def welch_t(n1, n2):
"""
Welch unequal variance two-sample t test, as proposed in:
B.L. Welch. 1947., The generalization of "Student"'s problem when
several different population variances are involved. Biometrika
34(1-2): 28-35.
# "Student"'s famous sleep data
>>> group1 = [0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0]
>>> group2 = [1.9, 0.8, 1.1, 0.1, -0.1, 4.4, 5.5, 1.6, 4.6, 3.4]
>>> t_results = welch_t(group1, group2)
>>> round(t_results['df.t'], 4)
17.7765
>>> round(t_results['t'], 4)
-1.8608
>>> round(t_results['p.t'], 4)
0.0794
"""
retval = {}
sigma1 = sample_variance(n1)
sigma2 = sample_variance(n2)
scaled = sigma1 / len(n1) + sigma2 / len(n2)
t = (mean(n1) - mean(n2)) / sqrt(scaled)
df = (scaled ** 2) / ((sigma1 ** 2 / ((len(n1) - 1) * len(n1) ** 2)) +
(sigma2 ** 2 / ((len(n2) - 1) * len(n2) ** 2)))
p = p_t(t, df)
if p > .5:
p = 1. - p
p *= 2.
return {'t': t, 'p.t': p, 'df.t': df}
def cohen_d(n1, n2):
"""
Compute Cohen's $d$ for two independent samples n1, n2, defined as:
d = \frac{\=x1 - \=x2}{s}
where $s$, the pooled standard deviation, is
s = \sqrt{\frac{\sum_{i=1}^{n} (x_{1,i} - \=x1) ^ 2 +
\sum_{i=1}^{n} (x_{2,i} - \=x2) ^ 2}{n1 + n2}}
This latter definition comes from:
J. Hartung, G. Knapp, & B.K. Sinha. 2008. Statistical meta-analysis
with application. Hoboken, NJ: Wiley. (p. 14)
>>> from csv import DictReader
>>> from collections import defaultdict
>>> species2petal_width = defaultdict(list)
>>> for row in DictReader(open('iris.csv', 'r')):
... species = row['Species']
... width = row['Petal.Width']
... species2petal_width[species].append(float(width))
>>> round(cohen_d(*species2petal_width.values()), 3)
2.955
"""
mu1 = mean(n1)
mu2 = mean(n2)
diff = abs(mu1 - mu2)
size = len(n1) + len(n2)
return diff / sqrt((sse(n1, mu1) + sse(n2, mu2)) / size)
def chisquare(n):
"""
Compute one-way chi-square statistic
"""
mu = mean(n)
chisq = sse(n, mu) / mu
return chisq
def spearman_rho_tr(m, n):
"""
rho for tied ranks, checked by comparison with Pycluster
>>> x = [2, 8, 5, 4, 2, 6, 1, 4, 5, 7, 4]
>>> y = [3, 9, 4, 3, 1, 7, 2, 5, 6, 8, 3]
>>> print round(spearman_rho_tr(x, y), 3)
0.935
"""
assert len(m) == len(n), 'args must be the same length'
m = rank(m)
n = rank(n)
num = 0.
den_m = 0.
den_n = 0.
m_mean = mean(m)
n_mean = mean(n)
for (i, j) in zip(m, n):
i = i - m_mean
j = j - n_mean
num += i * j
den_m += i ** 2
den_n += j ** 2
return num / sqrt(den_m * den_n)
def glass_Delta(n1, n2):
"""
Compute Glass's $\Delta$, a variant on Cohen's $d$ for two independent
samples; the denominator is replaced with standard deviation for the
control group, which is the first sample here.
This definition comes from:
L.V. Hedges & I. Olkin. 1985. Staistical methods for meta-analysis.
Orlando: Academic Press. (p. 78)
>>> from csv import DictReader
>>> from collections import defaultdict
>>> species2petal_width = defaultdict(list)
>>> for row in DictReader(open('iris.csv', 'r')):
... species = row['Species']
... width = row['Petal.Width']
... species2petal_width[species].append(float(width))
>>> round(glass_Delta(*species2petal_width.values()), 3)
2.575
"""
mu1 = mean(n1)
mu2 = mean(n2)
return abs(mu1 - mu2) / sqrt(sse(n1, mu1) / len(n1))
def moment(n, moment=1):
if moment == 1:
return 0.
mu = mean(n)
return sum((x - mu) ** moment for x in n) / len(n)