#!/usr/bin/env python
"""Provides standard statistical tests. Tests produce statistic and P-value.
"""
from __future__ import division
import warnings
from cogent.maths.stats.distribution import (chi_high, z_low, z_high, zprob,
t_high, t_low, tprob, f_high, f_low, fprob, binomial_high, binomial_low,
ndtri)
from cogent.maths.stats.special import (lgam, log_one_minus, one_minus_exp,
MACHEP)
from cogent.maths.stats import chisqprob
from cogent.maths.stats.ks import psmirnov2x, pkstwo
from cogent.maths.stats.special import Gamma
from numpy import (absolute, arctanh, array, asarray, concatenate, transpose,
ravel, take, nonzero, log, sum, mean, cov, corrcoef, fabs, any,
reshape, tanh, clip, nan, isnan, isinf, sqrt, trace, exp,
median as _median, zeros, ones, unique, copy, searchsorted, var,
argsort, hstack, arange, empty, e, where)
#, std - currently incorrect
from numpy.random import permutation, randint, shuffle
from operator import add
from random import choice
__author__ = "Rob Knight"
__copyright__ = "Copyright 2007-2012, The Cogent Project"
__credits__ = ["Gavin Huttley", "Rob Knight", "Catherine Lozupone",
"Sandra Smit", "Micah Hamady", "Daniel McDonald",
"Greg Caporaso", "Jai Ram Rideout", "Michael Dwan",
"Will Van Treuren"]
__license__ = "GPL"
__version__ = "1.5.3-dev"
__maintainer__ = "Rob Knight"
__email__ = "rob@spot.colorado.edu"
__status__ = "Production"
class IndexOrValueError(IndexError, ValueError): pass
var = cov #cov will calculate variance if called on a vector
def std_(x, axis=None):
"""Returns standard deviations by axis (similiar to numpy.std)
The result is unbiased, matching the result from MLab.std
"""
x = asarray(x)
if axis is None:
d = x - mean(x)
return sqrt(sum(d**2)/(len(x)-1))
elif axis == 0:
result = []
for col in range(x.shape[1]):
vals = x[:,col]
d = vals - mean(vals)
result.append(sqrt(sum(d**2)/(len(x)-1)))
return result
elif axis == 1:
result = []
for row in range(x.shape[0]):
vals = x[row,:]
d = vals - mean(vals)
result.append(sqrt(sum(d**2)/(len(x)-1)))
return result
else:
raise ValueError, "axis out of bounds"
# tested only by std
def var(x, axis=None):
"""Returns unbiased standard deviations over given axis.
Similar with numpy.std, except that it is unbiased. (var = SS/n-1)
x: a float ndarray or asarray(x) is a float ndarray.
axis=None: computed for the flattened array by default, or compute along an
integer axis.
Implementation Notes:
Change the SS calculation from:
SumSq(x-x_bar) to SumSq(x) - SqSum(x)/n
See p. 37 of Zar (1999) Biostatistical Analysis.
"""
x = asarray(x)
#figure out sample size along the axis
if axis is None:
n = x.size
else:
n = x.shape[axis]
#compute the sum of squares from the mean(s)
sample_SS = sum(x**2, axis) - sum(x, axis)**2/n
return sample_SS / (n-1)
def std(x, axis=None):
"""computed unbiased standard deviations along given axis or flat array.
Similar with numpy.std, except that it is unbiased. (var = SS/n-1)
x: a float ndarray or asarray(x) is a float ndarray.
axis=None: computed for the flattened array by default, or compute along an
given integer axis.
"""
try:
sample_variance = var(x, axis=axis)
except IndexError, e: #just to avoid breaking the old test code
raise IndexOrValueError(e)
return sqrt(sample_variance)
def median(m, axis=None):
"""Returns medians by axis (similiar to numpy.mean)
numpy.median does not except an axis parameter. Is safe for substition for
numpy.median
"""
median_vals = []
rows, cols = m.shape
if axis is None:
return _median(ravel(m))
elif axis == 0:
for col in range(cols):
median_vals.append(_median(m[:,col]))
elif axis == 1 or axis == -1:
for row in range(rows):
median_vals.append(_median(m[row,:]))
else:
raise ValueError, "axis(=%s) out of bounds" % axis
return array(median_vals)
class ZeroExpectedError(ValueError):
"""Class for handling tests where an expected value was zero."""
pass
def G_2_by_2(a, b, c, d, williams=1, directional=1):
"""G test for independence in a 2 x 2 table.
Usage: G, prob = G_2_by_2(a, b, c, d, willliams, directional)
Cells are in the order:
a b
c d
a, b, c, and d can be int, float, or long.
williams is a boolean stating whether to do the Williams correction.
directional is a boolean stating whether the test is 1-tailed.
Briefly, computes sum(f ln f) for cells - sum(f ln f) for
rows and columns + f ln f for the table.
Always has 1 degree of freedom
To generalize the test to r x c, use the same protocol:
2*(cells - rows/cols + table), then with (r-1)(c-1) df.
Note that G is always positive: to get a directional test,
the appropriate ratio (e.g. a/b > c/d) must be tested
as a separate procedure. Find the probability for the
observed G, and then either halve or halve and subtract from
one depending on whether the directional prediction was
upheld.
The default test is now one-tailed (Rob Knight 4/21/03).
See Sokal & Rohlf (1995), ch. 17. Specifically, see box 17.6 (p731).
"""
cells = [a, b, c, d]
n = sum(cells)
#return 0 if table was empty
if not n:
return (0, 1)
#raise error if any counts were negative
if min(cells) < 0:
raise ValueError, \
"G_2_by_2 got negative cell counts(s): must all be >= 0."
G = 0
#Add x ln x for items, adding zero for items whose counts are zero
for i in filter(None, cells):
G += i * log(i)
#Find totals for rows and cols
ab = a + b
cd = c + d
ac = a + c
bd = b + d
rows_cols = [ab, cd, ac, bd]
#exit if we are missing a row or column entirely: result counts as
#never significant
if min(rows_cols) == 0:
return (0, 1)
#Subtract x ln x for rows and cols
for i in filter(None, rows_cols):
G -= i * log(i)
#Add x ln x for table
G += n * log(n)
#Result needs to be multiplied by 2
G *= 2
#apply Williams correction
if williams:
q = 1 + (( ( (n/ab) + (n/cd) ) -1 ) * ( ( (n/ac) + (n/bd) ) -1))/(6*n)
G /= q
p = chi_high(max(G,0), 1)
#find which tail we were in if the test was directional
if directional:
is_high = ((b == 0) or (d != 0 and (a/b > c/d)))
p = tail(p, is_high)
if not is_high:
G = -G
return G, p
def safe_sum_p_log_p(a, base=None):
"""Calculates p * log(p) safely for an array that may contain zeros."""
flat = ravel(a)
nz = take(flat, nonzero(flat)[0])
logs = log(nz)
if base:
logs /= log(base)
return sum(nz * logs, 0)
def G_ind(m, williams=False):
"""Returns G test for independence in an r x c table.
Requires input data as a numpy array. From Sokal and Rohlf p 738.
"""
f_ln_f_elements = safe_sum_p_log_p(m)
f_ln_f_rows = safe_sum_p_log_p(sum(m,0))
f_ln_f_cols = safe_sum_p_log_p(sum(m,1))
tot = sum(ravel(m))
f_ln_f_table = tot * log(tot)
df = (len(m)-1) * (len(m[0])-1)
G = 2*(f_ln_f_elements-f_ln_f_rows-f_ln_f_cols+f_ln_f_table)
if williams:
q = 1+((tot*sum(1.0/sum(m,1))-1)*(tot*sum(1.0/sum(m,0))-1)/ \
(6*tot*df))
G = G/q
return G, chi_high(max(G,0), df)
## Start functions for G goodness of fit
def williams_correction(n, a, G):
"""Return the Williams corrected G statistic for G goodness of fit test.
For discussion read Sokal and Rohlf Biometry pg 698,699.
Inputs:
n - int, sum of observed frequencies
a - int, number of groups that are being compared
G - float, uncorrected G statistic
"""
# q = 1. + (a**2 - 1)/(6.*n*a - 6.*n) == 1. + (a+1.)/(6.*n)
q = 1. + (a+1.)/(6.*n)
return G/q
def G_stat(data):
"""Calculate the G statistic for data.
