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import numpy as np
import pandas as pd
from scipy.stats.distributions import chi2, norm
from statsmodels.graphics import utils
def _calc_survfunc_right(time, status, weights=None, entry=None, compress=True,
retall=True):
"""
Calculate the survival function and its standard error for a single
group.
"""
# Convert the unique times to ranks (0, 1, 2, ...)
if entry is None:
utime, rtime = np.unique(time, return_inverse=True)
else:
tx = np.concatenate((time, entry))
utime, rtime = np.unique(tx, return_inverse=True)
rtime = rtime[0:len(time)]
# Number of deaths at each unique time.
ml = len(utime)
if weights is None:
d = np.bincount(rtime, weights=status, minlength=ml)
else:
d = np.bincount(rtime, weights=status*weights, minlength=ml)
# Size of risk set just prior to each event time.
if weights is None:
n = np.bincount(rtime, minlength=ml)
else:
n = np.bincount(rtime, weights=weights, minlength=ml)
if entry is not None:
n = np.cumsum(n) - n
rentry = np.searchsorted(utime, entry, side='left')
if weights is None:
n0 = np.bincount(rentry, minlength=ml)
else:
n0 = np.bincount(rentry, weights=weights, minlength=ml)
n0 = np.cumsum(n0) - n0
n = n0 - n
else:
n = np.cumsum(n[::-1])[::-1]
# Only retain times where an event occurred.
if compress:
ii = np.flatnonzero(d > 0)
d = d[ii]
n = n[ii]
utime = utime[ii]
# The survival function probabilities.
sp = 1 - d / n.astype(np.float64)
ii = sp < 1e-16
sp[ii] = 1e-16
sp = np.log(sp)
sp = np.cumsum(sp)
sp = np.exp(sp)
sp[ii] = 0
if not retall:
return sp, utime, rtime, n, d
# Standard errors
if weights is None:
# Greenwood's formula
denom = n * (n - d)
denom = np.clip(denom, 1e-12, np.inf)
se = d / denom.astype(np.float64)
se[(n == d) | (n == 0)] = np.nan
se = np.cumsum(se)
se = np.sqrt(se)
locs = np.isfinite(se) | (sp != 0)
se[locs] *= sp[locs]
se[~locs] = np.nan
else:
# Tsiatis' (1981) formula
se = d / (n * n).astype(np.float64)
se = np.cumsum(se)
se = np.sqrt(se)
return sp, se, utime, rtime, n, d
def _calc_incidence_right(time, status, weights=None):
"""
Calculate the cumulative incidence function and its standard error.
"""
# Calculate the all-cause survival function.
status0 = (status >= 1).astype(np.float64)
sp, utime, rtime, n, d = _calc_survfunc_right(time, status0, weights,
compress=False, retall=False)
ngrp = int(status.max())
# Number of cause-specific deaths at each unique time.
d = []
for k in range(ngrp):
status0 = (status == k + 1).astype(np.float64)
if weights is None:
d0 = np.bincount(rtime, weights=status0, minlength=len(utime))
else:
d0 = np.bincount(rtime, weights=status0*weights,
minlength=len(utime))
d.append(d0)
# The cumulative incidence function probabilities.
ip = []
sp0 = np.r_[1, sp[:-1]] / n
for k in range(ngrp):
ip0 = np.cumsum(sp0 * d[k])
ip.append(ip0)
# The standard error of the cumulative incidence function.
if weights is not None:
return ip, None, utime
se = []
da = sum(d)
for k in range(ngrp):
ra = da / (n * (n - da))
v = ip[k]**2 * np.cumsum(ra)
v -= 2 * ip[k] * np.cumsum(ip[k] * ra)
v += np.cumsum(ip[k]**2 * ra)
ra = (n - d[k]) * d[k] / n
v += np.cumsum(sp0**2 * ra)
ra = sp0 * d[k] / n
v -= 2 * ip[k] * np.cumsum(ra)
v += 2 * np.cumsum(ip[k] * ra)
se.append(np.sqrt(v))
return ip, se, utime
def _checkargs(time, status, entry, freq_weights, exog):
if len(time) != len(status):
raise ValueError("time and status must have the same length")
if entry is not None and (len(entry) != len(time)):
msg = "entry times and event times must have the same length"
raise ValueError(msg)
if entry is not None and np.any(entry >= time):
msg = "Entry times must not occur on or after event times"
raise ValueError(msg)
if freq_weights is not None and (len(freq_weights) != len(time)):
raise ValueError("weights, time and status must have the same length")
if exog is not None and (exog.shape[0] != len(time)):
raise ValueError("the rows of exog should align with time")
class CumIncidenceRight:
"""
Estimation and inference for a cumulative incidence function.
