Estimating univariate models¶

In the previous section, we introduced the use of survival analysis, the need, and the mathematical objects on which it relies. In this article, we will work with real data and the lifelines library to estimate these mathematical objects.

Estimating the survival function using Kaplan-Meier¶

For this example, we will be investigating the lifetimes of political leaders around the world. A political leader, in this case, is defined by a single individual’s time in office who controls the ruling regime. This political leader could be an elected president, unelected dictator, monarch, etc. The birth event is the start of the individual’s tenure, and the death event is the retirement of the individual. Censoring can occur if they are a) still in offices at the time of dataset compilation (2008), or b) die while in power (this includes assassinations).

For example, the Bush regime began in 2000 and officially ended in 2008 upon his retirement, thus this regime’s lifespan was eight years, and there was a “death” event observed. On the other hand, the JFK regime lasted 2 years, from 1961 and 1963, and the regime’s official death event was not observed – JFK died before his official retirement.

(This is an example that has gladly redefined the birth and death events, and in fact completely flips the idea upside down by using deaths as the censoring event. This is also an example where the current time is not the only cause of censoring; there are the alternative events (e.g., death in office) that can be the cause of censoring.

To estimate the survival function, we first will use the Kaplan-Meier Estimate, defined:

$\hat{S}(t) = \prod_{t_i \lt t} \frac{n_i - d_i}{n_i}$

where $$d_i$$ are the number of death events at time $$t$$ and $$n_i$$ is the number of subjects at risk of death just prior to time $$t$$.

Let’s bring in our dataset.

from lifelines.datasets import load_dd

data = load_dd()
data.head()

ctryname cowcode2 politycode un_region_name un_continent_name ehead leaderspellreg democracy regime start_year duration observed
Afghanistan 700 700 Southern Asia Asia Mohammad Zahir Shah Mohammad Zahir Shah.Afghanistan.1946.1952.Monarchy Non-democracy Monarchy 1946 7 1
Afghanistan 700 700 Southern Asia Asia Sardar Mohammad Daoud Sardar Mohammad Daoud.Afghanistan.1953.1962.Civilian Dict Non-democracy Civilian Dict 1953 10 1
Afghanistan 700 700 Southern Asia Asia Mohammad Zahir Shah Mohammad Zahir Shah.Afghanistan.1963.1972.Monarchy Non-democracy Monarchy 1963 10 1
Afghanistan 700 700 Southern Asia Asia Sardar Mohammad Daoud Sardar Mohammad Daoud.Afghanistan.1973.1977.Civilian Dict Non-democracy Civilian Dict 1973 5 0
Afghanistan 700 700 Southern Asia Asia Nur Mohammad Taraki Nur Mohammad Taraki.Afghanistan.1978.1978.Civilian Dict Non-democracy Civilian Dict 1978 1 0

From the lifelines library, we’ll need the KaplanMeierFitter for this exercise:

from lifelines import KaplanMeierFitter
kmf = KaplanMeierFitter()


Note

Other ways to estimate the survival function in lifelines are discussed below.

For this estimation, we need the duration each leader was/has been in office, and whether or not they were observed to have left office (leaders who died in office or were in office in 2008, the latest date this data was record at, do not have observed death events)

We next use the KaplanMeierFitter method fit to fit the model to the data. (This is similar to, and inspired by, scikit-learn’s fit/predict API).

Below we fit our data with the KaplanMeierFitter:

T = data["duration"]
E = data["observed"]

kmf.fit(T, event_observed=E)


After calling the fit method, the KaplanMeierFitter has a property called survival_function_ (again, we follow the styling of scikit-learn, and append an underscore to all properties that were computational estimated). The property is a Pandas DataFrame, so we can call plot on it:

kmf.survival_function_.plot()
plt.title('Survival function of political regimes');


How do we interpret this? The y-axis represents the probability a leader is still around after $$t$$ years, where $$t$$ years is on the x-axis. We see that very few leaders make it past 20 years in office. Of course, like all good stats, we need to report how uncertain we are about these point estimates, i.e., we need confidence intervals. They are computed in the call to fit, and located under the confidence_interval_ property. (The method uses exponential Greenwood confidence interval. The mathematics are found in these notes.)