For discussion read Sokal and Rohlf Biometry pg. 695-699. The G tests is
normally applied to data where you have only one observation of any given
sample class (e.g. you observe 90 wildtype and 30 mutants). In microbial
ecology it is normal to have multiple samples which contain a given feature
where those samples share a metadata class (e.g. you observe OTUX at certain
frequencies in 12 samples, 6 of which are treatment samples, 6 of which are
control samples). To reconcile these approaches this function averages the
frequency of the given feature (OTU) across all samples in the metadata
class (e.g. in the 6 treatment samples, the value for OTUX is averaged, and
this forms the average frequency which represents all treatment samples in
aggregate). This means that this version of the G stat cannot detect sample
heterogeneity as a replicated goodness of fit test would be able to.
In addition, this function assumes the extrinsic hypothesis is that the
mean frequency in all the samples groups is the same.
Inputs:
data - list of arrays, each array is 1D with any length. each array
represents the observed frequencies of a given OTU in one of the sample
classes.
"""
# G = 2*sum(f_i*ln(f_i/f_i_hat)) over all i phenotypes/sample classes
# calculate the total number of observations under the consideration that
# multiple observations in a given group are averaged.
n = sum([arr.mean() for arr in data])
a = len(data) #a is number of phenotypes or sample classes
obs_freqs = array([sample_type.mean() for sample_type in data]) #f_i vals
exp_freqs = zeros(a)+(n/float(a)) #f_i_hat vals
G = 2.*(obs_freqs*log(obs_freqs/exp_freqs)).sum()
return G
def G_fit(data, williams=True):
"""Calculate G statistic and compare to one tailed chi-squared distribution.
For discussion read Sokal and Rohlf Biometry pg. 695-699. This function
compares the calculated G statistic (with Williams correction by default) to
the chi-squared distribution with the appropriate number of degrees of
freedom.
Inputs:
data - list of arrays, each array is 1D with any length. each array
represents the observed frequencies of a given OTU in one of the sample
classes.
williams - boolean, whether or not to apply williams correction before
comparing to the chi-squared dsitribution.
"""
# sanity checks on the data to return nans if conditions are not met
if not all([(i>=0).all() for i in data]):
# G_fit: data contains negative values. G test would be
# undefined. Ignoring this OTU.
return nan, nan
if not all([i.sum()>0 for i in data]):
# G_fit: data contains sample group with zero only values. This
# means that the given OTU was never observed in this sample class
# The G test fails in this case because we would be forced to
# take log(0). Ignoring this OTU.
return nan, nan
G = G_stat(data)
a = len(data) #a is number of phenotypes or sample classes
if williams:
# calculate the total number of observations under the consideration
# that multiple observations in a given group are averaged.
n = sum([arr.mean() for arr in data]) #total observations
G = williams_correction(n, a, G)
return G, chi_high(G, a-1) #a-1 degrees of freedom because of sum constraint
## End functions for G goodness of fit test
## Start functions for kruskal_wallis test
def _corr_kw(n):
"""Return n**3-n. Used for correction of Kruskal Wallis."""
return n**3 - n
def ssl_ssr_sx(x):
"""Return searchsorted right and left indices of x and sorted copy of x."""
y = copy(x)
y.sort()
ssl = searchsorted(y, x, 'left')
ssr = searchsorted(y, x, 'right')
return ssl, ssr, y
def tie_correction(sx):
"""Correct for ties in Kruskal Wallis."""
ux = unique(sx)
uxl = searchsorted(sx, ux, 'left')
uxr = searchsorted(sx, ux, 'right')
return 1.-_corr_kw(uxr-uxl).sum()/float(_corr_kw(len(sx)))
def kruskal_wallis(data):
"""Calculates corrected Kruskal Wallis statistic (Sokal and Rolhf pg. 423).
Implementation taken from Wikipedia and Sokal and Rohlf Biometry pg. 423.
H = [12/n(n+1) * sum(T_i^2/n_i)] - 3(n+1) = the Kruskal Wallis value, the
expected value of the variance of the sum of the ranks. Summation occurs
over all groups (samples)
T_i = sum of the ranks (with ties resolved by the Kruskal Wallis procedure)
of the values (or variates) in the ith group (sample).
n_i = number of values in the ith group.
n = total number of samples in all groups being compared.
D = 1 - sum(T_j^3-T_j)/(n^3-n) = correction factor for ties.
T_j = number of ties in the jth group of ties.
Inputs:
data - list of arrays, each array is 1D with any length. each array
represents the observed frequencies of a given OTU in one of the sample
classes.
Outputs:
H/D
"""
# record number of groups for comparison
num_groups = len(data)
x = hstack(data)
# calculate searchsorted right and searchsroted left indices.
ssl, ssr, sx = ssl_ssr_sx(x)
# calculate H
start = 0
stop = 0
tot = 0
for group in data:
stop += len(group)
# To average the ranks for tied entries we compute leftmost rank of
# value i, minus rightmost rank of value i (and divide by 2). Since
# python indexes to 0, ssl ranks are 1 lower than they shoud be (i.e.
# the smallest value has rank 0 instead of 1). The +1 below corrects for
# this and .5 averages.
ranks = (ssr[start:stop]+ssl[start:stop]+1)*.5
tot+=(ranks.sum()**2)/float(len(group))
start += len(group)
n = len(x)
a = 12./(n*(n+1))
b = -3.*(n+1)
H = (a*tot + b)
# correct for ties by calulating D
D = tie_correction(sx)
# Because of the way the chisqprob function in pycogent works, if it gets
# H/D = 0/0 it will fail to exit the loop and hang indefintitely.
if D == 0:
return nan, nan
else:
# give chisqprob the kw statistic, degrees of freedom = (num groups - 1)
p_value = chisqprob(H/D, num_groups-1)
return H/D, p_value
## End functions for kruskal_wallis test
## Begin functinos for Kendalls Tau
def rank_with_ties(v1):
'''Return ranked values of 1D vector v1 with averages for tied entries.'''
tmp_v1 = array(v1).astype(float)
tmp_v1.sort()
return (searchsorted(tmp_v1,v1,'left')+searchsorted(tmp_v1,v1,'right')+1)*.5
def count_occurrences(x):
"""Count occurrences of each entry in sorted(unique(x))."""
tmp_x = copy(x)
tmp_x.sort()
ux = unique(x)
return searchsorted(tmp_x, ux, 'right') - searchsorted(tmp_x, ux, 'left')
def kendall(v1, v2):
'''Compute Kendalls Tau correlation between v1 and v2.
This function calculates tie adjusted Kendalls Tau statistic according to
Wikipedia and Sokal and Rohlf Biometry pg 594-595. Sokal and Rohlf's
implementation is significantly more confusing than that from Wikipedia and
the results are the same. In short, we calculate:
tau = n_c - n_d / ((n_o - n_1)(n_o - n_2))^.5
n_c = sum concordant pairs
n_d = sum discordant pairs
n_o = n(n-1)/2 where n is the length either input vector (len(v1)=len(v2))
n_1 = sum of ti(ti-1)/2 where ti is the number of ties in the ith group of
ties for v1
n_2 = same as n_1 but for v2.
Inputs:
v1, v2 = 1D array like, vectors to be correlated
'''
v1r = rank_with_ties(array(v1))
v2r = rank_with_ties(array(v2))
# sort both vectors according to v1r's order
sort_inds = argsort(v1r)
v1rs = v1r[sort_inds]
v2rs = v2r[sort_inds]
# iterate through vectors counting concordant and discordant pairs
c, d = 0. , 0.
n = len(v1r)
for i in range(n-1):
diff = (v1rs[i+1:] - v1rs[i]) * (v2rs[i+1:] - v2rs[i])
c += (diff > 0).sum() # concordant pairs
d += (diff < 0).sum() # discordant pairs
# if product=0, one or more of the pairs was tied and should be ignored
# count ties in both vectors for correction factor calculation
t = count_occurrences(v1)
u = count_occurrences(v2)
n_o = n*(n-1)*.5
denom = ((n_o - (t*(t-1)).sum()*.5) * (n_o - (u*(u-1)).sum()*.5))**.5
tau = (c - d)/denom
return tau
def kendall_pval(tau, n):
'''Calculate the p-value for the passed tau and vector length n.'''
test_stat = tau/((2*(2*n+5))/float(9*n*(n-1)))**.5
return zprob(test_stat)
## End functions for Kendall Tau
def pearson(v1, v2):
'''Using numpy's builtin corrcoef. Faster, and well tested.'''
v1, v2 = array(v1), array(v2)
if not (v1.size==v2.size>1):
raise ValueError("One or more vectors isn't long enough"+\
" to correlate or they have unequal lengths. Can't continue.")
return corrcoef(v1,v2)[0][1] # 2x2 symmetric unit matrix
def spearman(v1, v2):
'''Calculate Spearmans rho.'''
return pearson(rank_with_ties(v1), rank_with_ties(v2))
def likelihoods(d_given_h, priors):
"""Calculate likelihoods through marginalization, given Pr(D|H) and priors.
