Kaplan-Meier¶
API Reference
For inputs and outputs see the API reference.
The Kaplan-Meier estimator provides a method by which to estimate the survival function (reliability function) of a population without assuming that the data comes from a particular distribution. Due to the lack of parameters required in this model, it is a non-parametric method of obtaining the survival function. With a few simple transformations, the survival function (SF) can be used to obtain the cumulative hazard function (CHF) and the cumulative distribution function (CDF). It is not possible to obtain a useful version of the probability density function (PDF) or hazard function (HF) as this would require the differentiation of the CDF and CHF respectively, which results in a very spikey plot due to the non-continuous nature of these plots.
The Kaplan-Meier estimator is very similar in result (but quite different in method) to the Nelson-Aalen estimator and Rank Adjustment estimator. While none of the three has been proven to be more accurate than the others, the Kaplan-Meier estimator is generally more popular as a non-parametric means of estimating the SF. Confidence intervals are provided using the Greenwood method with Normal approximation.
The Kaplan-Meier estimator can be used with both complete and right censored data. This function can be accessed from reliability.Nonparametric.KaplanMeier.
Example 1¶
In this first example, we will provide Kaplan-Meier with a list of failure times and right censored times. By leaving everything else unspecified, the plot will be shown with the confidence intervals shaded. We will layer this first Kaplan-Meier plot with a second one using just the failure data. As can be seen in the example below, the importance of including censored data is paramount to obtain an accurate estimate of the reliability, because without it the population’s survivors are not included so the reliability will appear much lower than it truly is.
from reliability.Nonparametric import KaplanMeier
import matplotlib.pyplot as plt
f = [5248, 7454, 16890, 17200, 38700, 45000, 49390, 69040, 72280, 131900]
rc = [3961, 4007, 4734, 6054, 7298, 10190, 23060, 27160, 28690, 37100, 40060, 45670, 53000, 67000, 69630, 77350, 78470, 91680, 105700, 106300, 150400]
KaplanMeier(failures=f, right_censored=rc, label='Failures + right censored')
KaplanMeier(failures=f, label='Failures only')
plt.title('Kaplan-Meier estimates showing the\nimportance of including censored data')
plt.xlabel('Miles to failure')
plt.legend()
plt.show()
'''
Results from KaplanMeier (95% CI):
Failure times Censoring code (censored=0) Items remaining Kaplan-Meier Estimate Lower CI bound Upper CI bound
3961 0 31 1 1 1
4007 0 30 1 1 1
4734 0 29 1 1 1
5248 1 28 0.964286 0.895548 1
6054 0 27 0.964286 0.895548 1
7298 0 26 0.964286 0.895548 1
7454 1 25 0.925714 0.826513 1
10190 0 24 0.925714 0.826513 1
16890 1 23 0.885466 0.76317 1
17200 1 22 0.845217 0.705334 0.985101
23060 0 21 0.845217 0.705334 0.985101
27160 0 20 0.845217 0.705334 0.985101
28690 0 19 0.845217 0.705334 0.985101
37100 0 18 0.845217 0.705334 0.985101
38700 1 17 0.795499 0.633417 0.95758
40060 0 16 0.795499 0.633417 0.95758
45000 1 15 0.742465 0.560893 0.924037
45670 0 14 0.742465 0.560893 0.924037
49390 1 13 0.685353 0.48621 0.884496
53000 0 12 0.685353 0.48621 0.884496
67000 0 11 0.685353 0.48621 0.884496
69040 1 10 0.616817 0.396904 0.836731
69630 0 9 0.616817 0.396904 0.836731
72280 1 8 0.539715 0.300949 0.778481
77350 0 7 0.539715 0.300949 0.778481
78470 0 6 0.539715 0.300949 0.778481
91680 0 5 0.539715 0.300949 0.778481
105700 0 4 0.539715 0.300949 0.778481
106300 0 3 0.539715 0.300949 0.778481
131900 1 2 0.269858 0 0.662446
150400 0 1 0.269858 0 0.662446
Results from KaplanMeier (95% CI):
Failure times Censoring code (censored=0) Items remaining Kaplan-Meier Estimate Lower CI bound Upper CI bound
5248 1 10 0.9 0.714061 1
7454 1 9 0.8 0.552082 1
16890 1 8 0.7 0.415974 0.984026
17200 1 7 0.6 0.296364 0.903636
38700 1 6 0.5 0.190102 0.809898
45000 1 5 0.4 0.0963637 0.703636
49390 1 4 0.3 0.0159742 0.584026
69040 1 3 0.2 0 0.447918
72280 1 2 0.1 0 0.285939
131900 1 1 0 0 0
'''
Example 2¶
In this second example, we will create some data from a Weibull distribution, and then right censor the data above our chosen threshold. We will then fit a Weibull_2P distribution to the censored data, and also obtain the Kaplan-Meier estimate of this data. Using the results from the Fit_Weibull_2P and the Kaplan-Meier estimate, we will plot the CDF, SF, and CHF, for both the Weibull and Kaplan-Meier results. Note that the default plot from KaplanMeier will only give you the SF, but the results object provides everything you need to reconstruct the SF plot yourself, as well as what we need to plot the CDF and CHF.
from reliability.Distributions import Weibull_Distribution
from reliability.Fitters import Fit_Weibull_2P
from reliability.Nonparametric import KaplanMeier
from reliability.Other_functions import make_right_censored_data
import matplotlib.pyplot as plt
dist = Weibull_Distribution(alpha=5, beta=2) # create a distribution
raw_data = dist.random_samples(100, seed=2) # get some data from the distribution. Seeded for repeatability
data = make_right_censored_data(raw_data, threshold=9)
wbf = Fit_Weibull_2P(failures=data.failures, right_censored=data.right_censored, show_probability_plot=False, print_results=False) # Fit the Weibull_2P
# Create the subplots and in each subplot we will plot the parametric distribution and obtain the Kaplan Meier fit.
# Note that the plot_type is being changed each time
plt.figure(figsize=(12, 5))
plt.subplot(131)
KaplanMeier(failures=data.failures, right_censored=data.right_censored, plot_type='SF', print_results=False, label='Kaplan-Meier')
wbf.distribution.SF(label='Parametric')
plt.legend()
plt.title('SF')
plt.subplot(132)
KaplanMeier(failures=data.failures, right_censored=data.right_censored, plot_type='CDF', print_results=False, label='Kaplan-Meier')
wbf.distribution.CDF(label='Parametric')
plt.legend()
plt.title('CDF')
plt.subplot(133)
KaplanMeier(failures=data.failures, right_censored=data.right_censored, plot_type='CHF', print_results=False, label='Kaplan-Meier')
wbf.distribution.CHF(label='Parametric')
plt.legend()
plt.title('CHF')
plt.subplots_adjust(left=0.07, right=0.95, top=0.92, wspace=0.25) # format the plot layout
plt.show()
