Source code for statsmodels.stats.multitest
'''Multiple Testing and P-Value Correction
Author: Josef Perktold
License: BSD-3
'''
import numpy as np
from statsmodels.stats._knockoff import RegressionFDR
__all__ = ['fdrcorrection', 'fdrcorrection_twostage', 'local_fdr',
'multipletests', 'NullDistribution', 'RegressionFDR']
# ==============================================
#
# Part 1: Multiple Tests and P-Value Correction
#
# ==============================================
def _ecdf(x):
'''no frills empirical cdf used in fdrcorrection
'''
nobs = len(x)
return np.arange(1,nobs+1)/float(nobs)
multitest_methods_names = {'b': 'Bonferroni',
's': 'Sidak',
'h': 'Holm',
'hs': 'Holm-Sidak',
'sh': 'Simes-Hochberg',
'ho': 'Hommel',
'fdr_bh': 'FDR Benjamini-Hochberg',
'fdr_by': 'FDR Benjamini-Yekutieli',
'fdr_tsbh': 'FDR 2-stage Benjamini-Hochberg',
'fdr_tsbky': 'FDR 2-stage Benjamini-Krieger-Yekutieli',
'fdr_gbs': 'FDR adaptive Gavrilov-Benjamini-Sarkar'
}
_alias_list = [['b', 'bonf', 'bonferroni'],
['s', 'sidak'],
['h', 'holm'],
['hs', 'holm-sidak'],
['sh', 'simes-hochberg'],
['ho', 'hommel'],
['fdr_bh', 'fdr_i', 'fdr_p', 'fdri', 'fdrp'],
['fdr_by', 'fdr_n', 'fdr_c', 'fdrn', 'fdrcorr'],
['fdr_tsbh', 'fdr_2sbh'],
['fdr_tsbky', 'fdr_2sbky', 'fdr_twostage'],
['fdr_gbs']
]
multitest_alias = {}
for m in _alias_list:
multitest_alias[m[0]] = m[0]
for a in m[1:]:
multitest_alias[a] = m[0]
[docs]
def multipletests(pvals, alpha=0.05, method='hs',
maxiter=1,
is_sorted=False,
returnsorted=False):
"""
Test results and p-value correction for multiple tests
Parameters
----------
pvals : array_like, 1-d
uncorrected p-values. Must be 1-dimensional.
alpha : float
FWER, family-wise error rate, e.g. 0.1
method : str
Method used for testing and adjustment of pvalues. Can be either the
full name or initial letters. Available methods are:
- `bonferroni` : one-step correction
- `sidak` : one-step correction
- `holm-sidak` : step down method using Sidak adjustments
- `holm` : step-down method using Bonferroni adjustments
- `simes-hochberg` : step-up method (independent)
- `hommel` : closed method based on Simes tests (non-negative)
- `fdr_bh` : Benjamini/Hochberg (non-negative)
- `fdr_by` : Benjamini/Yekutieli (negative)
- `fdr_tsbh` : two stage fdr correction (non-negative)
- `fdr_tsbky` : two stage fdr correction (non-negative)
maxiter : int or bool
Maximum number of iterations for two-stage fdr, `fdr_tsbh` and
`fdr_tsbky`. It is ignored by all other methods.
maxiter=1 (default) corresponds to the two stage method.
maxiter=-1 corresponds to full iterations which is maxiter=len(pvals).
maxiter=0 uses only a single stage fdr correction using a 'bh' or 'bky'
prior fraction of assumed true hypotheses.
is_sorted : bool
If False (default), the p_values will be sorted, but the corrected
pvalues are in the original order. If True, then it assumed that the
pvalues are already sorted in ascending order.
returnsorted : bool
not tested, return sorted p-values instead of original sequence
Returns
-------
reject : ndarray, boolean
true for hypothesis that can be rejected for given alpha
pvals_corrected : ndarray
p-values corrected for multiple tests
alphacSidak : float
corrected alpha for Sidak method
alphacBonf : float
corrected alpha for Bonferroni method
Notes
-----
There may be API changes for this function in the future.
Except for 'fdr_twostage', the p-value correction is independent of the
alpha specified as argument. In these cases the corrected p-values
can also be compared with a different alpha. In the case of 'fdr_twostage',
the corrected p-values are specific to the given alpha, see
``fdrcorrection_twostage``.