Usage: scores = likelihoods(d_given_h, priors)
d_given_h and priors are equal-length lists of probabilities. Returns
a list of the same length of numbers (not probabilities).
"""
#check that the lists of Pr(D|H_i) and priors are equal
length = len(d_given_h)
if length != len(priors):
raise ValueError, "Lists not equal lengths."
#find weighted sum of Pr(H_i) * Pr(D|H_i)
wt_sum = 0
for d, p in zip(d_given_h, priors):
wt_sum += d * p
#divide each Pr(D|H_i) by the weighted sum and multiply by its prior
#to get its likelihood
return [d/wt_sum for d in d_given_h]
def posteriors(likelihoods, priors):
"""Calculate posterior probabilities given priors and likelihoods.
Usage: probabilities = posteriors(likelihoods, priors)
likelihoods is a list of numbers. priors is a list of probabilities.
Returns a list of probabilities (0-1).
"""
#Check that there is a prior for each likelihood
if len(likelihoods) != len(priors):
raise ValueError, "Lists not equal lengths."
#Posterior probability is defined as prior * likelihood
return [l * p for l, p in zip(likelihoods, priors)]
def bayes_updates(ds_given_h, priors = None):
"""Successively apply lists of Pr(D|H) to get Pr(H|D) by marginalization.
Usage: final_probs = bayes_updates(ds_given_h, [priors])
ds_given_h is a list (for each form of evidence) of lists of probabilities.
priors is optionally a list of the prior probabilities.
Returns a list of posterior probabilities.
"""
try:
first_list = ds_given_h[0]
length = len(first_list)
#calculate flat prior if none was passed
if not priors:
priors = [1/length] * length
#apply each form of data to the priors to get posterior probabilities
for index, d in enumerate(ds_given_h):
#first, ignore the form of data if all the d's are the same
all_the_same = True
first_element = d[0]
for i in d:
if i != first_element:
all_the_same = False
break
if not all_the_same: #probabilities won't change
if len(d) != length:
raise ValueError, "bayes_updates requires equal-length lists."
liks = likelihoods(d, priors)
pr = posteriors(liks, priors)
priors = pr
return priors #posteriors after last calculation are 'priors' for next
#return column of zeroes if anything went wrong, e.g. if the sum of one of
#the ds_given_h is zero.
except (ZeroDivisionError, FloatingPointError):
return [0] * length
def t_paired(a,b, tails=None, exp_diff=0):
"""Returns t and prob for TWO RELATED samples of scores a and b.
From Sokal and Rohlf (1995), p. 354.
Calculates the vector of differences and compares it to exp_diff
using the 1-sample t test.
Usage: t, prob = t_paired(a, b, tails, exp_diff)
t is a float; prob is a probability.
a and b should be equal-length lists of paired observations (numbers).
tails should be None (default), 'high', or 'low'.
exp_diff should be the expected difference in means (a-b); 0 by default.
"""
n = len(a)
if n != len(b):
raise ValueError, 'Unequal length lists in ttest_paired.'
try:
diffs = array(a) - array(b)
return t_one_sample(diffs, popmean=exp_diff, tails=tails)
except (ZeroDivisionError, ValueError, AttributeError, TypeError, \
FloatingPointError):
return (nan, nan)
def t_one_sample(a,popmean=0, tails=None):
"""Returns t for ONE group of scores a, given a population mean.
Usage: t, prob = t_one_sample(a, popmean, tails)
t is a float; prob is a probability.
a should support Mean, StandardDeviation, and Count.
popmean should be the expected mean; 0 by default.
tails should be None (default), 'high', or 'low'.
"""
try:
n = len(a)
t = (mean(a) - popmean)/(std(a)/sqrt(n))
except (ZeroDivisionError, ValueError, AttributeError, TypeError, \
FloatingPointError):
return nan, nan
if isnan(t) or isinf(t):
return nan, nan
prob = t_tailed_prob(t, n-1, tails)
return t, prob
def t_two_sample(a, b, tails=None, exp_diff=0, none_on_zero_variance=True):
"""Returns t, prob for two INDEPENDENT samples of scores a, and b.
From Sokal and Rohlf, p 223.
Usage: t, prob = t_two_sample(a,b, tails, exp_diff)
t is a float; prob is a probability.
a and b should be sequences of observations (numbers). Need not be equal
lengths.
tails should be None (default), 'high', or 'low'.
exp_diff should be the expected difference in means (a-b); 0 by default.
none_on_zero_variance: if True, will return (None,None) if both a and b
have zero variance (e.g. a=[1,1,1] and b=[2,2,2]). If False, the
following values will be returned:
Two-tailed test (tails=None):
a < b: (-inf,0.0)
a > b: (+inf,0.0)
One-tailed 'high':
a < b: (-inf,1.0)
a > b: (+inf,0.0)
One-tailed 'low':
a < b: (-inf,0.0)
a > b: (+inf,1.0)
If a and b both have no variance and have the same single value (e.g.
a=[1,1,1] and b=[1,1,1]), (None,None) will always be returned.
"""
if tails is not None and tails != 'high' and tails != 'low':
raise ValueError("Invalid tail type '%s'. Must be either None, "
"'high', or 'low'." % tails)
try:
#see if we need to back off to the single-observation for single-item
#groups
n1 = len(a)
if n1 < 2:
return t_one_observation(sum(a), b, tails, exp_diff,
none_on_zero_variance=none_on_zero_variance)
n2 = len(b)
if n2 < 2:
t, prob = t_one_observation(sum(b), a, reverse_tails(tails),
exp_diff, none_on_zero_variance=none_on_zero_variance)
# Negate the t-statistic because we swapped the order of the inputs
# in the t_one_observation call, as well as tails.
if t != 0:
t = -t
return (t, prob)
#otherwise, calculate things properly
x1 = mean(a)
x2 = mean(b)
var1 = var(a)
var2 = var(b)
if var1 == 0 and var2 == 0:
# Both lists do not vary.
if x1 == x2 or none_on_zero_variance:
result = (nan, nan)
else:
result = _t_test_no_variance(x1, x2, tails)
else:
# At least one list varies.
df = n1+n2-2
svar = ((n1-1)*var1 + (n2-1)*var2)/df
t = (x1-x2-exp_diff)/sqrt(svar*(1/n1 + 1/n2))
if isnan(t) or isinf(t):
result = (nan, nan)
else:
prob = t_tailed_prob(t, df, tails)
result = (t, prob)
except (ZeroDivisionError, ValueError, AttributeError, TypeError,
FloatingPointError), e:
#invalidate if the sample sizes are wrong, the values aren't numeric or
#aren't present, etc.
result = (nan, nan)
return result
def _t_test_no_variance(mean1, mean2, tails):
"""Handles case where two distributions have no variance."""
if tails is not None and tails != 'high' and tails != 'low':
raise ValueError("Invalid tail type '%s'. Must be either None, "
"'high', or 'low'." % tails)
if tails is None:
if mean1 < mean2:
result = (float('-inf'), 0.0)
else:
result = (float('inf'), 0.0)
elif tails == 'high':
if mean1 < mean2:
result = (float('-inf'), 1.0)
else:
result = (float('inf'), 0.0)
else:
if mean1 < mean2:
result = (float('-inf'), 0.0)
else:
result = (float('inf'), 1.0)
return result
def mc_t_two_sample(x_items, y_items, tails=None, permutations=999,
exp_diff=0):
"""Performs a two-sample t-test with Monte Carlo permutations.
x_items and y_items must be INDEPENDENT observations (sequences of
numbers). They do not need to be of equal length.
Returns the observed t statistic, the parametric p-value, a list of t
statistics obtained through Monte Carlo permutations, and the nonparametric
p-value obtained from the Monte Carlo permutations test.
This code is partially based on Jeremy Widmann's
qiime.make_distance_histograms.monte_carlo_group_distances code.