$S(t) = Pr( T > t)$

Alternatively, we can call plot on the KaplanMeierFitter itself to plot both the KM estimate and its confidence intervals:

kmf.plot()


The median time in office, which defines the point in time where on average 1/2 of the population has expired, is a property:

kmf.median_
#   4.0


Interesting that it is only four years. That means, around the world, elected leaders have a 50% chance of cessation in four years or less!

Let’s segment on democratic regimes vs non-democratic regimes. Calling plot on either the estimate itself or the fitter object will return an axis object, that can be used for plotting further estimates:

ax = plt.subplot(111)

dem = (data["democracy"] == "Democracy")

kmf.fit(T[dem], event_observed=E[dem], label="Democratic Regimes")
kmf.plot(ax=ax)
kmf.fit(T[~dem], event_observed=E[~dem], label="Non-democratic Regimes")
kmf.plot(ax=ax)

plt.ylim(0, 1);
plt.title("Lifespans of different global regimes");


We might be interested in estimating the probabilities in between some points. We can do that with the timeline argument. We specify the times we are interested in and are returned a DataFrame with the probabilities of survival at those points:

ax = plt.subplot(111)

t = np.linspace(0, 50, 51)
kmf.fit(T[dem], event_observed=E[dem], timeline=t, label="Democratic Regimes")
ax = kmf.plot(ax=ax)
print("Median survival time of democratic:", kmf.median_)

kmf.fit(T[~dem], event_observed=E[~dem], timeline=t, label="Non-democratic Regimes")
ax = kmf.plot(ax=ax)
print("Median survival time of non-democratic:", kmf.median_)

plt.ylim(0, 1)
plt.title("Lifespans of different global regimes");

"""
Median survival time of democratic: 3
Median survival time of non-democratic: 6
"""


It is incredible how much longer these non-democratic regimes exist for. A democratic regime does have a natural bias towards death though: both via elections and natural limits (the US imposes a strict eight-year limit). The median of a non-democratic is only about twice as large as a democratic regime, but the difference is apparent in the tails: if you’re a non-democratic leader, and you’ve made it past the 10 year mark, you probably have a long life ahead. Meanwhile, a democratic leader rarely makes it past ten years, and then have a very short lifetime past that.

Here the difference between survival functions is very obvious, and performing a statistical test seems pedantic. If the curves are more similar, or we possess less data, we may be interested in performing a statistical test. In this case, lifelines contains routines in lifelines.statistics to compare two survival curves. Below we demonstrate this routine. The function logrank_test is a common statistical test in survival analysis that compares two event series’ generators. If the value returned exceeds some pre-specified value, then we rule that the series have different generators.

from lifelines.statistics import logrank_test

results = logrank_test(T[dem], T[~dem], E[dem], E[~dem], alpha=.99)

results.print_summary()

"""

<lifelines.StatisticalResult>
t_0 = -1
null_distribution = chi squared
degrees_of_freedom = 1
alpha = 0.99

---
test_statistic      p  -log2(p)
260.47  <0.005    192.23
""""


Lets compare the different types of regimes present in the dataset:

regime_types = data['regime'].unique()

for i, regime_type in enumerate(regime_types):
ax = plt.subplot(2, 3, i + 1)

ix = data['regime'] == regime_type
kmf.fit(T[ix], E[ix], label=regime_type)
kmf.plot(ax=ax, legend=False)

plt.title(regime_type)
plt.xlim(0, 50)

if i==0:
plt.ylabel('Frac. in power after $n$ years')

plt.tight_layout()


There are alternative (and sometimes better) tests of survival curves, and we explain more here: Statistically compare two populations

Getting data into the right format¶

lifelines data format is consistent across all estimator class and functions: an array of individual durations, and the individuals event observation (if any). These are often denoted T and E respectively. For example:

T = [0,3,3,2,1,2]
E = [1,1,0,0,1,1]
kmf.fit(T, event_observed=E)