The 'fdr_gbs' procedure is not verified against another package, p-values
are derived from scratch and are not derived in the reference. In Monte
Carlo experiments the method worked correctly and maintained the false
discovery rate.
All procedures that are included, control FWER or FDR in the independent
case, and most are robust in the positively correlated case.
`fdr_gbs`: high power, fdr control for independent case and only small
violation in positively correlated case
**Timing**:
Most of the time with large arrays is spent in `argsort`. When
we want to calculate the p-value for several methods, then it is more
efficient to presort the pvalues, and put the results back into the
original order outside of the function.
Method='hommel' is very slow for large arrays, since it requires the
evaluation of n partitions, where n is the number of p-values.
"""
import gc
pvals = np.asarray(pvals)
alphaf = alpha # Notation ?
if not is_sorted:
sortind = np.argsort(pvals)
pvals = np.take(pvals, sortind)
ntests = len(pvals)
alphacSidak = 1 - np.power((1. - alphaf), 1./ntests)
alphacBonf = alphaf / float(ntests)
if method.lower() in ['b', 'bonf', 'bonferroni']:
reject = pvals <= alphacBonf
pvals_corrected = pvals * float(ntests)
elif method.lower() in ['s', 'sidak']:
reject = pvals <= alphacSidak
pvals_corrected = -np.expm1(ntests * np.log1p(-pvals))
elif method.lower() in ['hs', 'holm-sidak']:
alphacSidak_all = 1 - np.power((1. - alphaf),
1./np.arange(ntests, 0, -1))
notreject = pvals > alphacSidak_all
del alphacSidak_all
nr_index = np.nonzero(notreject)[0]
if nr_index.size == 0:
# nonreject is empty, all rejected
notrejectmin = len(pvals)
else:
notrejectmin = np.min(nr_index)
notreject[notrejectmin:] = True
reject = ~notreject
del notreject
# It's eqivalent to 1 - np.power((1. - pvals),
# np.arange(ntests, 0, -1))
# but prevents the issue of the floating point precision
pvals_corrected_raw = -np.expm1(np.arange(ntests, 0, -1) *
np.log1p(-pvals))
pvals_corrected = np.maximum.accumulate(pvals_corrected_raw)
del pvals_corrected_raw
elif method.lower() in ['h', 'holm']:
notreject = pvals > alphaf / np.arange(ntests, 0, -1)
nr_index = np.nonzero(notreject)[0]
if nr_index.size == 0:
# nonreject is empty, all rejected
notrejectmin = len(pvals)
else:
notrejectmin = np.min(nr_index)
notreject[notrejectmin:] = True
reject = ~notreject
pvals_corrected_raw = pvals * np.arange(ntests, 0, -1)
pvals_corrected = np.maximum.accumulate(pvals_corrected_raw)
del pvals_corrected_raw
gc.collect()
elif method.lower() in ['sh', 'simes-hochberg']:
alphash = alphaf / np.arange(ntests, 0, -1)
reject = pvals <= alphash
rejind = np.nonzero(reject)
if rejind[0].size > 0:
rejectmax = np.max(np.nonzero(reject))
reject[:rejectmax] = True
pvals_corrected_raw = np.arange(ntests, 0, -1) * pvals
pvals_corrected = np.minimum.accumulate(pvals_corrected_raw[::-1])[::-1]
del pvals_corrected_raw
elif method.lower() in ['ho', 'hommel']:
# we need a copy because we overwrite it in a loop
a = pvals.copy()
for m in range(ntests, 1, -1):
cim = np.min(m * pvals[-m:] / np.arange(1,m+1.))
