def mwu(x, y, tail='two-sided'):
"""Mann-Whitney U Test (= Wilcoxon rank-sum test). It is the non-parametric
version of the independent T-test.
Parameters
----------
x, y : array_like
First and second set of observations. x and y must be independent.
tail : string
Specify whether to return 'one-sided' or 'two-sided' p-value.
Returns
-------
stats : pandas DataFrame
Test summary ::
'U-val' : U-value
'p-val' : p-value
'RBC' : rank-biserial correlation (effect size)
'CLES' : common language effect size
Notes
-----
mwu tests the hypothesis that data in x and y are samples from continuous
distributions with equal medians. The test assumes that x and y
are independent. This test corrects for ties and by default
uses a continuity correction.
Examples
--------
1. Compare the medians of two independent samples.
>>> import numpy as np
>>> from pingouin import mwu
>>> np.random.seed(123)
>>> x = np.random.uniform(low=0, high=1, size=20)
>>> y = np.random.uniform(low=0.2, high=1.2, size=20)
>>> print("Medians = %.2f - %.2f" % (np.median(x), np.median(y)))
>>> mwu(x, y, tail='two-sided')
U-val p-val RBC CLES
97.0 0.006 0.51 0.75
"""
from scipy.stats import mannwhitneyu
x = np.asarray(x)
y = np.asarray(y)
# Remove NA
x, y = _remove_na(x, y, paired=False)
nx = x.size
ny = y.size
# Compute test
if tail == 'one-sided':
tail = 'less' if np.median(x) < np.median(y) else 'greater'
uval, pval = mannwhitneyu(x, y, use_continuity=True, alternative=tail)
# Effect size 1: common language effect size (McGraw and Wong 1992)
c = np.array([(a, b) for a in x for b in y])
num = max((c[:, 0] < c[:, 1]).sum(), (c[:, 0] > c[:, 1]).sum())
cles = num / (nx * ny)
# Effect size 2: rank biserial correlation (Wendt 1972)
rbc = 1 - (2 * uval) / (nx * ny)
# Fill output DataFrame
stats = pd.DataFrame({}, index=['MWU'])
stats['U-val'] = uval.round(3)
stats['p-val'] = pval
stats['RBC'] = rbc
stats['CLES'] = cles
col_order = ['U-val', 'p-val', 'RBC', 'CLES']
stats = stats.reindex(columns=col_order)
return stats
def circ_corrcc(x, y, tail='two-sided'):
"""Correlation coefficient between two circular variables.
Parameters
----------
x : np.array
First circular variable (expressed in radians)
y : np.array
Second circular variable (expressed in radians)
tail : string
Specify whether to return 'one-sided' or 'two-sided' p-value.
Returns
-------
r : float
Correlation coefficient
pval : float
Uncorrected p-value
Notes
-----
Adapted from the CircStats MATLAB toolbox (Berens 2009).
Use the np.deg2rad function to convert angles from degrees to radians.
Please note that NaN are automatically removed.
Examples
--------
Compute the r and p-value of two circular variables
>>> from pingouin import circ_corrcc
>>> x = [0.785, 1.570, 3.141, 3.839, 5.934]
>>> y = [0.593, 1.291, 2.879, 3.892, 6.108]
>>> r, pval = circ_corrcc(x, y)
>>> print(r, pval)
0.942, 0.066
"""
from scipy.stats import norm
x = np.asarray(x)
y = np.asarray(y)
# Check size
if x.size != y.size:
raise ValueError('x and y must have the same length.')
# Remove NA
x, y = _remove_na(x, y, paired=True)
n = x.size
# Compute correlation coefficient
x_sin = np.sin(x - circmean(x))
y_sin = np.sin(y - circmean(y))
r = np.sum(x_sin * y_sin) / np.sqrt(np.sum(x_sin**2) * np.sum(y_sin**2))
# Compute T- and p-values
tval = np.sqrt((n * (x_sin**2).mean() *
(y_sin**2).mean()) / np.mean(x_sin**2 * y_sin**2)) * r
# Approximately distributed as a standard normal
pval = 2 * norm.sf(abs(tval))
pval = pval / 2 if tail == 'one-sided' else pval
return np.round(r, 3), pval
def wilcoxon(x, y, tail='two-sided'):
"""Wilcoxon signed-rank test. It is the non-parametric version of the
paired T-test.