If J = 1, 2, ... indicates the event type, the cumulative
incidence function for cause j is:
I(t, j) = P(T <= t and J=j)
Only right censoring is supported. If frequency weights are provided,
the point estimate is returned without a standard error.
Parameters
----------
time : array_like
An array of times (censoring times or event times)
status : array_like
If status >= 1 indicates which event occurred at time t. If
status = 0, the subject was censored at time t.
title : str
Optional title used for plots and summary output.
freq_weights : array_like
Optional frequency weights
exog : array_like
Optional, if present used to account for violation of
independent censoring.
bw_factor : float
Band-width multiplier for kernel-based estimation. Only
used if exog is provided.
dimred : bool
If True, proportional hazards regression models are used to
reduce exog to two columns by predicting overall events and
censoring in two separate models. If False, exog is used
directly for calculating kernel weights without dimension
reduction.
Attributes
----------
times : array_like
The distinct times at which the incidence rates are estimated
cinc : list of arrays
cinc[k-1] contains the estimated cumulative incidence rates
for outcome k=1,2,...
cinc_se : list of arrays
The standard errors for the values in `cinc`. Not available when
exog and/or frequency weights are provided.
Notes
-----
When exog is provided, a local estimate of the cumulative incidence
rate around each point is provided, and these are averaged to
produce an estimate of the marginal cumulative incidence
functions. The procedure is analogous to that described in Zeng
(2004) for estimation of the marginal survival function. The
approach removes bias resulting from dependent censoring when the
censoring becomes independent conditioned on the columns of exog.
References
----------
The Stata stcompet procedure:
http://www.stata-journal.com/sjpdf.html?articlenum=st0059
Dinse, G. E. and M. G. Larson. 1986. A note on semi-Markov models
for partially censored data. Biometrika 73: 379-386.
Marubini, E. and M. G. Valsecchi. 1995. Analysing Survival Data
from Clinical Trials and Observational Studies. Chichester, UK:
John Wiley & Sons.
D. Zeng (2004). Estimating marginal survival function by
adjusting for dependent censoring using many covariates. Annals
of Statistics 32:4.
https://arxiv.org/pdf/math/0409180.pdf
"""
def __init__(self, time, status, title=None, freq_weights=None,
exog=None, bw_factor=1., dimred=True):
_checkargs(time, status, None, freq_weights, None)
time = self.time = np.asarray(time)
status = self.status = np.asarray(status)
if freq_weights is not None:
freq_weights = self.freq_weights = np.asarray(freq_weights)
if exog is not None:
from ._kernel_estimates import _kernel_cumincidence
exog = self.exog = np.asarray(exog)
nobs = exog.shape[0]
kw = nobs**(-1/3.0) * bw_factor
kfunc = lambda x: np.exp(-x**2 / kw**2).sum(1)
x = _kernel_cumincidence(time, status, exog, kfunc, freq_weights,
dimred)
self.times = x[0]
self.cinc = x[1]
return
x = _calc_incidence_right(time, status, freq_weights)
self.cinc = x[0]
self.cinc_se = x[1]
self.times = x[2]
self.title = "" if not title else title
class SurvfuncRight:
"""
Estimation and inference for a survival function.
The survival function S(t) = P(T > t) is the probability that an
event time T is greater than t.
This class currently only supports right censoring.