Arguments:
x_items - the first list of observations
y_items - the second list of observations
tails - if None (the default), a two-sided test is performed. 'high'
or 'low' for one-tailed tests
permutations - the number of permutations to use in calculating the
nonparametric p-value. Must be a number greater than or equal to 0.
If 0, the nonparametric test will not be performed. In this case,
the list of t statistics obtained from permutations will be empty,
and the nonparametric p-value will be None
exp_diff - the expected difference in means (x_items - y_items)
"""
if tails is not None and tails != 'high' and tails != 'low':
raise ValueError("Invalid tail type '%s'. Must be either None, "
"'high', or 'low'." % tails)
if permutations < 0:
raise ValueError("Invalid number of permutations: %d. Must be greater "
"than or equal to zero." % permutations)
if (len(x_items) == 1 and len(y_items) == 1) or \
(len(x_items) < 1 or len(y_items) < 1):
raise ValueError("At least one of the sequences of observations is "
"empty, or the sequences each contain only a single "
"observation. Cannot perform the t-test.")
# Perform t-test using original observations.
obs_t, param_p_val = t_two_sample(x_items, y_items, tails=tails,
exp_diff=exp_diff,
none_on_zero_variance=False)
# Only perform the Monte Carlo test if we got a sane answer back from the
# initial t-test and we have been specified permutations.
nonparam_p_val = nan
perm_t_stats = []
if permutations > 0 and not (isnan(obs_t) or isnan(param_p_val)):
# Permute observations between x_items and y_items the specified number
# of times.
perm_x_items, perm_y_items = _permute_observations(x_items, y_items,
permutations)
perm_t_stats = [t_two_sample(perm_x_items[n], perm_y_items[n],
tails=tails, exp_diff=exp_diff,
none_on_zero_variance=False)[0]
for n in range(permutations)]
# Compute nonparametric p-value based on the permuted t-test results.
if tails is None:
better = (absolute(array(perm_t_stats)) >= absolute(obs_t)).sum()
elif tails == 'low':
better = (array(perm_t_stats) <= obs_t).sum()
elif tails == 'high':
better = (array(perm_t_stats) >= obs_t).sum()
nonparam_p_val = (better+1) / (permutations+1)
return obs_t, param_p_val, perm_t_stats, nonparam_p_val
def _permute_observations(x, y, num_perms):
"""Return num_perms pairs of permuted vectors x,y."""
vals = hstack([array(x), array(y)])
lenx = len(x)
# sorting step is unnecessary for this code, but it ensure that test code
# which relies on seeding the prng works (if we dont do this then different
# observation orders in x and y for eg. the mc_t_two_sample test will fail
# to produce the same results)
vals.sort()
inds = arange(vals.size)
xs, ys = [], []
for i in range(num_perms):
shuffle(inds)
xs.append(vals[inds[:lenx]])
ys.append(vals[inds[lenx:]])
return xs, ys
def t_one_observation(x, sample, tails=None, exp_diff=0,
none_on_zero_variance=True):
"""Returns t-test for significance of single observation versus a sample.
Equation for 1-observation t (Sokal and Rohlf 1995 p 228):
t = obs - mean - exp_diff / (var * sqrt((n+1)/n))
df = n - 1
none_on_zero_variance: see t_two_sample for details. If sample has no
variance and its single value is the same as x (e.g. x=1 and
sample=[1,1,1]), (None,None) will always be returned
"""
try:
sample_mean = mean(sample)
sample_std = std(sample)
if sample_std == 0:
# The list does not vary.
if sample_mean == x or none_on_zero_variance:
result = (None, None)
else:
result = _t_test_no_variance(x, sample_mean, tails)
else:
# The list varies.
n = len(sample)
t = (x - sample_mean - exp_diff)/sample_std/sqrt((n+1)/n)
prob = t_tailed_prob(t, n-1, tails)
result = (t, prob)
except (ZeroDivisionError, ValueError, AttributeError, TypeError,
FloatingPointError):
result = (None, None)
return result
def correlation(x_items, y_items):
"""Returns Pearson correlation between x and y, and its significance.
WARNING: x_items and y_items must be same length!
This function is retained for backwards-compatibility. Please use
correlation_test() for more control over how the test is performed.
"""
return correlation_test(x_items, y_items, method='pearson', tails=None,
permutations=0)[:2]
def correlation_test(x_items, y_items, method='pearson', tails=None,
permutations=999, confidence_level=0.95):
"""Computes the correlation between two vectors and its significance.
Computes a parametric p-value by using Student's t-distribution with df=n-2
to perform the test of significance, as well as a nonparametric p-value
obtained by permuting one of the input vectors the specified number of
times given by the permutations parameter. A confidence interval is also
computed using Fisher's Z transform if the number of observations is
greater than 3. Please see Sokal and Rohlf pp. 575-580 and pg. 598-601 for
more details regarding these techniques.
Warning: the parametric p-value is unreliable when the method is spearman's
and there are less than 11 observations in each vector.
Returns the correlation coefficient (r or rho), the parametric p-value, a
list of the r or rho values obtained from permuting the input, the
nonparametric p-value, and a tuple for the confidence interval, with the
first element being the lower bound of the confidence interval and the
second element being the upper bound for the confidence interval. The
confidence interval will be (None, None) if the number of observations is
not greater than 3.
x_items and y_items must be the same length, and cannot have fewer than 2
elements each. If one or both of the input vectors do not have any
variation, r or rho will be 0.0.
Note: the parametric portion of this function is based on the correlation
function in this module.
Arguments:
x_items - the first list of observations
y_items - the second list of observations
method - 'pearson' or 'spearman'
tails - if None (the default), a two-sided test is performed. 'high'
for a one-tailed test for positive association, or 'low' for a
one-tailed test for negative association. This parameter affects
both the parametric and nonparametric tests, but the confidence
interval will always be two-sided
permutations - the number of permutations to use in the nonparametric
test. Must be a number greater than or equal to 0. If 0, the
nonparametric test will not be performed. In this case, the list of
correlation coefficients obtained from permutations will be empty,
and the nonparametric p-value will be None
confidence_level - the confidence level to use when constructing the
confidence interval. Must be between 0 and 1 (exclusive)
"""
# Perform some initial error checking.
if method == 'pearson':
corr_fn = pearson
elif method == 'spearman':
corr_fn = spearman
else:
raise ValueError("Invalid method '%s'. Must be either 'pearson' or "
"'spearman'." % method)
if tails is not None and tails != 'high' and tails != 'low':
raise ValueError("Invalid tail type '%s'. Must be either None, "
"'high', or 'low'." % tails)
if permutations < 0:
raise ValueError("Invalid number of permutations: %d. Must be greater "
"than or equal to zero." % permutations)
if confidence_level <= 0 or confidence_level >= 1:
raise ValueError("Invalid confidence level: %.4f. Must be between "
"zero and one." % confidence_level)
# Calculate the correlation coefficient.
corr_coeff = corr_fn(x_items, y_items)
# Perform the parametric test first.
x_items, y_items = array(x_items), array(y_items)
n = len(x_items)
df = n - 2
if n < 3:
parametric_p_val = 1
else:
try:
t = corr_coeff / sqrt((1 - (corr_coeff * corr_coeff)) / df)
parametric_p_val = t_tailed_prob(t, df, tails)
except (ZeroDivisionError, FloatingPointError):
# r/rho was presumably 1.
parametric_p_val = 0
# Perform the nonparametric test.
permuted_corr_coeffs = []
nonparametric_p_val = None
better = 0
for i in range(permutations):
permuted_y_items = y_items[permutation(n)]
permuted_corr_coeff = corr_fn(x_items, permuted_y_items)
permuted_corr_coeffs.append(permuted_corr_coeff)
if tails is None:
if abs(permuted_corr_coeff) >= abs(corr_coeff):
better += 1
elif tails == 'high':
if permuted_corr_coeff >= corr_coeff:
better += 1
elif tails == 'low':
if permuted_corr_coeff <= corr_coeff:
better += 1
else:
# Not strictly necessary since this was checked above, but included
# for safety in case the above check gets removed or messed up. We
# don't want to return a p-value of 0 if someone passes in a bogus
# tail type somehow.
raise ValueError("Invalid tail type '%s'. Must be either None, "
"'high', or 'low'." % tails)
if permutations > 0:
nonparametric_p_val = (better + 1) / (permutations + 1)
# Compute the confidence interval for corr_coeff using Fisher's Z
# transform.
z_crit = abs(ndtri((1 - confidence_level) / 2))
ci_low, ci_high = None, None
if n > 3:
try:
ci_low = tanh(arctanh(corr_coeff) - (z_crit / sqrt(n - 3)))
ci_high = tanh(arctanh(corr_coeff) + (z_crit / sqrt(n - 3)))
except (ZeroDivisionError, FloatingPointError):
# r/rho was presumably 1 or -1. Match what R does in this case.
ci_low, ci_high = corr_coeff, corr_coeff
return (corr_coeff, parametric_p_val, permuted_corr_coeffs,
nonparametric_p_val, (ci_low, ci_high))
def correlation_matrix(series, as_rows=True):
"""Returns pairwise correlations between each pair of series.
