

3 Selection Bias Is Not The Same As The Effect

Under coin-flip treatment, the observed difference is estimating the ATE. Under targeted treatment, the observed difference mixes causal effects with positive selection on untreated potential outcomes.

selection_bias = y0[z_perfect == 1.0].mean() - y0[z_perfect == 0.0].mean()

pd.DataFrame(
    {
        "quantity": [
            "ATE",
            "Observed difference under perfect doctor",
            "Average effect among treated under perfect doctor",
            "Selection bias in untreated outcomes",
        ],
        "value": [
            ate,
            observed_difference(y_perfect, z_perfect),
            tau[z_perfect == 1.0].mean(),
            selection_bias,
        ],
    }
)

	quantity	value
0	ATE	-0.420168
1	Observed difference under perfect doctor	2.305471
2	Average effect among treated under perfect doctor	0.623081
3	Selection bias in untreated outcomes	1.682389

4 Takeaway

Potential outcomes make it obvious why the same observed contrast can have different meanings under different assignment rules. Chapter 2 is mostly estimand bookkeeping, but it is the bookkeeping that the rest of the Ding material depends on.