multiple-hypothesis-testing 0.1.15

Several methods of combining P-values

MultiTest — Global Tests for Multiple Hypothesis Testing

MultiTest provides several methods for combining p-values with a focus on detecting rare and weak effects.

Higher Criticism variants

Each HC variant is exposed as its own method so that the standardization choice is explicit rather than a constructor argument.

Method	Standardization	P-value range considered
MultiTest.hc	Donoho-Jin 2008 [2] – theoretical uniform std (default)	(0, γ]
MultiTest.hc_dj2004	Donoho-Jin 2004 [1] – observed p-value std	(0, γ]
MultiTest.hc_dj2008	Donoho-Jin 2008 [2] – theoretical uniform std	(0, γ]
MultiTest.hc_beta	Beta-distribution std	(0, γ]
MultiTest.hc_star	Beta-distribution std	(1/n, γ] (HCdagger [1])

Every HC method accepts:

• gamma – upper fraction of sorted p-values to consider. Only the p-values ranked in positions 1 through ⌊γ·n⌋ (i.e. the smallest γ fraction) enter the HC statistic. Defaults to 'auto' (see below).
• return_threshold – if True, returns (hc_score, threshold_pval); otherwise returns just the score (default False).

Default gamma and why it matters

When gamma='auto' the upper limit is set to

γ = log(n) / sqrt(n)

This keeps HC focused on the regime where it has the most power. Signals denser than roughly 1/√n features are detectable by simpler methods (e.g. the average z-score), so extending HC beyond γ = log(n)/√n adds noise without gaining power. The log(n) factor provides a small safety margin above the 1/√n threshold.

Other methods

• MultiTest.berkjones / berkjones_plus – Berk-Jones statistic [3]
• MultiTest.fdr – False-discovery rate functional
• MultiTest.fdr_control – Benjamini-Hochberg FDR control
• MultiTest.bonferroni / neg_log_minp – Bonferroni-style inference
• MultiTest.fisher – Fisher's method to combine p-values

In all cases, reject the null for large values of the test statistic.

Quick start

import numpy as np
from scipy.stats import norm
from multitest import MultiTest

n = 100
z = np.random.randn(n)
pvals = 2 * norm.cdf(-np.abs(z))

mt = MultiTest(pvals)

# Default HC score
hc = mt.hc()

# HC score + threshold p-value
hc, hct = mt.hc(return_threshold=True)

# Berk-Jones
bj = mt.berkjones()

ii = np.arange(n)
print(f"HC = {hc:.3f}, features below HCT: {ii[pvals <= hct]}")
print(f"Berk-Jones = {bj:.3f}")

Choosing an HC variant

• hc is the recommended starting point. It is an alias for hc_dj2008 with the default gamma setting.
• hc_dj2008 uses the expected (theoretical) uniform mean and std for normalization, matching the formulation in [2].
• hc_beta is nearly identical to hc_dj2008 for large n but uses the exact beta-distribution moments of order statistics.
• hc_dj2004 uses the observed p-value standard deviation as denominator. This makes it more sensitive to extreme p-values but also increases variance under the null.
• hc_star ignores p-values below 1/n (sample-size adjusted, HCdagger [1]).

Use cases

This package was used to obtain evaluations reported in [4] and [5].

References

[1] Donoho, David L. and Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures." The Annals of Statistics 32, no. 3 (2004): 962-994.
[2] Donoho, David L. and Jin, Jiashun. "Higher criticism thresholding: Optimal feature selection when useful features are rare and weak." Proceedings of the National Academy of Sciences, 2008.
[3] Moscovich, Amit, Boaz Nadler, and Clifford Spiegelman. "On the exact Berk-Jones statistics and their p-value calculation." Electronic Journal of Statistics 10 (2016): 2329-2354.
[4] Donoho, David L. and Alon Kipnis. "Higher criticism to compare two large frequency tables, with sensitivity to possible rare and weak differences." The Annals of Statistics 50, no. 3 (2022): 1447-1472.
[5] Kipnis, Alon. "Unification of rare/weak detection models using moderate deviations analysis and log-chisquared p-values." Statistica Sinica, 2025.