a[-m:] = np.maximum(a[-m:], cim)
a[:-m] = np.maximum(a[:-m], np.minimum(m * pvals[:-m], cim))
pvals_corrected = a
reject = a <= alphaf
elif method.lower() in ['fdr_bh', 'fdr_i', 'fdr_p', 'fdri', 'fdrp']:
# delegate, call with sorted pvals
reject, pvals_corrected = fdrcorrection(pvals, alpha=alpha,
method='indep',
is_sorted=True)
elif method.lower() in ['fdr_by', 'fdr_n', 'fdr_c', 'fdrn', 'fdrcorr']:
# delegate, call with sorted pvals
reject, pvals_corrected = fdrcorrection(pvals, alpha=alpha,
method='n',
is_sorted=True)
elif method.lower() in ['fdr_tsbky', 'fdr_2sbky', 'fdr_twostage']:
# delegate, call with sorted pvals
reject, pvals_corrected = fdrcorrection_twostage(pvals, alpha=alpha,
method='bky',
maxiter=maxiter,
is_sorted=True)[:2]
elif method.lower() in ['fdr_tsbh', 'fdr_2sbh']:
# delegate, call with sorted pvals
reject, pvals_corrected = fdrcorrection_twostage(pvals, alpha=alpha,
method='bh',
maxiter=maxiter,
is_sorted=True)[:2]
elif method.lower() in ['fdr_gbs']:
#adaptive stepdown in Gavrilov, Benjamini, Sarkar, Annals of Statistics 2009
## notreject = pvals > alphaf / np.arange(ntests, 0, -1) #alphacSidak
## notrejectmin = np.min(np.nonzero(notreject))
## notreject[notrejectmin:] = True
## reject = ~notreject
ii = np.arange(1, ntests + 1)
q = (ntests + 1. - ii)/ii * pvals / (1. - pvals)
pvals_corrected_raw = np.maximum.accumulate(q) #up requirementd
pvals_corrected = np.minimum.accumulate(pvals_corrected_raw[::-1])[::-1]
del pvals_corrected_raw
reject = pvals_corrected <= alpha
else:
raise ValueError('method not recognized')
if pvals_corrected is not None: #not necessary anymore
pvals_corrected[pvals_corrected>1] = 1
if is_sorted or returnsorted:
return reject, pvals_corrected, alphacSidak, alphacBonf
else:
pvals_corrected_ = np.empty_like(pvals_corrected)
pvals_corrected_[sortind] = pvals_corrected
del pvals_corrected
reject_ = np.empty_like(reject)
reject_[sortind] = reject
return reject_, pvals_corrected_, alphacSidak, alphacBonf
[docs]
def fdrcorrection(pvals, alpha=0.05, method='indep', is_sorted=False):
'''
pvalue correction for false discovery rate.
This covers Benjamini/Hochberg for independent or positively correlated and
Benjamini/Yekutieli for general or negatively correlated tests.
Parameters
----------
pvals : array_like, 1d
Set of p-values of the individual tests.
alpha : float, optional
Family-wise error rate. Defaults to ``0.05``.
method : {'i', 'indep', 'p', 'poscorr', 'n', 'negcorr'}, optional
Which method to use for FDR correction.
``{'i', 'indep', 'p', 'poscorr'}`` all refer to ``fdr_bh``
(Benjamini/Hochberg for independent or positively
correlated tests). ``{'n', 'negcorr'}`` both refer to ``fdr_by``
(Benjamini/Yekutieli for general or negatively correlated tests).
Defaults to ``'indep'``.
is_sorted : bool, optional
If False (default), the p_values will be sorted, but the corrected
pvalues are in the original order. If True, then it assumed that the
pvalues are already sorted in ascending order.
Returns
-------
rejected : ndarray, bool
True if a hypothesis is rejected, False if not
pvalue-corrected : ndarray
pvalues adjusted for multiple hypothesis testing to limit FDR
Notes
-----
If there is prior information on the fraction of true hypothesis, then alpha
should be set to ``alpha * m/m_0`` where m is the number of tests,
given by the p-values, and m_0 is an estimate of the true hypothesis.
(see Benjamini, Krieger and Yekuteli)
The two-step method of Benjamini, Krieger and Yekutiel that estimates the number
of false hypotheses will be available (soon).
Both methods exposed via this function (Benjamini/Hochberg, Benjamini/Yekutieli)
are also available in the function ``multipletests``, as ``method="fdr_bh"`` and
``method="fdr_by"``, respectively.
