def linregress(x:List(float),y:List(float))->(float,float,float,float,float):
"""
Calculates a regression line on x,y pairs.
Usage: llinregress(x,y) x,y are equal-length lists of x-y coordinates
Returns: slope, intercept, r, two-tailed prob, sterr-of-estimate
"""
TINY = 1.0e-20
if len(x) != len(y):
raise ValueError('Input values not paired in linregress. Aborting.')
n = len(x)
x = list(map(float,x))
y = list(map(float,y))
xmean = central_tendency.mean(x)
ymean = central_tendency.mean(y)
r_num = float(n*(support.summult(x,y)) - sum(x)*sum(y))
r_den = sqrt((n*support.ss(x) - support.square_of_sums(x))*(n*support.ss(y)-support.square_of_sums(y)))
r = r_num / r_den
z = 0.5*log((1.0+r+TINY)/(1.0-r+TINY))
df = n-2
t = r*sqrt(df/((1.0-r+TINY)*(1.0+r+TINY)))
prob = probability.betai(0.5*df,0.5,df/(df+t*t))
slope = r_num / float(n*support.ss(x) - support.square_of_sums(x))
intercept = ymean - slope*xmean
sterrest = sqrt(1-r*r)*variability.samplestdev(y)
return slope, intercept, r, prob, sterrest
def kendalltau(x:List(float),y:List(float))->(float,float):
"""
Calculates Kendall's tau ... correlation of ordinal data. Adapted
from function kendl1 in Numerical Recipies. Needs good test-routine.@@@
Usage: lkendalltau(x,y)
Returns: Kendall's tau, two-tailed p-value
"""
n1 = 0
n2 = 0
iss = 0
for j in range(len(x)-1):
for k in range(j,len(y)):
a1 = x[j] - x[k]
a2 = y[j] - y[k]
aa = a1 * a2
if (aa): # neither list has a tie
n1 = n1 + 1
n2 = n2 + 1
if aa > 0:
iss = iss + 1
else:
iss = iss -1
else:
if (a1):
n1 = n1 + 1
else:
n2 = n2 + 1
tau = iss / sqrt(n1*n2)
svar = (4.0*len(x)+10.0) / (9.0*len(x)*(len(x)-1))
_z = tau / sqrt(svar)
prob = probability.erfcc(abs(_z)/1.4142136)
return tau, prob
def mannwhitneyu(x: List(float), y: List(float)) -> (float, float):
"""
Calculates a Mann-Whitney U statistic on the provided scores and
returns the result. Use only when the n in each condition is < 20 and
you have 2 independent samples of ranks. NOTE: Mann-Whitney U is
significant if the u-obtained is LESS THAN or equal to the critical
value of U found in the tables. Equivalent to Kruskal-Wallis H with
just 2 groups.
Usage: lmannwhitneyu(data)
Returns: u-statistic, one-tailed p-value (i.e., p(z(U)))
"""
n1 = len(x)
n2 = len(y)
ranked = support.rankdata(x + y)
rankx = ranked[0:n1] # get the x-ranks
ranky = ranked[n1:] # the rest are y-ranks
u1 = n1 * n2 + (n1 * (n1 + 1)) / 2.0 - sum(rankx) # calc U for x
u2 = n1 * n2 - u1 # remainder is U for y
bigu = max(u1, u2)
smallu = min(u1, u2)
proportion = bigu / float(n1 * n2)
T = sqrt(tiecorrect(ranked)) # correction factor for tied scores
if T == 0:
raise ValueError('All numbers are identical in lmannwhitneyu')
sd = sqrt(T * n1 * n2 * (n1 + n2 + 1) / 12.0)
z = abs((bigu - n1 * n2 / 2.0) / sd) # normal approximation for prob calc
return smallu, 1.0 - probability.zprob(z) #, proportion
def pointbiserialr(x:List(float),y:List(float))->(float,float):
"""
Calculates a point-biserial correlation coefficient and the associated
probability value. Taken from Heiman's Basic Statistics for the Behav.
Sci (1st), p.194.