"""
return corrcoef(series, rowvar=as_rows)
#unused codes below
if as_rows:
return corrcoef(transpose(array(series)))
else:
return corrcoef(array(series))
def regress(x, y):
"""Returns coefficients to the regression line "y=ax+b" from x[] and y[].
Specifically, returns (slope, intercept) as a tuple from the regression of
y on x, minimizing the error in y assuming that x is precisely known.
Basically, it solves
Sxx a + Sx b = Sxy
Sx a + N b = Sy
where Sxy = \sum_i x_i y_i, Sx = \sum_i x_i, and Sy = \sum_i y_i. The
solution is
a = (Sxy N - Sy Sx)/det
b = (Sxx Sy - Sx Sxy)/det
where det = Sxx N - Sx^2. In addition,
Var|a| = s^2 |Sxx Sx|^-1 = s^2 | N -Sx| / det
|b| |Sx N | |-Sx Sxx|
s^2 = {\sum_i (y_i - \hat{y_i})^2 \over N-2}
= {\sum_i (y_i - ax_i - b)^2 \over N-2}
= residual / (N-2)
R^2 = 1 - {\sum_i (y_i - \hat{y_i})^2 \over \sum_i (y_i - \mean{y})^2}
= 1 - residual/meanerror
Adapted from the following URL:
http://www.python.org/topics/scicomp/recipes_in_python.html
"""
x, y = array(x,'Float64'), array(y,'Float64')
N = len(x)
Sx = sum(x)
Sy = sum(y)
Sxx = sum(x*x)
Syy = sum(y*y)
Sxy = sum(x*y)
det = Sxx * N - Sx * Sx
return (Sxy * N - Sy * Sx)/det, (Sxx * Sy - Sx * Sxy)/det
def regress_origin(x,y):
"""Returns coefficients to regression "y=ax+b" passing through origin.
Requires vectors x and y of same length.
See p. 351 of Zar (1999) Biostatistical Analysis.
returns slope, intercept as a tuple.
"""
x, y = array(x,'Float64'), array(y,'Float64')
return sum(x*y)/sum(x*x), 0
def regress_R2(x, y):
"""Returns the R^2 value for the regression of x and y
Used the method explained on pg 334 ofJ.H. Zar, Biostatistical analysis,
fourth edition. 1999
"""
slope, intercept = regress(x,y)
coords = zip(x, y)
Sx = Sy = Syy = SXY = 0.0
n = float(len(y))
for x, y in coords:
SXY += x*y
Sx += x
Sy += y
Syy += y*y
Sxy = SXY - (Sx*Sy)/n
regSS = slope * Sxy
totSS = Syy - ((Sy*Sy)/n)
return regSS/totSS
def regress_residuals(x, y):
"""reports the residual (error) for each point from the linear regression"""
slope, intercept = regress(x, y)
coords = zip(x, y)
residuals = []
for x, y in coords:
e = y - (slope * x) - intercept
residuals.append(e)
return residuals
def stdev_from_mean(x):
"""returns num standard deviations from the mean of each val in x[]"""
x = array(x)
return (x - mean(x))/std(x)
def regress_major(x, y):
"""Returns major-axis regression line of y on x.
Use in cases where there is error in both x and y.
"""
x, y = array(x), array(y)
N = len(x)
Sx = sum(x)
Sy = sum(y)
Sxx = sum(x*x)
Syy = sum(y*y)
Sxy = sum(x*y)
var_y = (Syy-((Sy*Sy)/N))/(N-1)
var_x = (Sxx-((Sx*Sx)/N))/(N-1)
cov = (Sxy-((Sy*Sx)/N))/(N-1)
mean_y = Sy/N
mean_x = Sx/N
D = sqrt((var_y + var_x)*(var_y + var_x) - 4*(var_y*var_x - (cov*cov)))
eigen_1 = (var_y + var_x + D)/2
slope = cov/(eigen_1 - var_y)
intercept = mean_y - (mean_x * slope)
return (slope, intercept)
def z_test(a, popmean=0, popstdev=1, tails=None):
"""Returns z and probability score for a single sample of items.
Calculates the z-score on ONE sample of items with mean x, given a
population mean and standard deviation (parametric).
Usage: z, prob = z_test(a, popmean, popstdev, tails)
z is a float; prob is a probability.
a is a sample with Mean and Count.
popmean should be the parametric population mean; 0 by default.
popstdev should be the parametric population standard deviation, default=1.
tails should be None (default), 'high', or 'low'.
"""
try:
z = (mean(a) - popmean)/popstdev*sqrt(len(a))
return z, z_tailed_prob(z, tails)
except (ValueError, TypeError, ZeroDivisionError, AttributeError, \
FloatingPointError):
return None
def z_tailed_prob(z, tails):
"""Returns appropriate p-value for given z, depending on tails."""
if tails == 'high':
return z_high(z)
elif tails == 'low':
return z_low(z)
else:
return zprob(z)
def t_tailed_prob(t, df, tails):
"""Return appropriate p-value for given t and df, depending on tails."""
if tails == 'high':
return t_high(t, df)
elif tails == 'low':
return t_low(t, df)
else:
return tprob(t,df)
def reverse_tails(tails):
"""Swaps high for low or vice versa, leaving other values alone."""
if tails == 'high':
return 'low'
elif tails == 'low':
return 'high'
else:
return tails
def tail(prob, test):
"""If test is true, returns prob/2. Otherwise returns 1-(prob/2).
"""
prob /= 2
if test:
return prob
else:
return 1 - prob
def combinations(n, k):
"""Returns the number of ways of choosing k items from n.
"""
return exp(lgam(n+1) - lgam(k+1) - lgam(n-k+1))
def multiple_comparisons(p, n):
"""Corrects P-value for n multiple comparisons.
Calculates directly if p is large and n is small; resorts to logs
otherwise to avoid rounding (1-p) to 1
"""
if p > 1e-6: #if p is large and n small, calculate directly
return 1 - (1-p)**n
else:
return one_minus_exp(-n * p)
def multiple_inverse(p_final, n):
"""Returns p_initial for desired p_final with n multiple comparisons.
WARNING: multiple_inverse is not very reliable when p_final is very close
to 1 (say, within 1e-4) since we then take the ratio of two very similar
numbers.
"""
return one_minus_exp(log_one_minus(p_final)/n)
def multiple_n(p_initial, p_final):
"""Returns number of comparisons such that p_initial maps to p_final.
WARNING: not very accurate when p_final is very close to 1.
"""
return log_one_minus(p_final)/log_one_minus(p_initial)
def fisher(probs):
"""Uses Fisher's method to combine multiple tests of a hypothesis.
-2 * SUM(ln(P)) gives chi-squared distribution with 2n degrees of freedom.
"""
stat = -2*log(array(probs)).sum()
if isnan(stat):
return nan
else:
try:
return chi_high(stat, 2 * len(probs))
except OverflowError, e:
return nan
def f_value(a,b):
"""Returns the num df, the denom df, and the F value.
a, b: lists of values, must have Variance attribute (recommended to
make them Numbers objects.
The F value is always calculated by dividing the variance of a by the
variance of b, because R uses the same approach. In f_two_value it's
decided what p-value is returned, based on the relative sizes of the
variances.
"""
if not any(a) or not any(b) or len(a) <= 1 or len(b) <= 1:
raise ValueError, "Vectors should contain more than 1 element"
F = var(a)/var(b)
dfn = len(a)-1
dfd = len(b)-1
return dfn, dfd, F
def f_two_sample(a, b, tails=None):
"""Returns the dfn, dfd, F-value and probability for two samples a, and b.
a and b: should be independent samples of scores. Should be lists of
observations (numbers).
tails should be None(default, two-sided test), 'high' or 'low'.
This implementation returns the same results as the F test in R.
