def BrownForsytheTest(x, y):
# Data analysis
K = 2
Ni = np.array([len(x), len(y)])
N = Ni.sum()
# Transform data
Zi = np.abs(np.array([x - np.median(x), y - np.median(y)]))
Z_Bar = np.concatenate([Zi[0], Zi[1]]).mean()
Zj = np.array([Zi[0].mean(), Zi[1].mean()])
# Compute test result
Nominator, Denominator = 0, 0
for i in range(K):
Nominator += Ni[i] * (Zj[i] - Z_Bar)**2
for j in range(Ni[i]):
Denominator += (Zi[i][j] - Zj[i])**2
W = (N - K) / (K - 1) * Nominator / Denominator
# Compute p value
p = 1 - f.cdf(W, K - 1, N - K)
return W, p
def gelman_rubin(chains, return_cdf=False):
"""
Compute the Gelman-Rubin R-statistic from an ensemble of chains. `chains`
is expected to have shape `(nsteps, nchains)` if samples are one dimensional,
or `(nsteps, nchains, ndim)` if multidimensional. For multidimensional samples
R-statistics will be computed for each dimension.
:param chains:
An `(nsteps, nchains)` or `(nsteps, nchains, ndim)`-shaped array.
:param return_cdf: (optional)
If ``True``, the CDF of the R-statistic(s), assuming an F-distribution, are
returned in addition to the R-statistic(s).
"""
if len(chains.shape) > 2:
results = [
gelman_rubin(chains[..., param], return_cdf=return_cdf)
for param in range(chains.shape[-1])
]
if return_cdf:
return zip(*results)
else:
return results
nchains, nsteps = chains.shape[1], chains.shape[0]
chain_means = np.mean(chains, axis=0)
chain_vars = np.var(chains, axis=0)
# between-chain variance
interchain_var = np.sum((chain_means - np.mean(chains))**2) / (nchains - 1)
# within-chain variances
intrachain_vars = (chains - chain_means)**2 / (nsteps - 1)
intrachain_var = np.sum(intrachain_vars) / nchains
var = intrachain_var * (nsteps - 1) / nsteps + interchain_var
post_var = var + interchain_var / nchains
# The Statistic
R = np.sqrt(post_var / intrachain_var)
if return_cdf:
# R should be F-distributed
dof1 = nchains - 1
dof2 = 2 * intrachain_var**2 * nchains / np.var(intrachain_vars)
return R, f.cdf(R, dof1, dof2)
else:
return R
def gelman_rubin(chains, return_cdf=False):
"""
Compute the Gelman-Rubin R-statistic from an ensemble of chains. `chains`
is expected to have shape `(nsteps, nchains)` if samples are one dimensional,
or `(nsteps, nchains, ndim)` if multidimensional. For multidimensional samples
R-statistics will be computed for each dimension.
:param chains:
An `(nsteps, nchains)` or `(nsteps, nchains, ndim)`-shaped array.
:param return_cdf: (optional)
If ``True``, the CDF of the R-statistic(s), assuming an F-distribution, are
returned in addition to the R-statistic(s).
"""
if len(chains.shape) > 2:
results = [gelman_rubin(chains[..., param], return_cdf=return_cdf)
for param in range(chains.shape[-1])]
if return_cdf:
return zip(*results)
else:
return results
nchains, nsteps = chains.shape[1], chains.shape[0]
chain_means = np.mean(chains, axis=0)
chain_vars = np.var(chains, axis=0)
# between-chain variance
interchain_var = np.sum((chain_means - np.mean(chains)) ** 2) / (nchains - 1)
# within-chain variances
intrachain_vars = (chains - chain_means)**2 / (nsteps - 1)
intrachain_var = np.sum(intrachain_vars)/nchains
var = intrachain_var * (nsteps - 1) / nsteps + interchain_var
post_var = var + interchain_var / nchains
# The Statistic
R = np.sqrt(post_var / intrachain_var)
if return_cdf:
# R should be F-distributed
dof1 = nchains - 1
dof2 = 2*intrachain_var**2*nchains/np.var(intrachain_vars)
return R, f.cdf(R, dof1, dof2)
else:
return R
def ANOVA(tab):
"""
ANOVA test for table tab (tab should be a list of lists).
Values returned are in order:
Fs: Value for the variable F (F of snedecor).
glentre: degrees of freedom in between.
gldentro: degrees of fredom inside.
1-fsnede: left tail (p-value for the test).
"""
r = len(tab)
ni = [len(ele) for ele in tab]
xbi = [stats(ele)[0] for ele in tab]
N = sum(ni)
XB = sum([ni[ii]*xbi[ii] for ii in xrange(r)])/N
ssi = [sum([(ele - xbi[ii])**2 for ele in ele2]) for ii, ele2 in enumerate(tab)]
SSdentro = sum(ssi)
gldentro = N-r
MSdentro = SSdentro/gldentro
SSentre = sum([ni[ii]*(ele-XB)**2 for ii, ele in enumerate(xbi)])
glentre = r-1
MSentre = SSentre/glentre
Fs = MSentre/MSdentro
return [Fs, glentre, gldentro, 1.-fsnede.cdf(Fs,glentre,gldentro)]
def predict_functional(result,
focus_var,
summaries=None,
values=None,
summaries2=None,
values2=None,
alpha=0.05,
ci_method="pointwise",
linear=True,
num_points=10,
exog=None,
exog2=None,
**kwargs):
if ci_method not in ("pointwise", "scheffe", "simultaneous"):
raise ValueError('confidence band method must be one of '
'`pointwise`, `scheffe`, and `simultaneous`.')