Parameters
----------
time : array_like
An array of times (censoring times or event times)
status : array_like
Status at the event time, status==1 is the 'event'
(e.g. death, failure), meaning that the event
occurs at the given value in `time`; status==0
indicates that censoring has occurred, meaning that
the event occurs after the given value in `time`.
entry : array_like, optional An array of entry times for handling
left truncation (the subject is not in the risk set on or
before the entry time)
title : str
Optional title used for plots and summary output.
freq_weights : array_like
Optional frequency weights
exog : array_like
Optional, if present used to account for violation of
independent censoring.
bw_factor : float
Band-width multiplier for kernel-based estimation. Only used
if exog is provided.
Attributes
----------
surv_prob : array_like
The estimated value of the survivor function at each time
point in `surv_times`.
surv_prob_se : array_like
The standard errors for the values in `surv_prob`. Not available
if exog is provided.
surv_times : array_like
The points where the survival function changes.
n_risk : array_like
The number of subjects at risk just before each time value in
`surv_times`. Not available if exog is provided.
n_events : array_like
The number of events (e.g. deaths) that occur at each point
in `surv_times`. Not available if exog is provided.
Notes
-----
If exog is None, the standard Kaplan-Meier estimator is used. If
exog is not None, a local estimate of the marginal survival
function around each point is constructed, and these are then
averaged. This procedure gives an estimate of the marginal
survival function that accounts for dependent censoring as long as
the censoring becomes independent when conditioning on the
covariates in exog. See Zeng et al. (2004) for details.
References
----------
D. Zeng (2004). Estimating marginal survival function by
adjusting for dependent censoring using many covariates. Annals
of Statistics 32:4.
https://arxiv.org/pdf/math/0409180.pdf
"""
def __init__(self, time, status, entry=None, title=None,
freq_weights=None, exog=None, bw_factor=1.):
_checkargs(time, status, entry, freq_weights, exog)
time = self.time = np.asarray(time)
status = self.status = np.asarray(status)
if freq_weights is not None:
freq_weights = self.freq_weights = np.asarray(freq_weights)
if entry is not None:
entry = self.entry = np.asarray(entry)
if exog is not None:
if entry is not None:
raise ValueError("exog and entry cannot both be present")
from ._kernel_estimates import _kernel_survfunc
exog = self.exog = np.asarray(exog)
nobs = exog.shape[0]
kw = nobs**(-1/3.0) * bw_factor
kfunc = lambda x: np.exp(-x**2 / kw**2).sum(1)
x = _kernel_survfunc(time, status, exog, kfunc, freq_weights)
self.surv_prob = x[0]
self.surv_times = x[1]
return
x = _calc_survfunc_right(time, status, weights=freq_weights,
entry=entry)
self.surv_prob = x[0]
self.surv_prob_se = x[1]
self.surv_times = x[2]
self.n_risk = x[4]
self.n_events = x[5]
self.title = "" if not title else title
def plot(self, ax=None):
"""
Plot the survival function.
Examples
--------
Change the line color:
>>> import statsmodels.api as sm
>>> data = sm.datasets.get_rdataset("flchain", "survival").data
>>> df = data.loc[data.sex == "F", :]
>>> sf = sm.SurvfuncRight(df["futime"], df["death"])
>>> fig = sf.plot()
>>> ax = fig.get_axes()[0]
>>> li = ax.get_lines()
>>> li[0].set_color('purple')
>>> li[1].set_color('purple')
Do not show the censoring points:
>>> fig = sf.plot()
>>> ax = fig.get_axes()[0]
>>> li = ax.get_lines()
>>> li[1].set_visible(False)
"""
return plot_survfunc(self, ax)
def quantile(self, p):
"""
Estimated quantile of a survival distribution.
Parameters
----------
p : float
The probability point at which the quantile
is determined.
Returns the estimated quantile.
"""
# SAS uses a strict inequality here.
ii = np.flatnonzero(self.surv_prob < 1 - p)
if len(ii) == 0:
return np.nan
return self.surv_times[ii[0]]
def quantile_ci(self, p, alpha=0.05, method='cloglog'):
"""
Returns a confidence interval for a survival quantile.