The raw data is not always available in this format – lifelines includes some helper functions to transform data formats to lifelines format. These are located in the lifelines.utils sub-library. For example, the function datetimes_to_durations accepts an array or Pandas object of start times/dates, and an array or Pandas objects of end times/dates (or None if not observed):

from lifelines.utils import datetimes_to_durations

start_date = ['2013-10-10 0:00:00', '2013-10-09', '2013-10-10']
end_date = ['2013-10-13', '2013-10-10', None]
T, E = datetimes_to_durations(start_date, end_date, fill_date='2013-10-15')
print('T (durations): ', T)
print('E (event_observed): ', E)

T (durations):  [ 3.  1.  5.]
E (event_observed):  [ True  True False]


The function datetimes_to_durations is very flexible, and has many keywords to tinker with.

Estimating hazard rates using Nelson-Aalen¶

The survival curve is a great way to summarize and visualize the survival dataset, however it is not the only way. If we are curious about the hazard function $$h(t)$$ of a population, we unfortunately cannot transform the Kaplan Meier estimate – statistics doesn’t work quite that well. Fortunately, there is a proper non-parametric estimator of the cumulative hazard function:

$H(t) = \int_0^t \lambda(z) \;dz$

The estimator for this quantity is called the Nelson Aalen estimator:

$\hat{H}(t) = \sum_{t_i \le t} \frac{d_i}{n_i}$

where $$d_i$$ is the number of deaths at time $$t_i$$ and $$n_i$$ is the number of susceptible individuals.

In lifelines, this estimator is available as the NelsonAalenFitter. Let’s use the regime dataset from above:

T = data["duration"]
E = data["observed"]

from lifelines import NelsonAalenFitter
naf = NelsonAalenFitter()

naf.fit(T,event_observed=E)


After fitting, the class exposes the property cumulative_hazard_ as a DataFrame:

print(naf.cumulative_hazard_.head())
naf.plot()

   NA-estimate
0     0.000000
1     0.325912
2     0.507356
3     0.671251
4     0.869867

[5 rows x 1 columns]


The cumulative hazard has less immediate understanding than the survival curve, but the hazard curve is the basis of more advanced techniques in survival analysis. Recall that we are estimating cumulative hazard curve, $$H(t)$$. (Why? The sum of estimates is much more stable than the point-wise estimates.) Thus we know the rate of change of this curve is an estimate of the hazard function.

Looking at figure above, it looks like the hazard starts off high and gets smaller (as seen by the decreasing rate of change). Let’s break the regimes down between democratic and non-democratic, during the first 20 years:

Note

We are using the loc argument in the call to plot here: it accepts a slice and plots only points within that slice.

naf.fit(T[dem], event_observed=E[dem], label="Democratic Regimes")
ax = naf.plot(loc=slice(0, 20))

naf.fit(T[~dem], event_observed=E[~dem], label="Non-democratic Regimes")
naf.plot(ax=ax, loc=slice(0, 20))

plt.title("Cumulative hazard function of different global regimes");


Looking at the rates of change, I would say that both political philosophies have a constant hazard, albeit democratic regimes have a much higher constant hazard.

Smoothing the hazard curve¶

Interpretation of the cumulative hazard function can be difficult – it is not how we usually interpret functions. On the other hand, most survival analysis is done using the cumulative hazard function, so understanding it is recommended.

Alternatively, we can derive the more-interpretable hazard curve, but there is a catch. The derivation involves a kernel smoother (to smooth out the differences of the cumulative hazard curve) , and this requires us to specify a bandwidth parameter that controls the amount of smoothing. This functionality is in the smoothed_hazard_ and hazard_confidence_intervals_ methods. Why methods? They require an argument representing the bandwidth.

There is also a plot_hazard function (that also requires a bandwidth keyword) that will plot the estimate plus the confidence intervals, similar to the traditional plot functionality.

bandwidth = 3.

naf.fit(T[dem], event_observed=E[dem], label="Democratic Regimes")
ax = naf.plot_hazard(bandwidth=bandwidth)

naf.fit(T[~dem], event_observed=E[~dem], label="Non-democratic Regimes")
naf.plot_hazard(ax=ax, bandwidth=bandwidth)

plt.title("Hazard function of different global regimes | bandwidth=%.1f" % bandwidth);
plt.ylim(0, 0.4)
plt.xlim(0, 25);


It is more clear here which group has the higher hazard, and Non-democratic regimes appear to have a constant hazard.