Usage: lpointbiserialr(x,y) where x,y are equal-length lists
Returns: Point-biserial r, two-tailed p-value
"""
TINY = 1e-30
if len(x) != len(y):
raise ValueError('INPUT VALUES NOT PAIRED IN pointbiserialr. ABORTING.')
data = pstat.abut(x,y)
categories = pstat.unique(x)
if len(categories) != 2:
raise ValueError("Exactly 2 categories required for pointbiserialr().")
else: # there are 2 categories, continue
codemap = pstat.abut(categories,list(range(2)))
recoded = pstat.recode(data,codemap,0)
_x = pstat.linexand(data,0,categories[0])
_y = pstat.linexand(data,0,categories[1])
xmean = central_tendency.mean(pstat.colex(_x,1))
ymean = central_tendency.mean(pstat.colex(_y,1))
n = len(data)
adjust = sqrt((len(_x)/float(n))*(len(_y)/float(n)))
rpb = (ymean - xmean)/variability.samplestdev(pstat.colex(data,1))*adjust
df = n-2
t = rpb*sqrt(df/((1.0-rpb+TINY)*(1.0+rpb+TINY)))
prob = probability.betai(0.5*df,0.5,df/(df+t*t)) # t already a float
return rpb, prob
def chisqprob(chisq:float,df:int)->float:
"""
Returns the (1-tailed) probability value associated with the provided
chi-square value and df. Adapted from chisq.c in Gary Perlman's |Stat.
Usage: lchisqprob(chisq,df)
"""
BIG = 20.0
if chisq <=0 or df < 1:
return 1.0
a = 0.5 * chisq
if df%2 == 0:
even = 1
else:
even = 0
if df > 1:
y = ex(-a,BIG)
if even:
s = y
else:
s = 2.0 * zprob(-sqrt(chisq))
if (df > 2):
chisq = 0.5 * (df - 1.0)
if even:
_z = 1.0
else:
_z = 0.5
if a > BIG:
if even:
e = 0.0
else:
e = log(sqrt(pi))
c = log(a)
while (_z <= chisq):
e = log(_z) + e
s = s + ex(c*_z-a-e,BIG)
_z = _z + 1.0
return s
else:
if even:
e = 1.0
else:
e = 1.0 / sqrt(pi) / sqrt(a)
c = 0.0
while (_z <= chisq):
e = e * (a/float(_z))
c = c + e
_z = _z + 1.0
return (c*y+s)
else:
return s
def ttest_ind(a: List(float), b: List(float)) -> (float, float):
"""
Calculates the t-obtained T-test on TWO INDEPENDENT samples of
scores a, and b. From Numerical Recipies, p.483. If printit=1, results
are printed to the screen. If printit='filename', the results are output
to 'filename' using the given writemode (default=append). Returns t-value,
and prob.
Usage: lttest_ind(a,b,printit=0,name1='Samp1',name2='Samp2',writemode='a')
Returns: t-value, two-tailed prob
"""
printit = 0
name1 = 'Samp1'
name2 = 'Samp2'
writemode = 'a'
#bg: optional args
x1 = central_tendency.mean(a)
x2 = central_tendency.mean(b)
v1 = variability.stdev(a)**2
v2 = variability.stdev(b)**2
n1 = len(a)
n2 = len(b)
df = n1 + n2 - 2
svar = ((n1 - 1) * v1 + (n2 - 1) * v2) / float(df)
t = (x1 - x2) / sqrt(svar * (1.0 / n1 + 1.0 / n2))
prob = probability.betai(0.5 * df, 0.5, df / (df + t * t))
if printit != 0:
statname = 'Independent samples T-test.'
outputpairedstats(printit, writemode, name1, n1,
x1, v1, min(a), max(a), name2, n2, x2, v2, min(b),
max(b), statname, t, prob)
return t, prob
def ttest_1samp(a: List(float), popmean: int) -> (float, float):
"""
Calculates the t-obtained for the independent samples T-test on ONE group
of scores a, given a population mean. If printit=1, results are printed
to the screen. If printit='filename', the results are output to 'filename'
using the given writemode (default=append). Returns t-value, and prob.