Parameters
----------
p : float
The probability point for which a confidence interval is
determined.
alpha : float
The confidence interval has nominal coverage probability
1 - `alpha`.
method : str
Function to use for g-transformation, must be ...
Returns
-------
lb : float
The lower confidence limit.
ub : float
The upper confidence limit.
Notes
-----
The confidence interval is obtained by inverting Z-tests. The
limits of the confidence interval will always be observed
event times.
References
----------
The method is based on the approach used in SAS, documented here:
http://support.sas.com/documentation/cdl/en/statug/68162/HTML/default/viewer.htm#statug_lifetest_details03.htm
"""
tr = norm.ppf(1 - alpha / 2)
method = method.lower()
if method == "cloglog":
g = lambda x: np.log(-np.log(x))
gprime = lambda x: -1 / (x * np.log(x))
elif method == "linear":
g = lambda x: x
gprime = lambda x: 1
elif method == "log":
g = np.log
gprime = lambda x: 1 / x
elif method == "logit":
g = lambda x: np.log(x / (1 - x))
gprime = lambda x: 1 / (x * (1 - x))
elif method == "asinsqrt":
g = lambda x: np.arcsin(np.sqrt(x))
gprime = lambda x: 1 / (2 * np.sqrt(x) * np.sqrt(1 - x))
else:
raise ValueError("unknown method")
r = g(self.surv_prob) - g(1 - p)
r /= (gprime(self.surv_prob) * self.surv_prob_se)
ii = np.flatnonzero(np.abs(r) <= tr)
if len(ii) == 0:
return np.nan, np.nan
lb = self.surv_times[ii[0]]
if ii[-1] == len(self.surv_times) - 1:
ub = np.inf
else:
ub = self.surv_times[ii[-1] + 1]
return lb, ub
def summary(self):
"""
Return a summary of the estimated survival function.
The summary is a dataframe containing the unique event times,
estimated survival function values, and related quantities.
"""
df = pd.DataFrame(index=self.surv_times)
df.index.name = "Time"
df["Surv prob"] = self.surv_prob
df["Surv prob SE"] = self.surv_prob_se
df["num at risk"] = self.n_risk
df["num events"] = self.n_events
return df
def simultaneous_cb(self, alpha=0.05, method="hw", transform="log"):
"""
Returns a simultaneous confidence band for the survival function.
Parameters
----------
alpha : float
`1 - alpha` is the desired simultaneous coverage
probability for the confidence region. Currently alpha
must be set to 0.05, giving 95% simultaneous intervals.
method : str
The method used to produce the simultaneous confidence
band. Only the Hall-Wellner (hw) method is currently
implemented.
transform : str
The used to produce the interval (note that the returned
interval is on the survival probability scale regardless
of which transform is used). Only `log` and `arcsin` are
implemented.
Returns
-------
lcb : array_like
The lower confidence limits corresponding to the points
in `surv_times`.
ucb : array_like
The upper confidence limits corresponding to the points
in `surv_times`.
"""
method = method.lower()
if method != "hw":
msg = "only the Hall-Wellner (hw) method is implemented"
raise ValueError(msg)
if alpha != 0.05:
raise ValueError("alpha must be set to 0.05")
transform = transform.lower()
s2 = self.surv_prob_se**2 / self.surv_prob**2
nn = self.n_risk
if transform == "log":
denom = np.sqrt(nn) * np.log(self.surv_prob)
theta = 1.3581 * (1 + nn * s2) / denom
theta = np.exp(theta)
lcb = self.surv_prob**(1/theta)
ucb = self.surv_prob**theta
elif transform == "arcsin":
k = 1.3581
k *= (1 + nn * s2) / (2 * np.sqrt(nn))
k *= np.sqrt(self.surv_prob / (1 - self.surv_prob))
f = np.arcsin(np.sqrt(self.surv_prob))
v = np.clip(f - k, 0, np.inf)
lcb = np.sin(v)**2
v = np.clip(f + k, -np.inf, np.pi/2)
ucb = np.sin(v)**2
else:
raise ValueError("Unknown transform")
return lcb, ucb
def survdiff(time, status, group, weight_type=None, strata=None,
entry=None, **kwargs):
"""
Test for the equality of two survival distributions.