"""
dfn, dfd, F = f_value(a, b)
if tails == 'low':
return dfn, dfd, F, f_low(dfn, dfd, F)
elif tails == 'high':
return dfn, dfd, F, f_high(dfn, dfd, F)
else:
if var(a) >= var(b):
side='right'
else:
side='left'
return dfn, dfd, F, fprob(dfn, dfd, F, side=side)
def ANOVA_one_way(a):
"""Performs a one way analysis of variance
a is a list of lists of observed values. Each list is the values
within a category. The analysis must include 2 or more categories(lists).
Each category of the list, and overall list, is converted to a numpy array.
An F value is first calculated as the variance of the group means
divided by the mean of the within-group variances.
"""
#a = array(a)
group_means = []
group_variances = []
num_cases = 0 # total observations in all groups
all_vals = []
for i in a:
num_cases += len(i)
group_means.append(mean(i))
group_variances.append(i.var(ddof=1) * (len(i)-1))
all_vals.extend(i)
# Get within Group variances (denominator)
dfd = num_cases - len(group_means)
# need to add a check -- if the sum of the group variances is zero it will
# error, but only if the between_Groups value is not zero
within_Groups = sum(group_variances) / dfd
if within_Groups == 0.:
return nan, nan
# Get between Group variances (numerator)
all_vals = array(all_vals)
grand_mean = all_vals.mean()
between_Groups = 0
for i in a:
diff = i.mean() - grand_mean
diff_sq = diff * diff
x = diff_sq * len(i)
between_Groups += x
dfn = len(group_means) - 1
between_Groups = between_Groups/dfn
F = between_Groups/within_Groups
return F, f_high(dfn, dfd, F)
def MonteCarloP(value, rand_values, tail = 'high'):
"""takes a true value and a list of random values as
input and returns a p-value
tail indicates which side of the distribution to look at:
low = look for smaller values than expected by chance
high = look for larger values than expected by chance
"""
pop_size= len(rand_values)
rand_values.sort()
if tail == 'high':
num_better = pop_size
for i, curr_val in enumerate(rand_values):
if value <= curr_val:
num_better = i
break
p_val = 1-(num_better / pop_size)
elif tail == 'low':
num_better = pop_size
for i, curr_val in enumerate(rand_values):
if value < curr_val:
num_better = i
break
p_val = num_better / pop_size
return p_val
def sign_test(success, trials, alt="two sided"):
"""Returns the probability for the sign test.
Arguments:
- success: the number of successes
- trials: the number of trials
- alt: the alternate hypothesis, one of 'less', 'greater', 'two sided'
(default).
"""
lo = ["less", "lo", "lower", "l"]
hi = ["greater", "hi", "high", "h", "g"]
two = ["two sided", "2", 2, "two tailed", "two"]
alt = alt.lower().strip()
if alt in lo:
p = binomial_low(success, trials, 0.5)
elif alt in hi:
success -= 1
p = binomial_high(success, trials, 0.5)
elif alt in two:
success = min(success, trials-success)
hi = 1 - binomial_high(success, trials, 0.5)
lo = binomial_low(success, trials, 0.5)
p = hi+lo
else:
raise RuntimeError("alternate [%s] not in %s" % (lo+hi+two))
return p
def ks_test(x, y=None, alt="two sided", exact = None, warn_for_ties = True):
"""Returns the statistic and probability from the Kolmogorov-Smirnov test.
Arguments:
- x, y: vectors of numbers whose distributions are to be compared.
- alt: the alternative hypothesis, default is 2-sided.
- exact: whether to compute the exact probability
- warn_for_ties: warns when values are tied. This should left at True
unless a monte carlo variant, like ks_boot, is being used.
Note the 1-sample cases are not implemented, although their cdf's are
implemented in ks.py"""
# translation from R 2.4
num_x = len(x)
num_y = None
x = zip(x, zeros(len(x), int))
lo = ["less", "lo", "lower", "l", "lt"]
hi = ["greater", "hi", "high", "h", "g", "gt"]
two = ["two sided", "2", 2, "two tailed", "two", "two.sided"]
Pval = None
if y is not None: # in anticipation of actually implementing the 1-sample cases
num_y = len(y)
y = zip(y, ones(len(y), int))
n = num_x * num_y / (num_x + num_y)
combined = x + y
if len(set(combined)) < num_x + num_y:
ties = True
else:
ties = False
combined = array(combined, dtype=[('stat', float), ('sample', int)])
combined.sort(order='stat')
cumsum = zeros(combined.shape[0], float)
scales = array([1/num_x, -1/num_y])
indices = combined['sample']
cumsum = scales.take(indices)
cumsum = cumsum.cumsum()
if exact == None:
exact = num_x * num_y < 1e4
if alt in two:
stat = max(fabs(cumsum))
elif alt in lo:
stat = -cumsum.min()
elif alt in hi:
stat = cumsum.max()
else:
raise RuntimeError, "Unknown alt: %s" % alt
if exact and alt in two and not ties:
Pval = 1 - psmirnov2x(stat, num_x, num_y)
else:
raise NotImplementedError
if Pval == None:
if alt in two:
Pval = 1 - pkstwo(sqrt(n) * stat)
else:
Pval = exp(-2 * n * stat**2)
if ties and warn_for_ties:
warnings.warn("Cannot compute correct KS probability with ties")
try: # if numpy arrays were input, the Pval can be an array of len==1
Pval = Pval[0]
except (TypeError, IndexError):
pass
return stat, Pval
def _get_bootstrap_sample(x, y, num_reps):
"""yields num_reps random samples drawn with replacement from x and y"""
combined = array(list(x) + list(y))
total_obs = len(combined)
num_x = len(x)
for i in range(num_reps):
# sampling with replacement
indices = randint(0, total_obs, total_obs)
sampled = combined.take(indices)
# split into the two populations
sampled_x = sampled[:num_x]
sampled_y = sampled[num_x:]
yield sampled_x, sampled_y
def ks_boot(x, y, alt = "two sided", num_reps=1000):
"""Monte Carlo (bootstrap) variant of the Kolmogorov-Smirnov test. Useful
for when there are ties.
Arguments:
- x, y: vectors of numbers
- alt: alternate hypothesis, as per ks_test
- num_reps: number of replicates for the bootstrap"""
# based on the ks_boot method in the R Matching package
# see http://sekhon.berkeley.edu/matching/
# One important difference is I preserve the original sample sizes
# instead of making them equal
tol = MACHEP * 100
combined = array(list(x) + list(y))
observed_stat, _p = ks_test(x, y, exact=False, warn_for_ties=False)
total_obs = len(combined)
num_x = len(x)
num_greater = 0
for sampled_x, sampled_y in _get_bootstrap_sample(x, y, num_reps):
sample_stat, _p = ks_test(sampled_x, sampled_y, alt=alt, exact=False,
warn_for_ties=False)
if sample_stat >= (observed_stat - tol):
num_greater += 1
return observed_stat, num_greater / num_reps
def mw_test(n1, n2):
"""Compute Mann-Whitney U (equivalent to Wilcoxon ranked sum) stat.
This function computes the MWU statistic which is equivalent to the
Wilcoxon ranked sum test. It then computes the (two-tail) pval based on
the normal approximation. Two tails is appropriate because we do not know
which of our groups has a higher mean, thus our alternate hypothesis is that
the distributions from which the two samples come are not the same (FA!=FB)
and we must account for E[FA] > E[FB] and E[FB] < E[FA].
Implementation of test from Sokal and Rolhf, Biometry, pgs 427-431.
Specifically the algorithm is derived from pgs 429-430 under heading
'The Wilcoxon two-sample test'.
C = n1*n2 + n2(n2+1)/2 - sum(ranks of n2)
U = max(C, n1*n2 - C)
n1 and n2 are lists or arrays of numeric values.
"""
# find smaller sample, defined historically as n2. modify the names so we
# don't risk modifying data outside the scope of the function.
if len(n2) > len(n1):
sn1, sn2 = array(n2), array(n1)
else:
sn1, sn2 = array(n1), array(n2)
# sum the ranks of s2 by using the searchsorted magic. the logic is that we
# use a sorted copy of the data from both groups (n1 and n2) to figure out
# at what index we would insert the values from sample 2. by assessing the
# difference between the index that value x would be inserted in if we were
# doing left insertion versus right insertion, we can tell how many values
# are tied with x. this allows us to calculate the average ranks easily.
data = hstack([sn1,sn2])
data.sort()
ssl = searchsorted(data, sn2, 'left')
ssr = searchsorted(data, sn2, 'right')
sum_sn2_ranks = ((ssl+ssr+1)/2.).sum()
ln1, ln2 = sn1.size, sn2.size
C = (ln1*ln2)+ (ln2*(ln2+1)/2.) - sum_sn2_ranks
U = max(C, ln1*ln2 - C)
# now we calculate the pvalue using the normal approximation and the two
# tailed test. our formula corrects for ties, because in the case where
# there are no ties, the forumla on the bottom of pg 429=the formula on the
# bottom of pg 430.
numerator = (U - ln1*ln2/2.)