Usage: lttest_1samp(a,popmean,Name='Sample',printit=0,writemode='a')
Returns: t-value, two-tailed prob
"""
printit = 0 #bg: optional arg
name = 'Sample' #bg: optional arg
writemode = 'a' #bg: optional arg
x = central_tendency.mean(a)
v = variability.var(a)
n = len(a)
df = n - 1
svar = ((n - 1) * v) / float(df)
t = (x - popmean) / sqrt(svar * (1.0 / n))
prob = probability.betai(0.5 * df, 0.5, float(df) / (df + t * t))
if printit != 0:
statname = 'Single-sample T-test.'
outputpairedstats(printit, writemode,
'Population', '--', popmean, 0, 0, 0, name, n, x, v,
min(a), max(a), statname, t, prob)
return t, prob
def wilcoxont(x: List(float), y: List(float)) -> (float, float):
"""
Calculates the Wilcoxon T-test for related samples and returns the
result. A non-parametric T-test.
Usage: lwilcoxont(x,y)
Returns: a t-statistic, two-tail probability estimate
"""
if len(x) != len(y):
raise ValueError('Unequal N in wilcoxont. Aborting.')
d = []
for i in range(len(x)):
diff = x[i] - y[i]
if diff != 0:
d.append(diff)
count = len(d)
absd = list(map(abs, d))
absranked = support.rankdata(absd)
r_plus = 0.0
r_minus = 0.0
for i in range(len(absd)):
if d[i] < 0:
r_minus = r_minus + absranked[i]
else:
r_plus = r_plus + absranked[i]
wt = min(r_plus, r_minus)
mn = count * (count + 1) * 0.25
se = sqrt(count * (count + 1) * (2.0 * count + 1.0) / 24.0)
_z = fabs(wt - mn) / se
prob = 2 * (1.0 - probability.zprob(abs(_z)))
return wt, prob
def stdev(inlist: List(float)) -> float:
"""
Returns the standard deviation of the values in the passed list
using N-1 in the denominator (i.e., to estimate population stdev).
Usage: lstdev(inlist)
"""
return sqrt(var(inlist))
def sterr(inlist: List(float)) -> float:
"""
Returns the standard error of the values in the passed list using N-1
in the denominator (i.e., to estimate population standard error).
Usage: lsterr(inlist)
"""
return stdev(inlist) / float(sqrt(len(inlist)))
def samplestdev(inlist: List(float)) -> float:
"""
Returns the standard deviation of the values in the passed list using
N for the denominator (i.e., DESCRIBES the sample stdev only).
Usage: lsamplestdev(inlist)
"""
return sqrt(samplevar(inlist))
def sem(inlist: List(float)) -> float:
"""
Returns the estimated standard error of the mean (sx-bar) of the
values in the passed list. sem = stdev / sqrt(n)
Usage: lsem(inlist)
"""
sd = stdev(inlist)
n = len(inlist)
return sd / sqrt(n)
def pearsonr(x:List(float),y:List(float))->(float,float):
"""
Calculates a Pearson correlation coefficient and the associated
probability value. Taken from Heiman's Basic Statistics for the Behav.
Sci (2nd), p.195.
Usage: lpearsonr(x,y) where x and y are equal-length lists
Returns: Pearson's r value, two-tailed p-value
"""
TINY = 1.0e-30
if len(x) != len(y):
raise ValueError('Input values not paired in pearsonr. Aborting.',x,y)
n = len(x)
x = list(map(float,x))
y = list(map(float,y))
xmean = central_tendency.mean(x)
ymean = central_tendency.mean(y)
r_num = n*(support.summult(x,y)) - sum(x)*sum(y)
r_den = sqrt((n*support.ss(x) - support.square_of_sums(x))*(n*support.ss(y)-support.square_of_sums(y)))
r = (r_num / r_den) # denominator already a float
df = n-2
t = r*sqrt(df/((1.0-r+TINY)*(1.0+r+TINY)))
prob = probability.betai(0.5*df,0.5,df/float(df+t*t))
return r, prob
def ranksums(x: List(float), y: List(float)) -> (float, float):
"""
Calculates the rank sums statistic on the provided scores and
returns the result. Use only when the n in each condition is > 20 and you
have 2 independent samples of ranks.
