Matched-pairs experiments
Matched-pairs experiments replace unrestricted treatment permutations with sign flips inside pairs. The Darwin maize data in the source folder make that logic transparent.
from pathlib import Path
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
np.set_printoptions(precision=4, suppress=True)
def repo_root():
for candidate in [Path.cwd().resolve(), *Path.cwd().resolve().parents]:
if (candidate / "ding_w_source").exists():
return candidate
raise FileNotFoundError("could not locate ding_w_source from the current working directory")
data_dir = repo_root() / "ding_w_source"zea = pd.read_csv(data_dir / "ZeaMays.csv")
diff = zea["diff"].to_numpy(dtype=float)
n_pairs = diff.size
tau_hat = diff.mean()
se_hat = diff.std(ddof=1) / np.sqrt(n_pairs)
sign_patterns = np.array(
np.meshgrid(*([np.array([-1.0, 1.0])] * n_pairs), indexing="ij")
).reshape(n_pairs, -1).T
randomization_dist = sign_patterns @ np.abs(diff) / n_pairs
pvalue = np.mean(np.abs(randomization_dist) >= abs(tau_hat))
pd.DataFrame(
{
"estimate": [tau_hat],
"se": [se_hat],
"exact_randomization_pvalue": [pvalue],
}
)| estimate | se | exact_randomization_pvalue | |
|---|---|---|---|
| 0 | 2.616667 | 1.218195 | 0.052673 |
fig, ax = plt.subplots(figsize=(6, 4))
ax.hist(randomization_dist, bins=45, alpha=0.75)
ax.axvline(tau_hat, color="black", linestyle="--", linewidth=2.0)
ax.set_xlabel("Matched-pairs average effect under sign flips")
ax.set_ylabel("Count")
ax.set_title("Darwin maize data: exact matched-pairs randomization distribution")
fig.tight_layout()
Once the pairs are fixed, the experiment is no longer a general permutation problem. The admissible treatment reallocations are just the within-pair sign changes, and the mean pair difference is the natural estimator.