Parameters
----------
time : array_like
The event or censoring times.
status : array_like
The censoring status variable, status=1 indicates that the
event occurred, status=0 indicates that the observation was
censored.
group : array_like
Indicators of the two groups
weight_type : str
The following weight types are implemented:
None (default) : logrank test
fh : Fleming-Harrington, weights by S^(fh_p),
requires exponent fh_p to be provided as keyword
argument; the weights are derived from S defined at
the previous event time, and the first weight is
always 1.
gb : Gehan-Breslow, weights by the number at risk
tw : Tarone-Ware, weights by the square root of the number
at risk
strata : array_like
Optional stratum indicators for a stratified test
entry : array_like
Entry times to handle left truncation. The subject is not in
the risk set on or before the entry time.
Returns
-------
chisq : The chi-square (1 degree of freedom) distributed test
statistic value
pvalue : The p-value for the chi^2 test
"""
time = np.asarray(time)
status = np.asarray(status)
group = np.asarray(group)
gr = np.unique(group)
if strata is None:
obs, var = _survdiff(time, status, group, weight_type, gr,
entry, **kwargs)
else:
strata = np.asarray(strata)
stu = np.unique(strata)
obs, var = 0., 0.
for st in stu:
# could be more efficient?
ii = (strata == st)
obs1, var1 = _survdiff(time[ii], status[ii], group[ii],
weight_type, gr, entry, **kwargs)
obs += obs1
var += var1
chisq = obs.dot(np.linalg.solve(var, obs)) # (O - E).T * V^(-1) * (O - E)
pvalue = 1 - chi2.cdf(chisq, len(gr)-1)
return chisq, pvalue
def _survdiff(time, status, group, weight_type, gr, entry=None,
**kwargs):
# logrank test for one stratum
# calculations based on https://web.stanford.edu/~lutian/coursepdf/unit6.pdf
# formula for variance better to take from https://web.stanford.edu/~lutian/coursepdf/survweek3.pdf
# Get the unique times.
if entry is None:
utimes, rtimes = np.unique(time, return_inverse=True)
else:
utimes, rtimes = np.unique(np.concatenate((time, entry)),
return_inverse=True)
rtimes = rtimes[0:len(time)]
# Split entry times by group if present (should use pandas groupby)
tse = [(gr_i, None) for gr_i in gr]
if entry is not None:
for k, _ in enumerate(gr):
ii = (group == gr[k])
entry1 = entry[ii]
tse[k] = (gr[k], entry1)
# Event count and risk set size at each time point, per group and overall.
# TODO: should use Pandas groupby
nrisk, obsv = [], []
ml = len(utimes)
for g, entry0 in tse:
mk = (group == g)
n = np.bincount(rtimes, weights=mk, minlength=ml)
ob = np.bincount(rtimes, weights=status*mk, minlength=ml)
obsv.append(ob)
if entry is not None:
n = np.cumsum(n) - n
rentry = np.searchsorted(utimes, entry0, side='left')
n0 = np.bincount(rentry, minlength=ml)
n0 = np.cumsum(n0) - n0
nr = n0 - n
else:
nr = np.cumsum(n[::-1])[::-1]
nrisk.append(nr)
obs = sum(obsv)
nrisk_tot = sum(nrisk)
ix = np.flatnonzero(nrisk_tot > 1)
weights = None
if weight_type is not None:
weight_type = weight_type.lower()
if weight_type == "gb":
weights = nrisk_tot
elif weight_type == "tw":
weights = np.sqrt(nrisk_tot)
elif weight_type == "fh":
if "fh_p" not in kwargs:
msg = "weight_type type 'fh' requires specification of fh_p"
raise ValueError(msg)
fh_p = kwargs["fh_p"]
# Calculate the survivor function directly to avoid the
# overhead of creating a SurvfuncRight object
sp = 1 - obs / nrisk_tot.astype(np.float64)
sp = np.log(sp)
sp = np.cumsum(sp)
sp = np.exp(sp)
weights = sp**fh_p
weights = np.roll(weights, 1)
weights[0] = 1
else:
raise ValueError("weight_type not implemented")
dfs = len(gr) - 1
r = np.vstack(nrisk) / np.clip(nrisk_tot, 1e-10, np.inf)[None, :] # each line is timeseries of r's. line per group
