exact-tests

This package implements the binomial, Hardy-Weinberg equilibrium, and Fisher's (2x2 and 2x3 so far) exact tests, along with corresponding discrete distributions (binomial, hypergeometric). scipy- and R-style interfaces are provided. cdf-like functions offer a fast 'approx' mode.

As of this writing, these functions (even in 'approx' mode) are more accurate, and often simultaneously more efficient, than their scipy counterparts: see the MPFR-comparison and benchmark scripts under utils/ . E.g.

$ utils/pbinom_accuracy.py
n in [2^5, 2^6): errRMS=0  approxErrRMS=7.31e-17  scipyErrRMS=0
n in [2^20, 2^21): errRMS=0  approxErrRMS=0  scipyErrRMS=2.18e-15
n in [2^35, 2^36): errRMS=0  approxErrRMS=0  scipyErrRMS=1.15e-14
$ utils/pbinom_accuracy.py -p 0.1  # scipy accuracy degrades more quickly with most values of p != 0.5, or k further from np
n in [2^5, 2^6): errRMS=0  approxErrRMS=1.06e-16  scipyErrRMS=4.22e-16
n in [2^20, 2^21): errRMS=0  approxErrRMS=0  scipyErrRMS=1.23e-11
n in [2^35, 2^36): errRMS=0  approxErrRMS=0  scipyErrRMS=3.47e-07
$ utils/pbinom_accuracy.py --z-score -2
n in [2^5, 2^6): errRMS=6.51e-17  approxErrRMS=2.48e-16  scipyErrRMS=1.96e-16
n in [2^20, 2^21): errRMS=0  approxErrRMS=0  scipyErrRMS=1.01e-13
n in [2^35, 2^36): errRMS=0  approxErrRMS=4.82e-17  scipyErrRMS=1.66e-11
$ utils/pbinom_benchmark.py
n=(2^5)-1: base=7.54e-07  approx=3.12e-07  scipy=2.91e-05 sec/iter
n=(2^20)-1: base=2.94e-06  approx=1.24e-06  scipy=3.06e-05 sec/iter
n=(2^35)-1: base=1.63e-06  approx=1.12e-06  scipy=9.54e-05 sec/iter

$ utils/binomtest_accuracy.py --z-score 1
n in [2^5, 2^6): errRMS=1.44e-16  scipyErrRMS=1.96e-16
n in [2^20, 2^21): errRMS=3.03e-16  scipyErrRMS=6e-14
n in [2^35, 2^36): errRMS=7.32e-16  scipyErrRMS=7.53e-12
$ utils/binomtest_benchmark.py --z-score 1  # ~100-250x speedup, better accuracy
n=(2^5)-1: base=1.02e-06  scipy=0.00023 sec/iter
n=(2^20)-1: base=4.14e-06  scipy=0.00053 sec/iter
n=(2^35)-1: base=5.98e-06  scipy=0.000852 sec/iter

$ utils/phyper_accuracy.py --z-score -0.5  # scipy accuracy degrades quickly
n in [2^5, 2^6): errRMS=0  approxErrRMS=1.72e-16  scipyErrRMS=2.84e-16
n in [2^20, 2^21): errRMS=0  approxErrRMS=2.75e-15  scipyErrRMS=1.73e-10
n in [2^35, 2^36): errRMS=0  approxErrRMS=4.13e-14  scipyErrRMS=1.05e-05
$ utils/phyper_benchmark.py --z-score -0.5  # scipy speed is ok, except...
n=(2^5)-1: base=7.12e-07  approx=2.13e-07  scipy=3.46e-05 sec/iter
n=(2^20)-1: base=6.97e-05  approx=5.36e-06  scipy=4.34e-05 sec/iter
n=(2^35)-1: base=0.0146  approx=0.000851  scipy=0.00194 sec/iter
$ utils/phyper_benchmark.py --z-score 0.0001  # ...it blows up for large n when z approaches 0.
n=(2^5)-1: base=8.67e-07  approx=2.13e-07  scipy=3.68e-05 sec/iter
n=(2^20)-1: base=6.93e-05  approx=5.32e-06  scipy=0.000495 sec/iter
n=(2^35)-1: base=0.0149  approx=0.000908  scipy=2.7 sec/iter
$ utils/phyper_benchmark.py
n=(2^5)-1: base=1.48e-06  approx=2.33e-07  scipy=3.98e-05 sec/iter
n=(2^20)-1: base=7.82e-05  approx=5.33e-06  scipy=0.000522 sec/iter
n=(2^35)-1: base=0.0149  approx=0.00087  scipy=14.9 sec/iter