There is no obvious way to choose a bandwidth, and different bandwidths produce different inferences, so it’s best to be very careful here. My advice: stick with the cumulative hazard function.

bandwidth = 8.0

naf.fit(T[dem], event_observed=E[dem], label="Democratic Regimes")
ax = naf.plot_hazard(bandwidth=bandwidth)

naf.fit(T[~dem], event_observed=E[~dem], label="Non-democratic Regimes")
naf.plot_hazard(ax=ax, bandwidth=bandwidth)

plt.title("Hazard function of different global regimes | bandwidth=%.1f" % bandwidth);


Estimating cumulative hazards using parametric models¶

Fitting to a Weibull model¶

Note

The parameterization of the Weibull and Exponential model changed in lifelines 0.19.0, released in Feb. 2019.

Another very popular model for survival data is the Weibull model. In contrast the the Nelson-Aalen estimator, this model is a parametric model, meaning it has a functional form with parameters that we are fitting the data to. (The Nelson-Aalen estimator has no parameters to fit to). The survival function looks like:

$S(t) = \exp\left(-\left(\frac{t}{\lambda}\right)^\rho\right), \lambda >0, \rho > 0,$

A priori, we do not know what $$\lambda$$ and $$\rho$$ are, but we use the data on hand to estimate these parameters. We model and estimate the cumulative hazard rate instead of the survival function (this is different than the Kaplan-Meier estimator):

$H(t) = \left(\frac{t}{\lambda}\right)^\rho$

In lifelines, estimation is available using the WeibullFitter class. The plot method will plot the cumulative hazard.

from lifelines import WeibullFitter
from lifelines.datasets import load_waltons

data = load_waltons()

T = data['T']
E = data['E']

wf = WeibullFitter().fit(T, E)

wf.print_summary()
wf.plot()

"""
<lifelines.WeibullFitter: fitted with 163 observations, 7 censored>
number of subjects = 163
number of events = 156
log-likelihood = -672.062
hypothesis = lambda != 1, rho != 1

---
coef  se(coef)  lower 0.95  upper 0.95      p  -log2(p)
lambda_  0.02      0.00        0.02        0.02 <0.005       inf
rho_     3.45      0.24        2.97        3.93 <0.005     76.83
"""


Other parametric models: Exponential, Log-Logistic & Log-Normal¶

Similarly, there are other parametric models in lifelines. Generally, which parametric model to choose is determined by either knowledge of the distribution of durations, or some sort of model goodness-of-fit. Below are the built-in parametric models, and the Nelson-Aalen nonparametric model, of the same data.

from lifelines import WeibullFitter
from lifelines import ExponentialFitter
from lifelines import LogNormalFitter
from lifelines import LogLogisticFitter
from lifelines import NelsonAalenFitter
from lifelines import PiecewiseExponentialFitter

from lifelines.datasets import load_waltons
data = load_waltons()

fig, axes = plt.subplots(2, 3, figsize=(9, 5))

T = data['T']
E = data['E']

wbf = WeibullFitter().fit(T, E, label='WeibullFitter')
exf = ExponentialFitter().fit(T, E, label='ExponentalFitter')
lnf = LogNormalFitter().fit(T, E, label='LogNormalFitter')
naf = NelsonAalenFitter().fit(T, E, label='NelsonAalenFitter')
llf = LogLogisticFitter().fit(T, E, label='LogLogisticFitter')
pwf = PiecewiseExponentialFitter([40, 60]).fit(T, E, label='PiecewiseExponentialFitter')

wbf.plot_cumulative_hazard(ax=axes[0][0])
exf.plot_cumulative_hazard(ax=axes[0][1])
lnf.plot_cumulative_hazard(ax=axes[0][2])
naf.plot_cumulative_hazard(ax=axes[1][0])
llf.plot_cumulative_hazard(ax=axes[1][1])
pwf.plot_cumulative_hazard(ax=axes[1][2])


lifelines can also be used to define your own parametric model. There is a tutorial on this available, see Piecewise Exponential Models and Creating Custom Models.