See also
--------
multipletests
'''
pvals = np.asarray(pvals)
assert pvals.ndim == 1, "pvals must be 1-dimensional, that is of shape (n,)"
if not is_sorted:
pvals_sortind = np.argsort(pvals)
pvals_sorted = np.take(pvals, pvals_sortind)
else:
pvals_sorted = pvals # alias
if method in ['i', 'indep', 'p', 'poscorr']:
ecdffactor = _ecdf(pvals_sorted)
elif method in ['n', 'negcorr']:
cm = np.sum(1./np.arange(1, len(pvals_sorted)+1)) #corrected this
ecdffactor = _ecdf(pvals_sorted) / cm
## elif method in ['n', 'negcorr']:
## cm = np.sum(np.arange(len(pvals)))
## ecdffactor = ecdf(pvals_sorted)/cm
else:
raise ValueError('only indep and negcorr implemented')
reject = pvals_sorted <= ecdffactor*alpha
if reject.any():
rejectmax = max(np.nonzero(reject)[0])
reject[:rejectmax] = True
pvals_corrected_raw = pvals_sorted / ecdffactor
pvals_corrected = np.minimum.accumulate(pvals_corrected_raw[::-1])[::-1]
del pvals_corrected_raw
pvals_corrected[pvals_corrected>1] = 1
if not is_sorted:
pvals_corrected_ = np.empty_like(pvals_corrected)
pvals_corrected_[pvals_sortind] = pvals_corrected
del pvals_corrected
reject_ = np.empty_like(reject)
reject_[pvals_sortind] = reject
return reject_, pvals_corrected_
else:
return reject, pvals_corrected
[docs]
def fdrcorrection_twostage(pvals, alpha=0.05, method='bky',
maxiter=1,
iter=None,
is_sorted=False):
'''(iterated) two stage linear step-up procedure with estimation of number of true
hypotheses
Benjamini, Krieger and Yekuteli, procedure in Definition 6
Parameters
----------
pvals : array_like
set of p-values of the individual tests.
alpha : float
error rate
method : {'bky', 'bh')
see Notes for details
* 'bky' - implements the procedure in Definition 6 of Benjamini, Krieger
and Yekuteli 2006
* 'bh' - the two stage method of Benjamini and Hochberg
maxiter : int or bool
Maximum number of iterations.
maxiter=1 (default) corresponds to the two stage method.
maxiter=-1 corresponds to full iterations which is maxiter=len(pvals).
maxiter=0 uses only a single stage fdr correction using a 'bh' or 'bky'
prior fraction of assumed true hypotheses.
Boolean maxiter is allowed for backwards compatibility with the
deprecated ``iter`` keyword.
maxiter=False is two-stage fdr (maxiter=1)
maxiter=True is full iteration (maxiter=-1 or maxiter=len(pvals))
.. versionadded:: 0.14
Replacement for ``iter`` with additional features.
iter : bool
``iter`` is deprecated use ``maxiter`` instead.
If iter is True, then only one iteration step is used, this is the
two-step method.
If iter is False, then iterations are stopped at convergence which
occurs in a finite number of steps (at most len(pvals) steps).
.. deprecated:: 0.14
Use ``maxiter`` instead of ``iter``.
Returns
-------
rejected : ndarray, bool
True if a hypothesis is rejected, False if not
pvalue-corrected : ndarray
pvalues adjusted for multiple hypotheses testing to limit FDR
m0 : int
ntest - rej, estimated number of true (not rejected) hypotheses
alpha_stages : list of floats
A list of alphas that have been used at each stage
Notes
-----
The returned corrected p-values are specific to the given alpha, they
cannot be used for a different alpha.
The returned corrected p-values are from the last stage of the fdr_bh
linear step-up procedure (fdrcorrection0 with method='indep') corrected
for the estimated fraction of true hypotheses.
This means that the rejection decision can be obtained with
``pval_corrected <= alpha``, where ``alpha`` is the original significance
level.
(Note: This has changed from earlier versions (<0.5.0) of statsmodels.)
BKY described several other multi-stage methods, which would be easy to implement.
However, in their simulation the simple two-stage method (with iter=False) was the
most robust to the presence of positive correlation
TODO: What should be returned?
'''
pvals = np.asarray(pvals)
if iter is not None:
import warnings
msg = "iter keyword is deprecated, use maxiter keyword instead."
warnings.warn(msg, FutureWarning)
if iter is False:
maxiter = 1
elif iter is True or maxiter in [-1, None] :
maxiter = len(pvals)
# otherwise we use maxiter
if not is_sorted:
pvals_sortind = np.argsort(pvals)
pvals = np.take(pvals, pvals_sortind)
ntests = len(pvals)
if method == 'bky':
fact = (1.+alpha)
alpha_prime = alpha / fact
elif method == 'bh':
fact = 1.
