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Module « scipy.stats »

Fonction logrank - module scipy.stats

Signature de la fonction logrank

def logrank(x: 'npt.ArrayLike | CensoredData', y: 'npt.ArrayLike | CensoredData', alternative: Literal['two-sided', 'less', 'greater'] = 'two-sided') -> scipy.stats._survival.LogRankResult 

Description

help(scipy.stats.logrank)

Compare the survival distributions of two samples via the logrank test.

Parameters
----------
x, y : array_like or CensoredData
    Samples to compare based on their empirical survival functions.
alternative : {'two-sided', 'less', 'greater'}, optional
    Defines the alternative hypothesis.

    The null hypothesis is that the survival distributions of the two
    groups, say *X* and *Y*, are identical.

    The following alternative hypotheses [4]_ are available (default is
    'two-sided'):

    * 'two-sided': the survival distributions of the two groups are not
      identical.
    * 'less': survival of group *X* is favored: the group *X* failure rate
      function is less than the group *Y* failure rate function at some
      times.
    * 'greater': survival of group *Y* is favored: the group *X* failure
      rate function is greater than the group *Y* failure rate function at
      some times.

Returns
-------
res : `~scipy.stats._result_classes.LogRankResult`
    An object containing attributes:

    statistic : float ndarray
        The computed statistic (defined below). Its magnitude is the
        square root of the magnitude returned by most other logrank test
        implementations.
    pvalue : float ndarray
        The computed p-value of the test.

See Also
--------
scipy.stats.ecdf

Notes
-----
The logrank test [1]_ compares the observed number of events to
the expected number of events under the null hypothesis that the two
samples were drawn from the same distribution. The statistic is

.. math::

    Z_i = \frac{\sum_{j=1}^J(O_{i,j}-E_{i,j})}{\sqrt{\sum_{j=1}^J V_{i,j}}}
    \rightarrow \mathcal{N}(0,1)

where

.. math::

    E_{i,j} = O_j \frac{N_{i,j}}{N_j},
    \qquad
    V_{i,j} = E_{i,j} \left(\frac{N_j-O_j}{N_j}\right)
    \left(\frac{N_j-N_{i,j}}{N_j-1}\right),

:math:`i` denotes the group (i.e. it may assume values :math:`x` or
:math:`y`, or it may be omitted to refer to the combined sample)
:math:`j` denotes the time (at which an event occurred),
:math:`N` is the number of subjects at risk just before an event occurred,
and :math:`O` is the observed number of events at that time.

The ``statistic`` :math:`Z_x` returned by `logrank` is the (signed) square
root of the statistic returned by many other implementations. Under the
null hypothesis, :math:`Z_x**2` is asymptotically distributed according to
the chi-squared distribution with one degree of freedom. Consequently,
:math:`Z_x` is asymptotically distributed according to the standard normal
distribution. The advantage of using :math:`Z_x` is that the sign
information (i.e. whether the observed number of events tends to be less
than or greater than the number expected under the null hypothesis) is
preserved, allowing `scipy.stats.logrank` to offer one-sided alternative
hypotheses.

References
----------
.. [1] Mantel N. "Evaluation of survival data and two new rank order
       statistics arising in its consideration."
       Cancer Chemotherapy Reports, 50(3):163-170, PMID: 5910392, 1966
.. [2] Bland, Altman, "The logrank test", BMJ, 328:1073,
       :doi:`10.1136/bmj.328.7447.1073`, 2004
.. [3] "Logrank test", Wikipedia,
       https://en.wikipedia.org/wiki/Logrank_test
.. [4] Brown, Mark. "On the choice of variance for the log rank test."
       Biometrika 71.1 (1984): 65-74.
.. [5] Klein, John P., and Melvin L. Moeschberger. Survival analysis:
       techniques for censored and truncated data. Vol. 1230. New York:
       Springer, 2003.

Examples
--------
Reference [2]_ compared the survival times of patients with two different
types of recurrent malignant gliomas. The samples below record the time
(number of weeks) for which each patient participated in the study. The
`scipy.stats.CensoredData` class is used because the data is
right-censored: the uncensored observations correspond with observed deaths
whereas the censored observations correspond with the patient leaving the
study for another reason.

>>> from scipy import stats
>>> x = stats.CensoredData(
...     uncensored=[6, 13, 21, 30, 37, 38, 49, 50,
...                 63, 79, 86, 98, 202, 219],
...     right=[31, 47, 80, 82, 82, 149]
... )
>>> y = stats.CensoredData(
...     uncensored=[10, 10, 12, 13, 14, 15, 16, 17, 18, 20, 24, 24,
...                 25, 28,30, 33, 35, 37, 40, 40, 46, 48, 76, 81,
...                 82, 91, 112, 181],
...     right=[34, 40, 70]
... )

We can calculate and visualize the empirical survival functions
of both groups as follows.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> ax = plt.subplot()
>>> ecdf_x = stats.ecdf(x)
>>> ecdf_x.sf.plot(ax, label='Astrocytoma')
>>> ecdf_y = stats.ecdf(y)
>>> ecdf_y.sf.plot(ax, label='Glioblastoma')
>>> ax.set_xlabel('Time to death (weeks)')
>>> ax.set_ylabel('Empirical SF')
>>> plt.legend()
>>> plt.show()

Visual inspection of the empirical survival functions suggests that the
survival times tend to be different between the two groups. To formally
assess whether the difference is significant at the 1% level, we use the
logrank test.

>>> res = stats.logrank(x=x, y=y)
>>> res.statistic
-2.73799
>>> res.pvalue
0.00618

The p-value is less than 1%, so we can consider the data to be evidence
against the null hypothesis in favor of the alternative that there is a
difference between the two survival functions.

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