Parametric models can also be used to create and plot the survival function, too. Below we compare the parametric models versus the non-parametric Kaplan-Meier estimate:

from lifelines import KaplanMeierFitter

fig, axes = plt.subplots(2, 3, figsize=(9, 5))

T = data['T']
E = data['E']

kmf = KaplanMeierFitter().fit(T, E, label='KaplanMeierFitter')
wbf = WeibullFitter().fit(T, E, label='WeibullFitter')
exf = ExponentialFitter().fit(T, E, label='ExponentalFitter')
lnf = LogNormalFitter().fit(T, E, label='LogNormalFitter')
llf = LogLogisticFitter().fit(T, E, label='LogLogisticFitter')
pwf = PiecewiseExponentialFitter([40, 60]).fit(T, E, label='PiecewiseExponentialFitter')

wbf.plot_survival_function(ax=axes[0][0])
exf.plot_survival_function(ax=axes[0][1])
lnf.plot_survival_function(ax=axes[0][2])
kmf.plot_survival_function(ax=axes[1][0])
llf.plot_survival_function(ax=axes[1][1])
pwf.plot_survival_function(ax=axes[1][2])


With parametric models, we have a functional form that allows us to extend the survival function (or hazard or cumulative hazard) past our maximum observed duration. This is called extrapolation. We can do this in a few ways.

timeline = np.linspace(0, 100, 200)

# directly compute the survival function, these return a pandas Series
wbf = WeibullFitter().fit(T, E)
wbf.survival_function_at_times(timeline)
wbf.hazard_at_times(timeline)
wbf.cumulative_hazard_at_times(timeline)

# use the timeline kwarg in fit
# by default, all functions and properties will use
# these values provided
wbf = WeibullFitter().fit(T, E, timeline=timeline)

wbf.plot_survival_function()


To aid model selection, _lifelines_ has provided qq-plots, Selecting a parametric model using QQ plots.

Other types of censoring¶

Left censored data and non-detection¶

We’ve mainly been focusing on right-censoring, which describes cases where we do not observe the death event. This situation is the most common one. Alternatively, there are situations where we do not observe the birth event occurring. Consider the case where a doctor sees a delayed onset of symptoms of an underlying disease. The doctor is unsure when the disease was contracted (birth), but knows it was before the discovery.

Another situation where we have left-censored data is when measurements have only an upper bound, that is, the measurements instruments could only detect the measurement was less than some upper bound. This bound is often called the limit of detection (LOD). In practice, there could be more than one LOD. One very important statistical lesson: don’t “fill-in” this value naively. It’s tempting to use something like one-half the LOD, but this will cause lots of bias in downstream analysis. An example dataset is below:

from lifelines.datasets import load_nh4
df = load_nh4()[['NH4.Orig.mg.per.L', 'NH4.mg.per.L', 'Censored']]
print(df.head())

"""
NH4.Orig.mg.per.L  NH4.mg.per.L  Censored
1            <0.006         0.006      True
2            <0.006         0.006      True
3             0.006         0.006     False
4             0.016         0.016     False
5            <0.006         0.006      True
"""


lifelines has support for left-censored datasets in most univariate models, including the KaplanMeierFitter class, by adding the keyword left_censoring=True (default False) to the call to fit.

T, E = df['NH4.mg.per.L'], ~df['Censored']

kmf = KaplanMeierFitter()
kmf.fit(T, E, left_censorship=True)


Instead of producing a survival function, left-censored data analysis is more interested in the cumulative density function. This is available as the cumulative_density_ property after fitting the data.

print(kmf.cumulative_density_.head())

kmf.plot() #will plot the CDF
plt.xlabel("Concentration of NH_4")

"""
KM_estimate
timeline
0.000        0.379897
0.006        0.401002
0.007        0.464319
0.008        0.478828
0.009        0.536868
"""