alpha_prime = alpha
else:
raise ValueError("only 'bky' and 'bh' are available as method")
alpha_stages = [alpha_prime]
rej, pvalscorr = fdrcorrection(pvals, alpha=alpha_prime, method='indep',
is_sorted=True)
r1 = rej.sum()
if (r1 == 0) or (r1 == ntests):
# return rej, pvalscorr * fact, ntests - r1, alpha_stages
reject = rej
pvalscorr *= fact
ri = r1
else:
ri_old = ri = r1
ntests0 = ntests # needed if maxiter=0
# while True:
for it in range(maxiter):
ntests0 = 1.0 * ntests - ri_old
alpha_star = alpha_prime * ntests / ntests0
alpha_stages.append(alpha_star)
#print ntests0, alpha_star
rej, pvalscorr = fdrcorrection(pvals, alpha=alpha_star, method='indep',
is_sorted=True)
ri = rej.sum()
if (it >= maxiter - 1) or ri == ri_old:
break
elif ri < ri_old:
# prevent cycles and endless loops
raise RuntimeError(" oops - should not be here")
ri_old = ri
# make adjustment to pvalscorr to reflect estimated number of Non-Null cases
# decision is then pvalscorr < alpha (or <=)
pvalscorr *= ntests0 * 1.0 / ntests
if method == 'bky':
pvalscorr *= (1. + alpha)
pvalscorr[pvalscorr>1] = 1
if not is_sorted:
pvalscorr_ = np.empty_like(pvalscorr)
pvalscorr_[pvals_sortind] = pvalscorr
del pvalscorr
reject = np.empty_like(rej)
reject[pvals_sortind] = rej
return reject, pvalscorr_, ntests - ri, alpha_stages
else:
return rej, pvalscorr, ntests - ri, alpha_stages
[docs]
def local_fdr(zscores, null_proportion=1.0, null_pdf=None, deg=7,
nbins=30, alpha=0):
"""
Calculate local FDR values for a list of Z-scores.
Parameters
----------
zscores : array_like
A vector of Z-scores
null_proportion : float
The assumed proportion of true null hypotheses
null_pdf : function mapping reals to positive reals
The density of null Z-scores; if None, use standard normal
deg : int
The maximum exponent in the polynomial expansion of the
density of non-null Z-scores
nbins : int
The number of bins for estimating the marginal density
of Z-scores.
alpha : float
Use Poisson ridge regression with parameter alpha to estimate
the density of non-null Z-scores.
Returns
-------
fdr : array_like
A vector of FDR values
References
----------
B Efron (2008). Microarrays, Empirical Bayes, and the Two-Groups
Model. Statistical Science 23:1, 1-22.
Examples
--------
Basic use (the null Z-scores are taken to be standard normal):
>>> from statsmodels.stats.multitest import local_fdr
>>> import numpy as np
>>> zscores = np.random.randn(30)
>>> fdr = local_fdr(zscores)
Use a Gaussian null distribution estimated from the data:
>>> null = EmpiricalNull(zscores)
>>> fdr = local_fdr(zscores, null_pdf=null.pdf)
"""
from statsmodels.genmod.generalized_linear_model import GLM
from statsmodels.genmod.generalized_linear_model import families
from statsmodels.regression.linear_model import OLS
# Bins for Poisson modeling of the marginal Z-score density
minz = min(zscores)
maxz = max(zscores)
bins = np.linspace(minz, maxz, nbins)
# Bin counts
zhist = np.histogram(zscores, bins)[0]
# Bin centers
zbins = (bins[:-1] + bins[1:]) / 2
# The design matrix at bin centers
dmat = np.vander(zbins, deg + 1)
# Rescale the design matrix
sd = dmat.std(0)
ii = sd >1e-8
dmat[:, ii] /= sd[ii]
start = OLS(np.log(1 + zhist), dmat).fit().params
# Poisson regression
if alpha > 0:
md = GLM(zhist, dmat, family=families.Poisson()).fit_regularized(L1_wt=0, alpha=alpha, start_params=start)
else:
md = GLM(zhist, dmat, family=families.Poisson()).fit(start_params=start)
# The design matrix for all Z-scores
dmat_full = np.vander(zscores, deg + 1)
dmat_full[:, ii] /= sd[ii]
# The height of the estimated marginal density of Z-scores,
# evaluated at every observed Z-score.
fz = md.predict(dmat_full) / (len(zscores) * (bins[1] - bins[0]))
# The null density.
if null_pdf is None:
f0 = np.exp(-0.5 * zscores**2) / np.sqrt(2 * np.pi)
else:
f0 = null_pdf(zscores)
# The local FDR values
fdr = null_proportion * f0 / fz
fdr = np.clip(fdr, 0, 1)
return fdr
[docs]
class NullDistribution:
"""
Estimate a Gaussian distribution for the null Z-scores.