contrast = (values2 is not None) or (summaries2 is not None)
if contrast and not linear:
raise ValueError("`linear` must be True for computing contrasts")
model = result.model
if exog is not None:
if any(x is not None
for x in [summaries, summaries2, values, values2]):
raise ValueError("if `exog` is provided then do not "
"provide `summaries` or `values`")
fexog = exog
dexog = patsy.dmatrix(model.data.design_info,
fexog,
return_type='dataframe')
fvals = exog[focus_var]
if exog2 is not None:
fexog2 = exog
dexog2 = patsy.dmatrix(model.data.design_info,
fexog2,
return_type='dataframe')
fvals2 = fvals
else:
values, summaries, values2, summaries2 = _check_args(
values, summaries, values2, summaries2)
dexog, fexog, fvals = _make_exog(result, focus_var, summaries, values,
num_points)
if len(summaries2) + len(values2) > 0:
dexog2, fexog2, fvals2 = _make_exog(result, focus_var, summaries2,
values2, num_points)
from statsmodels.genmod.generalized_linear_model import GLM
from statsmodels.genmod.generalized_estimating_equations import GEE
if isinstance(result.model, (GLM, GEE)):
kwargs_pred = kwargs.copy()
kwargs_pred.update({"linear": True})
else:
kwargs_pred = kwargs
pred = result.predict(exog=fexog, **kwargs_pred)
if contrast:
pred2 = result.predict(exog=fexog2, **kwargs_pred)
pred = pred - pred2
dexog = dexog - dexog2
if ci_method == 'pointwise':
t_test = result.t_test(dexog)
cb = t_test.conf_int(alpha=alpha)
elif ci_method == 'scheffe':
t_test = result.t_test(dexog)
sd = t_test.sd
cb = np.zeros((num_points, 2))
# Scheffe's method
from scipy.stats.distributions import f as fdist
df1 = result.model.exog.shape[1]
df2 = result.model.exog.shape[0] - df1
qf = fdist.cdf(1 - alpha, df1, df2)
fx = sd * np.sqrt(df1 * qf)
cb[:, 0] = pred - fx
cb[:, 1] = pred + fx
elif ci_method == 'simultaneous':
sigma, c = _glm_basic_scr(result, dexog, alpha)
cb = np.zeros((dexog.shape[0], 2))
cb[:, 0] = pred - c * sigma
cb[:, 1] = pred + c * sigma
if not linear:
# May need to support other models with link-like functions.
link = result.family.link
pred = link.inverse(pred)
cb = link.inverse(cb)
return pred, cb, fvals
def predict_functional(result, focus_var, summaries=None, values=None,
summaries2=None, values2=None, alpha=0.05,
ci_method="pointwise", linear=True, num_points=10,
exog=None, exog2=None, **kwargs):
# docstring attached below
if ci_method not in ("pointwise", "scheffe", "simultaneous"):
raise ValueError('confidence band method must be one of `pointwise`, `scheffe`, and `simultaneous`.')
contrast = (values2 is not None) or (summaries2 is not None)
if contrast and not linear:
raise ValueError("`linear` must be True for computing contrasts")
model = result.model
if exog is not None:
if any(x is not None for x in [summaries, summaries2, values, values2]):
raise ValueError("if `exog` is provided then do not provide `summaries` or `values`")
fexog = exog
dexog = patsy.dmatrix(model.data.design_info.builder,
fexog, return_type='dataframe')
fvals = exog[focus_var]
if exog2 is not None:
fexog2 = exog
dexog2 = patsy.dmatrix(model.data.design_info.builder,
fexog2, return_type='dataframe')
fvals2 = fvals
else:
values, summaries, values2, summaries2 = _check_args(values,
summaries, values2, summaries2)
dexog, fexog, fvals = _make_exog(result, focus_var, summaries, values, num_points)
if len(summaries2) + len(values2) > 0:
dexog2, fexog2, fvals2 = _make_exog(result, focus_var, summaries2, values2, num_points)
from statsmodels.genmod.generalized_linear_model import GLM
from statsmodels.genmod.generalized_estimating_equations import GEE
if isinstance(result.model, (GLM, GEE)):
kwargs_pred = kwargs.copy()
kwargs_pred.update({"linear": True})
else:
kwargs_pred = kwargs
pred = result.predict(exog=fexog, **kwargs_pred)
if contrast:
pred2 = result.predict(exog=fexog2, **kwargs_pred)
pred = pred - pred2
dexog = dexog - dexog2
if ci_method == 'pointwise':
t_test = result.t_test(dexog)
cb = t_test.conf_int(alpha=alpha)
elif ci_method == 'scheffe':
t_test = result.t_test(dexog)
sd = t_test.sd
cb = np.zeros((num_points, 2))
# Scheffe's method
from scipy.stats.distributions import f as fdist
df1 = result.model.exog.shape[1]
df2 = result.model.exog.shape[0] - df1
qf = fdist.cdf(1 - alpha, df1, df2)
fx = sd * np.sqrt(df1 * qf)
cb[:, 0] = pred - fx
cb[:, 1] = pred + fx
elif ci_method == 'simultaneous':
sigma, c = _glm_basic_scr(result, dexog, alpha)
cb = np.zeros((dexog.shape[0], 2))
cb[:, 0] = pred - c*sigma
cb[:, 1] = pred + c*sigma
if not linear:
# May need to support other models with link-like functions.
link = result.family.link
pred = link.inverse(pred)
cb = link.inverse(cb)
return pred, cb, fvals