Alternatively, you can use a parametric model to model the data. This allows for you to “peer” below the LOD, however using a parametric model means you need to correctly specify the distribution. You can use plots like qq-plots to help invalidate some distributions, see Selecting a parametric model using QQ plots.

fig, axes = plt.subplots(3, 2, figsize=(9, 9))
timeline = np.linspace(0, 0.25, 100)

wf = WeibullFitter().fit(T, E, left_censorship=True, label="Weibull", timeline=timeline)
lnf = LogNormalFitter().fit(T, E, left_censorship=True, label="Log Normal", timeline=timeline)
lgf = LogLogisticFitter().fit(T, E, left_censorship=True, label="Log Logistic", timeline=timeline)

# plot what we just fit, along with the KMF estimate
kmf.plot_cumulative_density(ax=axes[0][0], ci_show=False)
wf.plot_cumulative_density(ax=axes[0][0], ci_show=False)
qq_plot(wf, ax=axes[0][1])

kmf.plot_cumulative_density(ax=axes[1][0], ci_show=False)
lnf.plot_cumulative_density(ax=axes[1][0], ci_show=False)
qq_plot(lnf, ax=axes[1][1])

kmf.plot_cumulative_density(ax=axes[2][0], ci_show=False)
lgf.plot_cumulative_density(ax=axes[2][0], ci_show=False)
qq_plot(lgf, ax=axes[2][1])


Based on the above, the log-normal distribution seems to fit well, and the Weibull not very well at all.

Note

Other types of censoring, like interval-censoring, are not implemented in lifelines yet.

Left truncated (late entry) data¶

Another form of bias that is introduced into a dataset is called left-truncation (or late entry). Left-truncation can occur in many situations. One situation is when individuals may have the opportunity to die before entering into the study. For example, if you are measuring time to death of prisoners in prison, the prisoners will enter the study at different ages. So it’s possible there are some counter-factual individuals who would have entered into your study (that is, went to prison), but instead died early.

All univariate fitters, like KaplanMeierFitter and any parametric models, have an optional argument for entry, which is an array of equal size to the duration array. It describes the time between actual “birth” (or “exposure”) to entering the study.

Note

Nothing changes in the duration array: it still measures time from “birth” to time exited study (either by death or censoring). That is, durations refers to the absolute death time rather than a duration relative to the study entry.

Another situation with left-truncation occurs when subjects are exposed before entry into study. For example, a study of time to all-cause mortality of AIDS patients that recruited individuals previously diagnosed with AIDS, possibly years before. In our example below we will use a dataset like this, called the Multicenter Aids Cohort Study. In the figure below, we plot the lifetimes of subjects. A solid line is when the subject was under our observation, and a dashed line represents the unobserved period between diagnosis and study entry. A solid dot at the end of the line represents death.

from lifelines.datasets import load_multicenter_aids_cohort_study
from lifelines.plotting import plot_lifetimes

df = load_multicenter_aids_cohort_study()

plot_lifetimes(
df["T"] - df["W"],
event_observed=df["D"],
entry=df["W"],
event_observed_color="#383838",
event_censored_color="#383838",
left_truncated=True,
)
plt.ylabel("Patient Number")
plt.xlabel("Years from AIDS diagnosis")


So subject #77, the subject at the top, was diagnosed with AIDS 7.5 years ago, but wasn’t in our study for the first 4.5 years. From this point-of-view, why can’t we “fill in” the dashed lines and say, for example, “subject #77 lived for 7.5 years”? If we did this, we would severely underestimate chance of dieing early on after diagnosis. Why? It’s possible that there were individuals who were diagnosed and then died shortly after, and never had a chance to enter our study. If we did manage to observe them however, they would have depressed the survival curve early on. Thus, “filling in” the dashed lines makes us over confident about what occurs in the early period after diagnosis. We can see this below when we model the survival curve with and without taking into account late entries.

from lifelines import KaplanMeierFitter

kmf = KaplanMeierFitter()
kmf.fit(df["T"], event_observed=df["D"], entry=df["W"], label='modeling late entries')
ax = kmf.plot()

kmf.fit(df["T"], event_observed=df["D"], label='ignoring late entries')
kmf.plot(ax=ax)