The observed Z-scores consist of both null and non-null values.
The fitted distribution of null Z-scores is Gaussian, but may have
non-zero mean and/or non-unit scale.
Parameters
----------
zscores : array_like
The observed Z-scores.
null_lb : float
Z-scores between `null_lb` and `null_ub` are all considered to be
true null hypotheses.
null_ub : float
See `null_lb`.
estimate_mean : bool
If True, estimate the mean of the distribution. If False, the
mean is fixed at zero.
estimate_scale : bool
If True, estimate the scale of the distribution. If False, the
scale parameter is fixed at 1.
estimate_null_proportion : bool
If True, estimate the proportion of true null hypotheses (i.e.
the proportion of z-scores with expected value zero). If False,
this parameter is fixed at 1.
Attributes
----------
mean : float
The estimated mean of the empirical null distribution
sd : float
The estimated standard deviation of the empirical null distribution
null_proportion : float
The estimated proportion of true null hypotheses among all hypotheses
References
----------
B Efron (2008). Microarrays, Empirical Bayes, and the Two-Groups
Model. Statistical Science 23:1, 1-22.
Notes
-----
See also:
http://nipy.org/nipy/labs/enn.html#nipy.algorithms.statistics.empirical_pvalue.NormalEmpiricalNull.fdr
"""
def __init__(self, zscores, null_lb=-1, null_ub=1, estimate_mean=True,
estimate_scale=True, estimate_null_proportion=False):
# Extract the null z-scores
ii = np.flatnonzero((zscores >= null_lb) & (zscores <= null_ub))
if len(ii) == 0:
raise RuntimeError("No Z-scores fall between null_lb and null_ub")
zscores0 = zscores[ii]
# Number of Z-scores, and null Z-scores
n_zs, n_zs0 = len(zscores), len(zscores0)
# Unpack and transform the parameters to the natural scale, hold
# parameters fixed as specified.
def xform(params):
mean = 0.
sd = 1.
prob = 1.
ii = 0
if estimate_mean:
mean = params[ii]
ii += 1
if estimate_scale:
sd = np.exp(params[ii])
ii += 1
if estimate_null_proportion:
prob = 1 / (1 + np.exp(-params[ii]))
return mean, sd, prob
from scipy.stats.distributions import norm
def fun(params):
"""
Negative log-likelihood of z-scores.
The function has three arguments, packed into a vector:
mean : location parameter
logscale : log of the scale parameter
logitprop : logit of the proportion of true nulls
The implementation follows section 4 from Efron 2008.
"""
d, s, p = xform(params)
# Mass within the central region
central_mass = (norm.cdf((null_ub - d) / s) -
norm.cdf((null_lb - d) / s))
# Probability that a Z-score is null and is in the central region
cp = p * central_mass
# Binomial term
rval = n_zs0 * np.log(cp) + (n_zs - n_zs0) * np.log(1 - cp)
# Truncated Gaussian term for null Z-scores
zv = (zscores0 - d) / s
rval += np.sum(-zv**2 / 2) - n_zs0 * np.log(s)
rval -= n_zs0 * np.log(central_mass)
return -rval
# Estimate the parameters
from scipy.optimize import minimize
# starting values are mean = 0, scale = 1, p0 ~ 1
mz = minimize(fun, np.r_[0., 0, 3], method="Nelder-Mead")
mean, sd, prob = xform(mz['x'])
self.mean = mean
self.sd = sd
self.null_proportion = prob
# The fitted null density function
[docs]
def pdf(self, zscores):
"""
Evaluates the fitted empirical null Z-score density.
Parameters
----------
zscores : scalar or array_like
The point or points at which the density is to be
evaluated.
Returns
-------
The empirical null Z-score density evaluated at the given
points.
"""
zval = (zscores - self.mean) / self.sd
return np.exp(-0.5*zval**2 - np.log(self.sd) - 0.5*np.log(2*np.pi))
Last update:
Oct 03, 2024