class scipy.stats.CensoredData(uncensored=None, *, left=None, right=None, interval=None)[source]#

Instances of this class represent censored data.

Instances may be passed to the fit method of continuous univariate SciPy distributions for maximum likelihood estimation. The only method of the univariate continuous distributions that understands CensoredData is the fit method. An instance of CensoredData can not be passed to methods such as pdf and cdf.

An observation is said to be censored when the precise value is unknown, but it has a known upper and/or lower bound. The conventional terminology is:

  • left-censored: an observation is below a certain value but it is unknown by how much.

  • right-censored: an observation is above a certain value but it is unknown by how much.

  • interval-censored: an observation lies somewhere on an interval between two values.

Left-, right-, and interval-censored data can be represented by CensoredData.

For convenience, the class methods left_censored and right_censored are provided to create a CensoredData instance from a single one-dimensional array of measurements and a corresponding boolean array to indicate which measurements are censored. The class method interval_censored accepts two one-dimensional arrays that hold the lower and upper bounds of the intervals.

uncensoredarray_like, 1D

Uncensored observations.

leftarray_like, 1D

Left-censored observations.

rightarray_like, 1D

Right-censored observations.

intervalarray_like, 2D, with shape (m, 2)

Interval-censored observations. Each row interval[k, :] represents the interval for the kth interval-censored observation.


In the input array interval, the lower bound of the interval may be -inf, and the upper bound may be inf, but at least one must be finite. When the lower bound is -inf, the row represents a left- censored observation, and when the upper bound is inf, the row represents a right-censored observation. If the length of an interval is 0 (i.e. interval[k, 0] == interval[k, 1], the observation is treated as uncensored. So one can represent all the types of censored and uncensored data in interval, but it is generally more convenient to use uncensored, left and right for uncensored, left-censored and right-censored observations, respectively.


In the most general case, a censored data set may contain values that are left-censored, right-censored, interval-censored, and uncensored. For example, here we create a data set with five observations. Two are uncensored (values 1 and 1.5), one is a left-censored observation of 0, one is a right-censored observation of 10 and one is interval-censored in the interval [2, 3].

>>> import numpy as np
>>> from scipy.stats import CensoredData
>>> data = CensoredData(uncensored=[1, 1.5], left=[0], right=[10],
...                     interval=[[2, 3]])
>>> print(data)
CensoredData(5 values: 2 not censored, 1 left-censored,
1 right-censored, 1 interval-censored)


>>> data = CensoredData(interval=[[1, 1],
...                               [1.5, 1.5],
...                               [-np.inf, 0],
...                               [10, np.inf],
...                               [2, 3]])
>>> print(data)
CensoredData(5 values: 2 not censored, 1 left-censored,
1 right-censored, 1 interval-censored)

A common case is to have a mix of uncensored observations and censored observations that are all right-censored (or all left-censored). For example, consider an experiment in which six devices are started at various times and left running until they fail. Assume that time is measured in hours, and the experiment is stopped after 30 hours, even if all the devices have not failed by that time. We might end up with data such as this:

Device  Start-time  Fail-time  Time-to-failure
   1         0         13           13
   2         2         24           22
   3         5         22           17
   4         8         23           15
   5        10        ***          >20
   6        12        ***          >18

Two of the devices had not failed when the experiment was stopped; the observations of the time-to-failure for these two devices are right-censored. We can represent this data with

>>> data = CensoredData(uncensored=[13, 22, 17, 15], right=[20, 18])
>>> print(data)
CensoredData(6 values: 4 not censored, 2 right-censored)

Alternatively, we can use the method CensoredData.right_censored to create a representation of this data. The time-to-failure observations are put the list ttf. The censored list indicates which values in ttf are censored.

>>> ttf = [13, 22, 17, 15, 20, 18]
>>> censored = [False, False, False, False, True, True]

Pass these lists to CensoredData.right_censored to create an instance of CensoredData.

>>> data = CensoredData.right_censored(ttf, censored)
>>> print(data)
CensoredData(6 values: 4 not censored, 2 right-censored)

If the input data is interval censored and already stored in two arrays, one holding the low end of the intervals and another holding the high ends, the class method interval_censored can be used to create the CensoredData instance.

This example creates an instance with four interval-censored values. The intervals are [10, 11], [0.5, 1], [2, 3], and [12.5, 13.5].

>>> a = [10, 0.5, 2, 12.5]  # Low ends of the intervals
>>> b = [11, 1.0, 3, 13.5]  # High ends of the intervals
>>> data = CensoredData.interval_censored(low=a, high=b)
>>> print(data)
CensoredData(4 values: 0 not censored, 4 interval-censored)

Finally, we create and censor some data from the weibull_min distribution, and then fit weibull_min to that data. We’ll assume that the location parameter is known to be 0.

>>> from scipy.stats import weibull_min
>>> rng = np.random.default_rng()

Create the random data set.

>>> x = weibull_min.rvs(2.5, loc=0, scale=30, size=250, random_state=rng)
>>> x[x > 40] = 40  # Right-censor values greater or equal to 40.

Create the CensoredData instance with the right_censored method. The censored values are those where the value is 40.

>>> data = CensoredData.right_censored(x, x == 40)
>>> print(data)
CensoredData(250 values: 215 not censored, 35 right-censored)

35 values have been right-censored.

Fit weibull_min to the censored data. We expect to shape and scale to be approximately 2.5 and 30, respectively.

>>>, floc=0)
(2.3575922823897315, 0, 30.40650074451254)



The number of values (censored and not censored).

interval_censored(low, high)

Create a CensoredData instance of interval-censored data.

left_censored(x, censored)

Create a CensoredData instance of left-censored data.


Number of censored values.

right_censored(x, censored)

Create a CensoredData instance of right-censored data.