$ utils/fisher_exact_22_accuracy.py --z-score -1
n in [2^5, 2^6): errRMS=9.71e-17  scipyErrRMS=2.39e-16
n in [2^20, 2^21): errRMS=1.33e-15  scipyErrRMS=1.85e-10
n in [2^35, 2^36): errRMS=4.52e-13
$ utils/fisher_exact_22_benchmark.py --z-score -1  # ~100-600x speedup, better accuracy
n=(2^5)-1: base=4.5e-07  scipy=0.000331 sec/iter
n=(2^20)-1: base=5.97e-06  scipy=0.000986 sec/iter
n=(2^35)-1: base=0.000906 sec/iter

$ utils/odds_ratio_accuracy.py --z-score 1
n in [2^5, 2^6): estErrRMS=5.25e-16  ciLowErrRMS=7.12e-16  ciHighErrRMS=7.23e-16  scipyEstErrRMS=1.3e-15  scipyCiLowErrRMS=1.25e-14  scipyCiHighErrRMS=1.42e-15
n in [2^10, 2^11): estErrRMS=1.43e-16  ciLowErrRMS=2e-15  ciHighErrRMS=1.88e-15  scipyEstErrRMS=1.2e-14  scipyCiLowErrRMS=2.33e-14  scipyCiHighErrRMS=3.13e-14
n in [2^15, 2^16): estErrRMS=8.46e-17  ciLowErrRMS=7.79e-15  ciHighErrRMS=7.73e-15  scipyEstErrRMS=1.52e-12  scipyCiLowErrRMS=1.2e-11  scipyCiHighErrRMS=6.64e-13
n in [2^35, 2^36): estErrRMS=7.56e-17  ciLowErrRMS=8.35e-12  ciHighErrRMS=8.35e-12
$ utils/odds_ratio_benchmark.py --z-score 1  # 200x and larger speedups, better accuracy
n=(2^5)-1: est=8.75e-07  ci=8.21e-07  scipy_est=0.000456  scipy_ci=0.00254 sec/iter
n=(2^10)-1: est=7.71e-07  ci=1.62e-06  scipy_est=0.000512  scipy_ci=0.0273 sec/iter
n=(2^15)-1: est=3.03e-06  ci=6.29e-06  scipy_est=0.000702  scipy_ci=2.24 sec/iter
n=(2^35)-1: est=0.000943  ci=0.00172 sec/iter

$ utils/HWE_accuracy.py --maf 0.05 --z-score 1
n in [2^10, 2^11): errRMS=1.95e-16  snphweErrRMS=2.04e-16
n in [2^15, 2^16): errRMS=2.9e-16  snphweErrRMS=6.48e-16
n in [2^20, 2^21): errRMS=6.25e-16  snphweErrRMS=1.6e-15
$ utils/HWE_benchmark.py --maf 0.05 --z-score 1  # snphwe.snphwe() is accurate, but slow for biobank-scale cases (we're >200x as fast for n~=1m)
n=(2^10)-1: base=2.67e-07  snphwe=6.17e-07 sec/iter
n=(2^15)-1: base=4.08e-07  snphwe=1.28e-05 sec/iter
n=(2^20)-1: base=1.56e-06  snphwe=0.00038 sec/iter

The central building block is a high-precision log-factorial function utilizing the QD library (https://github.com/BL-highprecision/QD ).

Build instructions

PyPI:

python -m pip install 'pip>=20.3'
python -m pip install exact_tests

GitHub:

python -m pip install 'pip>=20.3'
python -m pip install -e 'git+https://github.com/chrchang/stats.git#egg=exact_tests&subdirectory=Python'

Or install from a cloned copy:

# clone repo
git clone https://github.com/chrchang/stats
# go to python folder
cd stats/Python
# install the package
python -m pip install -e .

You can test the package with pytest.

Example usage:

import exact_tests

exact_tests.binomtest(3, n=15, p=0.1, alternative="greater")
exact_tests.binomtest(2, 29, 0.1)
exact_tests.binomtest(2, 29, "0.1")
exact_tests.binomtest(2, 29, "0.1", midp=True)
exact_tests.binomtest(100, 10000, "0.000001", logp=True)
exact_tests.fisher_exact([[4, 0], [0, 4]], alternative="greater", midp=True)
exact_tests.fisher_exact([[10000, 20000], [30000, 40000], [50000, 60000]], logp=True)