# follwing three lines give the T value in the formula on page 430. same
# logic as above; we calculate the left and right indices of the unique
# values for all combined data from both samples, then calculate ti**3-ti
# for each value.
ux = unique(data)
uxl = searchsorted(data, ux, 'left')
uxr = searchsorted(data, ux, 'right')
T = _corr_kw(uxr-uxl).sum()
denominator = sqrt(((ln1*ln2)/float((ln1+ln2)*(ln1+ln2-1)))*(((ln1+ln2)**3 \
- (ln1+ln2)-T)/12.))
if denominator == 0:
# Warning: probability of U can't be calculated by mw_test
# because all ranks of data were tied. Returning nan as pvalue.
return U, nan
else:
pval = zprob(numerator/float(denominator))
return U, pval
def mw_boot(x, y, num_reps=1000):
"""Monte Carlo (bootstrap) variant of the Mann-Whitney test.
Arguments:
- x, y: vectors of numbers
- num_reps: number of replicates for the bootstrap
Uses the same Monte-Carlo resampling code as kw_boot
"""
tol = MACHEP * 100
combined = array(list(x) + list(y))
observed_stat, obs_p = mw_test(x, y)
total_obs = len(combined)
num_x = len(x)
num_greater = 0
for sampled_x, sampled_y in _get_bootstrap_sample(x, y, num_reps):
sample_stat, sample_p = mw_test(sampled_x, sampled_y)
if sample_stat >= (observed_stat - tol):
num_greater += 1
return observed_stat, num_greater / num_reps
def permute_2d(m, p):
"""Performs 2D permutation of matrix m according to p."""
return m[p][:, p]
#unused below
m_t = transpose(m)
r_t = take(m_t, p, axis=0)
return take(transpose(r_t), p, axis=0)
def mantel(m1, m2, n):
"""Compares two distance matrices. Reports P-value for correlation.
The p-value is based on a two-sided test.
WARNING: The two distance matrices must be symmetric, hollow distance
matrices, as only the lower triangle (excluding the diagonal) will be used
in the calculations (matching R's vegan::mantel function).
This function is retained for backwards-compatibility. Please use
mantel_test() for more control over how the test is performed.
"""
return mantel_test(m1, m2, n)[0]
def mantel_test(m1, m2, n, alt="two sided",
suppress_symmetry_and_hollowness_check=False):
"""Runs a Mantel test on two distance matrices.
Returns the p-value, Mantel correlation statistic, and a list of Mantel
correlation statistics for each permutation test.
WARNING: The two distance matrices must be symmetric, hollow distance
matrices, as only the lower triangle (excluding the diagonal) will be used
in the calculations (matching R's vegan::mantel function).
Arguments:
m1 - the first distance matrix to use in the test (should be a numpy
array or convertible to a numpy array)
m2 - the second distance matrix to use in the test (should be a numpy
array or convertible to a numpy array)
n - the number of permutations to test when calculating the p-value
alt - the type of alternative hypothesis to test (can be either
'two sided' for a two-sided test, 'greater' or 'less' for one-sided
tests)
suppress_symmetry_and_hollowness_check - by default, the input distance
matrices will be checked for symmetry and hollowness. It is
recommended to leave this check in place for safety, as the check
is fairly fast. However, if you *know* you have symmetric and
hollow distance matrices, you can disable this check for small
performance gains on extremely large distance matrices
"""
# Perform some sanity checks on our input.
if alt not in ("two sided", "greater", "less"):
raise ValueError("Unrecognized alternative hypothesis. Must be either "
"'two sided', 'greater', or 'less'.")
m1, m2 = asarray(m1), asarray(m2)
if m1.shape != m2.shape:
raise ValueError("Both distance matrices must be the same size.")
if n < 1:
raise ValueError("The number of permutations must be greater than or "
"equal to one.")
if not suppress_symmetry_and_hollowness_check:
if not (is_symmetric_and_hollow(m1) and is_symmetric_and_hollow(m2)):
raise ValueError("Both distance matrices must be symmetric and "
"hollow.")
# Get a flattened list of lower-triangular matrix elements (excluding the
# diagonal) in column-major order. Use these values to calculate the
# correlation statistic.
m1_flat, m2_flat = _flatten_lower_triangle(m1), _flatten_lower_triangle(m2)
orig_stat = pearson(m1_flat, m2_flat)
# Run our permutation tests so we can calculate a p-value for the test.
size = len(m1)
better = 0
perm_stats = []
for i in range(n):
perm = permute_2d(m1, permutation(size))
perm_flat = _flatten_lower_triangle(perm)
r = pearson(perm_flat, m2_flat)
if alt == 'two sided':
if abs(r) >= abs(orig_stat):
better += 1
else:
if ((alt == 'greater' and r >= orig_stat) or
(alt == 'less' and r <= orig_stat)):
better += 1
perm_stats.append(r)
return (better + 1) / (n + 1), orig_stat, perm_stats
def is_symmetric_and_hollow(matrix):
return (matrix.T == matrix).all() and (trace(matrix) == 0)
def _flatten_lower_triangle(matrix):
"""Returns a list containing the flattened lower triangle of the matrix.
The returned list will contain the elements in column-major order. The
diagonal will be excluded.
Arguments:
matrix - numpy array containing the matrix data
"""
matrix = asarray(matrix)
flattened = []
for col_num in range(matrix.shape[1]):
for row_num in range(matrix.shape[0]):
if col_num < row_num:
flattened.append(matrix[row_num][col_num])
return flattened
## Start functions for distance_matrix_permutation_test
def distance_matrix_permutation_test(matrix, cells, cells2=None,\
f=t_two_sample, tails=None, n=1000, return_scores=False,\
is_symmetric=True):
"""performs a monte carlo permutation test to determine if the
values denoted in cells are significantly different than the rest
of the values in the matrix
matrix: a numpy array
cells: a list of indices of special cells to compare to the rest of the
matrix
cells2: an optional list of indices to compare cells to. If set to None
(default), compares cells to the rest of the matrix
f: the statistical test used. Should take a "tails" parameter as input
tails: can be None(default), 'high', or 'low'. Input into f.
n: the number of replicates in the Monte Carlo simulations
is_symmetric: corrects if the matrix is symmetric. Need to only look at
one half otherwise the degrees of freedom value will be incorrect.
"""
#if matrix is symmetric convert all indices to lower trangular
if is_symmetric:
cells = get_ltm_cells(cells)
if cells2:
cells2 = get_ltm_cells(cells2)
# pull out the special values
special_values, other_values = \
get_values_from_matrix(matrix, cells, cells2, is_symmetric)
# calc the stat and parameteric p-value for real data
stat, p = f(special_values, other_values, tails)
#calc for randomized matrices
count_more_extreme = 0
stats = []
indices = range(len(matrix))
for k in range(n):
# shuffle the order of indices, and use those to permute the matrix
permuted_matrix = permute_2d(matrix,permutation(indices))
special_values, other_values = \
get_values_from_matrix(permuted_matrix, cells,\
cells2, is_symmetric)
# calc the stat and p for a random subset (we don't do anything
# with these p-values, we only use the current_stat value)
current_stat, current_p = f(special_values, other_values, tails)
stats.append(current_stat)
if tails == None:
if abs(current_stat) > abs(stat): count_more_extreme += 1
elif tails == 'low':
if current_stat < stat: count_more_extreme += 1
elif tails == 'high':
if current_stat > stat: count_more_extreme += 1
# pack up the parametric stat, parametric p, and empirical p; calc the
# the latter in the process
result = [stat, p, count_more_extreme/n]
# append the scores of the n tests if requested
if return_scores: result.append(stats)
return tuple(result)
def get_values_from_matrix(matrix, cells, cells2=None, is_symmetric=True):
"""get values from matrix positions in cells and cells2
matrix: the numpy array from which values should be taken
cells: indices of first set of requested values
cells2: indices of second set of requested values or None
if they should be randomly selected
is_symmetric: True if matrix is symmetric
"""
# pull cells values
cells_values = [matrix[i] for i in cells]
# pull cells2 values
if cells2:
cells2_values = [matrix[i] for i in cells2]
# or generate the indices and grab them if they
# weren't passed in
else:
cells2_values = []
for i, val_i in enumerate(matrix):
for j, val in enumerate(val_i):
if is_symmetric:
if (i,j) not in cells and i > j:
cells2_values.append(val)
else:
if (i,j) not in cells:
cells2_values.append(val)
return cells_values, cells2_values
def get_ltm_cells(cells):
"""converts matrix indices so all are below the diagonal
cells: list of indices into a 2D integer-indexable object
(typically a list or lists of array of arrays)
"""
new_cells = []
for cell in cells:
if cell[0] < cell[1]:
new_cells.append((cell[1], cell[0]))
elif cell[0] > cell[1]:
new_cells.append(cell)
#remove duplicates
new_cells = set(new_cells)
return list(new_cells)
## End functions for distance_matrix_permutation_test
def fdr_correction(pvals):
"""Adjust pvalues for multiple tests using the false discovery rate method.
