Source code for statsmodels.duration.survfunc

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 occured.
    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(object):
    """
    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 occured at time t.  If
        status = 0, the subject was censored at time t.
    title : string
        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 : boolean
        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


[docs]class SurvfuncRight(object): """ 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 occured, 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 : string 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
[docs] 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') Don't 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)
[docs] 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]]
[docs] 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 : string 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 = lambda x: np.log(x) 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
[docs] def summary(self): """ Return a summary of the estimated survival function. The summary is a datafram 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
[docs] 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 : string The method used to produce the simultaneous confidence band. Only the Hall-Wellner (hw) method is currently implemented. transform : string 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 occured, status=0 indicates that the observation was censored. group : array_like Indicators of the two groups weight_type : string 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 """ # TODO: extend to handle more than two groups time = np.asarray(time) status = np.asarray(status) group = np.asarray(group) gr = np.unique(group) if len(gr) != 2: raise ValueError("logrank only supports two groups") 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 zstat = obs / np.sqrt(var) # The chi^2 test statistic and p-value. chisq = zstat**2 pvalue = 1 - chi2.cdf(chisq, 1) return chisq, pvalue def _survdiff(time, status, group, weight_type, gr, entry=None, **kwargs): # logrank test for one stratum # 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[0], None), (gr[1], None)] if entry is not None: for k in 0, 1: 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) # The variance of event counts in the first group. r = nrisk[0] / np.clip(nrisk_tot, 1e-10, np.inf) denom = nrisk_tot - 1 denom = np.clip(denom, 1e-10, np.inf) var = obs * r * (1 - r) * (nrisk_tot - obs) / denom # The expected number of events in the first group. exp1 = obs * r 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") # The Z-scale test statistic (compare to normal reference # distribution). ix = np.flatnonzero(nrisk_tot > 1) if weights is None: obs = np.sum(obsv[0][ix] - exp1[ix]) var = np.sum(var[ix]) else: obs = np.dot(weights[ix], obsv[0][ix] - exp1[ix]) var = np.dot(weights[ix]**2, var[ix]) return obs, var 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]), '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 = 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