Parameters
----------
x, y : array_like
First and second set of observations. x and y must be related (e.g
repeated measures).
tail : string
Specify whether to return 'one-sided' or 'two-sided' p-value.
Returns
-------
stats : pandas DataFrame
Test summary ::
'W-val' : W-value
'p-val' : p-value
'RBC' : matched pairs rank-biserial correlation (effect size)
'CLES' : common language effect size
Notes
-----
The Wilcoxon signed-rank test tests the null hypothesis that two related
paired samples come from the same distribution.
A continuity correction is applied by default.
Examples
--------
1. Wilcoxon test on two related samples.
>>> import numpy as np
>>> from pingouin import wilcoxon
>>> x = [20, 22, 19, 20, 22, 18, 24, 20]
>>> y = [38, 37, 33, 29, 14, 12, 20, 22]
>>> print("Medians = %.2f - %.2f" % (np.median(x), np.median(y)))
>>> wilcoxon(x, y, tail='two-sided')
"""
from scipy.stats import wilcoxon
x = np.asarray(x)
y = np.asarray(y)
# Remove NA
x, y = _remove_na(x, y, paired=True)
nx = x.size
ny = y.size
# Compute test
wval, pval = wilcoxon(x, y, zero_method='wilcox', correction=False)
pval *= .5 if tail == 'one-sided' else pval
# Effect size 1: common language effect size (McGraw and Wong 1992)
c = np.array([(a, b) for a in x for b in y])
num = max((c[:, 0] < c[:, 1]).sum(), (c[:, 0] > c[:, 1]).sum())
cles = num / (nx * ny)
# Effect size 2: matched-pairs rank biserial correlation (Kerby 2014)
rank = np.arange(x.size, 0, -1)
rsum = rank.sum()
fav = rank[np.sign(y - x) > 0].sum()
unfav = rank[np.sign(y - x) < 0].sum()
rbc = fav / rsum - unfav / rsum
# Fill output DataFrame
stats = pd.DataFrame({}, index=['Wilcoxon'])
stats['W-val'] = wval.round(3)
stats['p-val'] = pval
stats['RBC'] = rbc
stats['CLES'] = cles
col_order = ['W-val', 'p-val', 'RBC', 'CLES']
stats = stats.reindex(columns=col_order)
return stats
def circ_corrcl(x, y, tail='two-sided'):
"""Correlation coefficient between one circular and one linear variable
random variables.
Parameters
----------
x : np.array
First circular variable (expressed in radians)
y : np.array
Second circular variable (linear)
tail : string
Specify whether to return 'one-sided' or 'two-sided' p-value.
Returns
-------
r : float
Correlation coefficient
pval : float
Uncorrected p-value
Notes
-----
Python code borrowed from brainpipe (based on the MATLAB toolbox CircStats)
Please note that NaN are automatically removed from datasets.
Examples
--------
Compute the r and p-value between one circular and one linear variables.
>>> from pingouin import circ_corrcl
>>> x = [0.785, 1.570, 3.141, 0.839, 5.934]
>>> y = [1.593, 1.291, -0.248, -2.892, 0.102]
>>> r, pval = circ_corrcl(x, y)
>>> print(r, pval)
0.109, 0.971
"""
from scipy.stats import pearsonr, chi2
x = np.asarray(x)
y = np.asarray(y)
# Check size
if x.size != y.size:
raise ValueError('x and y must have the same length.')
# Remove NA
x, y = _remove_na(x, y, paired=True)
n = x.size
# Compute correlation coefficent for sin and cos independently
rxs = pearsonr(y, np.sin(x))[0]
rxc = pearsonr(y, np.cos(x))[0]
rcs = pearsonr(np.sin(x), np.cos(x))[0]
# Compute angular-linear correlation (equ. 27.47)
r = np.sqrt((rxc**2 + rxs**2 - 2 * rxc * rxs * rcs) / (1 - rcs**2))
# Compute p-value
pval = chi2.sf(n * r**2, 2)
pval = pval / 2 if tail == 'one-sided' else pval
return np.round(r, 3), pval