Usage: lranksums(x,y)
Returns: a z-statistic, two-tailed p-value
"""
n1 = len(x)
n2 = len(y)
alldata = x + y
ranked = support.rankdata(alldata)
x = ranked[:n1]
y = ranked[n1:]
s = sum(x)
expected = n1 * (n1 + n2 + 1) / 2.0
_z = (s - expected) / sqrt(n1 * n2 * (n1 + n2 + 1) / 12.0)
prob = 2 * (1.0 - probability.zprob(abs(_z)))
return _z, prob
def spearmanr(x:List(float),y:List(float))->(float,float):
"""
Calculates a Spearman rank-order correlation coefficient. Taken
from Heiman's Basic Statistics for the Behav. Sci (1st), p.192.
Usage: lspearmanr(x,y) where x and y are equal-length lists
Returns: Spearman's r, two-tailed p-value
"""
TINY = 1e-30
if len(x) != len(y):
raise ValueError('Input values not paired in spearmanr. Aborting.')
n = len(x)
rankx = rankdata(x)
ranky = rankdata(y)
dsq = sumdiffsquared(rankx,ranky)
rs = 1 - 6*dsq / float(n*(n**2-1))
t = rs * sqrt((n-2) / ((rs+1.0)*(1.0-rs)))
df = n-2
probrs = probability.betai(0.5*df,0.5,df/(df+t*t)) # t already a float
# probability values for rs are from part 2 of the spearman function in
# Numerical Recipies, p.510. They are close to tables, but not exact. (?)
return rs, probrs
def ks_2samp(data1: List(float), data2: List(float)) -> (float, float):
"""
Computes the Kolmogorov-Smirnof statistic on 2 samples. From
Numerical Recipies in C, page 493.
Usage: lks_2samp(data1,data2) data1&2 are lists of values for 2 conditions
Returns: KS D-value, associated p-value
"""
j1 = 0
j2 = 0
fn1 = 0.0
fn2 = 0.0
n1 = len(data1)
n2 = len(data2)
en1 = n1
en2 = n2
d = 0.0
data1.sort()
data2.sort()
while j1 < n1 and j2 < n2:
d1 = data1[j1]
d2 = data2[j2]
if d1 <= d2:
fn1 = (j1) / float(en1)
j1 = j1 + 1
if d2 <= d1:
fn2 = (j2) / float(en2)
j2 = j2 + 1
dt = (fn2 - fn1)
if fabs(dt) > fabs(d):
d = dt
try:
en = sqrt(en1 * en2 / float(en1 + en2))
prob = ksprob((en + 0.12 + 0.11 / en) * abs(d))
except:
prob = 1.0
return d, prob
def ttest_rel(a: List(float), b: List(float)) -> (float, float):
"""
Calculates the t-obtained T-test on TWO RELATED samples of scores,
a and b. From Numerical Recipies, p.483. If printit=1, results are
printed to the screen. If printit='filename', the results are output to
'filename' using the given writemode (default=append). Returns t-value,
and prob.
Usage: lttest_rel(a,b,printit=0,name1='Sample1',name2='Sample2',writemode='a')
Returns: t-value, two-tailed prob
"""
printit = 0
name1 = 'Sample1'
name2 = 'Sample2'
writemode = 'a' #bg: optional arg
if len(a) != len(b):
raise ValueError('Unequal length lists in ttest_rel.')
x1 = central_tendency.mean(a)
x2 = central_tendency.mean(b)
v1 = variability.var(a)
v2 = variability.var(b)
n = len(a)
cov = 0
for i in range(len(a)):
cov = cov + (a[i] - x1) * (b[i] - x2)
df = n - 1
_cov = cov / float(df)
sd = sqrt((v1 + v2 - 2.0 * _cov) / float(n))
t = (x1 - x2) / sd
prob = probability.betai(0.5 * df, 0.5, df / (df + t * t))
if printit != 0:
statname = 'Related samples T-test.'
outputpairedstats(printit, writemode, name1, n, x1, v1, min(a), max(a),
name2, n, x2, v2, min(b), max(b), statname, t, prob)
return t, prob