In short: ranks the p-values in ascending order and multiplies each p-value
by the number of comparisons divided by the rank of the p-value in the
sorted list. Input is list of floats.
Does *not* assume pvals is sorted.
"""
tmp = array(pvals).astype(float) #this converts Nones to nans
return tmp*tmp.size/(1.+argsort(argsort(tmp)).astype(float))
def benjamini_hochberg_step_down(pvals):
"""Perform Benjamini and Hochberg's 1995 FDR step down procedure.
In short, compute the fdr adjusted pvals (ap_i's), and working from
the largest to smallest, compare ap_i to ap_i-1. If ap_i < ap_i-1 set ap_i-1
equal to ap_i.
Does *not* assume pvals is sorted
"""
tmp = fdr_correction(pvals)
corrected_vals = empty(len(pvals))
max_pval = 1.
for i in argsort(pvals)[::-1]:
if tmp[i]<max_pval:
corrected_vals[i] = tmp[i]
max_pval = tmp[i]
else:
corrected_vals[i] = max_pval
return corrected_vals
def bonferroni_correction(pvals):
"""Adjust pvalues for multiple tests using the Bonferroni method.
In short: multiply all pvals by the number of comparisons."""
return array(pvals, dtype=float)*len(pvals) #float conversion: Nones->nans
def fisher_z_transform(r):
"""Calculate the Fisher Z transform of a correlation coefficient.
Relies on formulation in Sokal and Rohlf Biometry pg 575.
"""
if abs(r) == 1: # fisher z transform is undefined, have to return nan
return nan
return .5*log((1.+r)/(1.-r))
def z_transform_pval(z, n):
'''Calculate two tailed probability of value more extreme than z given n.
Relies on formulation in Sokal and Rohlf Biometry pg 576.
'''
if n <= 3: #sample size must be greater than 3 otherwise this transform
# isn't supported.
return nan
return zprob(z*((n-3)**.5))
def inverse_fisher_z_transform(z):
"""Calculate the inverse of the Fisher Z transform on a z value.
Relies on formulation in Sokal and Rohlf Biometry pg 576.
"""
return ((e**(2*z))-1.)/((e**(2*z))+1.)
def fisher_population_correlation(corrcoefs, sample_sizes):
"""Calculate population rho, homogeneity from corrcoefs using Z transform.
Exclude pvals of nan.
"""
tmp_rs = array(corrcoefs)
tmp_ns = array(sample_sizes)
# make checks for nans and exclude them as they will cause things to break
rs = tmp_rs[~isnan(tmp_rs)]
ns = tmp_ns[~isnan(tmp_rs)]
if not (ns > 3).all():
# not all samples have size > 3 which causes 0 varaince estimation.
# thus we must return nan for pval and h_val
return nan, nan
if not len(ns) > 1:
# only one sample, because of reduced degrees of freedom must have at
# least two samples to calculate the homogeneity.
return nan, nan
# calculate zs
zs = array(map(fisher_z_transform, rs))
# calculate variance weighted z average = z_bar
z_bar = (zs*(ns-3)).sum()/float((ns-3).sum())
rho = inverse_fisher_z_transform(z_bar)
# calculate homogeneity
x_2 = ((ns-3)*(zs-z_bar)**2).sum()
h_val = chisqprob(x_2, len(ns)-1)
return rho, h_val
def assign_correlation_pval(corr, n, method, permutations=None,
perm_test_fn=None, v1=None, v2=None):
'''Assign pval to a correlation score with given method.
This function will assign significance to the correlation score passed given
the method that is passed. Some of the methods are appropriate only for
certain types of data and there is no way for this test to determine the
appropriateness, thus you must use this function only when with the proper
prior knowledge. The 'parametric_t_distribution' method is described in
Sokal and Rohlf Biometry pg. 576, the 'fisher_z_transform' method is
described on pg 576 and 577. The 'bootstrap' method calculates the given
correlation permutations number of times using perm_test_fn.
Also note, this does *not* take the place of FDR correction.
Inputs:
corr - float, correlation score from Kendall's Tau, Spearman's Rho, or
Pearson.
n - length of the vectors that were correlated.
method - str in ['parametric_t_distribution', 'fisher_z_transform',
'bootstrapped', 'kendall'].
permutations - int, number of permutations to use if bootstrapped selected.
perm_test_fn - function, to use to calculate correlation if permuation test
desired.
v1, v2 = 1D vectors of numerics, passed if method='bootstrapped'
'''
if method == 'parametric_t_distribution':
df = n-2
assert df>1, "Must have more than 1 degree of freedom. Can't Continue."
try:
ts = corr*((df/(1.-corr**2))**.5)
return tprob(ts, df) #two tailed test because H0 is corr=0
except (ValueError, FloatingPointError, ZeroDivisionError):
# something unpleasant happened, most likely r or rho where +- 1
# which means the parametric p val should be 1 or 0 or nan
return nan
elif method=='fisher_z_transform':
# Sokal and Rohlf indicate that for n<50, the Fisher Z transform for
# assigning correlation probabilities is not accurate. Currently no
# check is in place
z = fisher_z_transform(corr)
# the z transform pval compares against a t distribution with inf
# degrees of freedom which is equal to a z distribution.
return z_transform_pval(z, n)
elif method=='bootstrapped':
if any([v1==None, v2==None, permutations==None, perm_test_fn==None]):
raise ValueError('You must specify vectors, permutation '+\
'function, and number of permutations to calculate '+\
'bootstrapped pvalues. Cant continue.')
r = empty(permutations)
for i in range(permutations):
r[i] = perm_test_fn(v1,permutation(v2))
return (abs(r)>=abs(corr)).sum()/float(permutations)
elif method == 'kendall':
return kendall_pval(corr, n)
else:
raise ValueError("'%s' method is unknown." % method)
def fisher_confidence_intervals(test_stat, n, confidence_level=.95):
"""Calc confidence interval of test_stat using Fishers Z transform.
Fishers Z transform is described in Sokal and Rolhf.
Inputs:
test_stat - numeric, the correlation coefficient whose significance we are
to test.
n - int, length of vectors that were correlated.
confidence_level - float in (0,1), level of confidence we want for the
intervals.
"""
# compute confidence intervals using fishers z transform
z_crit = abs(ndtri((1 - confidence_level) / 2.))
ci_low, ci_high = None, None
if n > 3:
try:
ci_low = tanh(arctanh(test_stat) - (z_crit / sqrt(n - 3)))
ci_high = tanh(arctanh(test_stat) + (z_crit / sqrt(n - 3)))
except (ZeroDivisionError, FloatingPointError):
# r or rho was presumably 1 or -1. Match what R does in this case.
# feel like nan should be returned here given that we can't make
# the calculation
ci_low, ci_high = test_stat, test_stat
return ci_low, ci_high
def cscore(v1, v2):
'''Calculate C-score between v1 and v2 according to Stone and Roberts 1990.
This function calculates the C-score between equal length vectors v1 and v2
according to the formulation given in Stone and Roberts 1990, Oecologia
85: 74-79.
v1, v2 - 1d arrays of equal length.
'''
v1_b = v1.astype(bool)
v2_b = v2.astype(bool)
sij = (v1_b*v2_b).sum()
return (v1_b.sum()-sij)*(v2_b.sum()-sij)