# The variance of event counts in each group.
groups_oe = []
groups_var = []
var_denom = nrisk_tot - 1
var_denom = np.clip(var_denom, 1e-10, np.inf)
# use the first group as a reference
for g in range(1, dfs+1):
# Difference between observed and expected number of events in the group #g
oe = obsv[g] - r[g]*obs
# build one row of the dfs x dfs variance matrix
var_tensor_part = r[1:, :].T * (np.eye(1, dfs, g-1).ravel() - r[g, :, None]) # r*(1 - r) in multidim
var_scalar_part = obs * (nrisk_tot - obs) / var_denom
var = var_tensor_part * var_scalar_part[:, None]
if weights is not None:
oe = weights * oe
var = (weights**2)[:, None] * var
# sum over times and store
groups_oe.append(oe[ix].sum())
groups_var.append(var[ix].sum(axis=0))
obs_vec = np.hstack(groups_oe)
var_mat = np.vstack(groups_var)
return obs_vec, var_mat
def plot_survfunc(survfuncs, ax=None):
"""
Plot one or more survivor functions.
Parameters
----------
survfuncs : object or array_like
A single SurvfuncRight object, or a list or SurvfuncRight
objects that are plotted together.
Returns
-------
A figure instance on which the plot was drawn.
Examples
--------
Add a legend:
>>> import statsmodels.api as sm
>>> from statsmodels.duration.survfunc import plot_survfunc
>>> data = sm.datasets.get_rdataset("flchain", "survival").data
>>> df = data.loc[data.sex == "F", :]
>>> sf0 = sm.SurvfuncRight(df["futime"], df["death"])
>>> sf1 = sm.SurvfuncRight(3.0 * df["futime"], df["death"])
>>> fig = plot_survfunc([sf0, sf1])
>>> ax = fig.get_axes()[0]
>>> ax.set_position([0.1, 0.1, 0.64, 0.8])
>>> ha, lb = ax.get_legend_handles_labels()
>>> leg = fig.legend((ha[0], ha[1]), (lb[0], lb[1]), loc='center right')
Change the line colors:
>>> fig = plot_survfunc([sf0, sf1])
>>> ax = fig.get_axes()[0]
>>> ax.set_position([0.1, 0.1, 0.64, 0.8])
>>> ha, lb = ax.get_legend_handles_labels()
>>> ha[0].set_color('purple')
>>> ha[1].set_color('orange')
"""
fig, ax = utils.create_mpl_ax(ax)
# If we have only a single survival function to plot, put it into
# a list.
try:
assert type(survfuncs[0]) is SurvfuncRight
except:
survfuncs = [survfuncs]
for gx, sf in enumerate(survfuncs):
# The estimated survival function does not include a point at
# time 0, include it here for plotting.
surv_times = np.concatenate(([0], sf.surv_times))
surv_prob = np.concatenate(([1], sf.surv_prob))
# If the final times are censoring times they are not included
# in the survival function so we add them here
mxt = max(sf.time)
if mxt > surv_times[-1]:
surv_times = np.concatenate((surv_times, [mxt]))
surv_prob = np.concatenate((surv_prob, [surv_prob[-1]]))
label = getattr(sf, "title", "Group %d" % (gx + 1))
li, = ax.step(surv_times, surv_prob, '-', label=label, lw=2,
where='post')
# Plot the censored points.
ii = np.flatnonzero(np.logical_not(sf.status))
ti = np.unique(sf.time[ii])
jj = np.searchsorted(surv_times, ti) - 1
sp = surv_prob[jj]
ax.plot(ti, sp, '+', ms=12, color=li.get_color(),
label=label + " points")
ax.set_ylim(0, 1.01)
return fig