Survival analysis with lifelines¶
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. Censorship 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 censorship event. This is also an example where the current time is not the only cause of censorship; there are the alternative events (e.g., death in office) that can be the cause of censorship.
To estimate the survival function, we first will use the Kaplan-Meier Estimate, defined:
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.
import pandas as pd from lifelines.datasets import load_dd data = load_dd()
data.sample(6) #the boolean columns `observed` refers to whether the death (leaving office) #was observed or not.
|164||Bolivia||145||145.0||South America||Americas||Rene Barrientos Ortuno||Rene Barrientos Ortuno.Bolivia.1966.1968.Milit...||Non-democracy||Military Dict||1966||3||0|
|740||India||750||750.0||Southern Asia||Asia||Chandra Shekhar||Chandra Shekhar.India.1990.1990.Parliamentary Dem||Democracy||Parliamentary Dem||1990||1||1|
|220||Bulgaria||355||355.0||Eastern Europe||Europe||Todor Zhivkov||Todor Zhivkov.Bulgaria.1954.1988.Civilian Dict||Non-democracy||Civilian Dict||1954||35||1|
|772||Ireland||205||205.0||Northern Europe||Europe||Charles Haughey||Charles Haughey.Ireland.1979.1980.Mixed Dem||Democracy||Mixed Dem||1979||2||1|
|1718||United States of America||2||2.0||Northern America||Americas||Gerald Ford||Gerald Ford.United States of America.1974.1976...||Democracy||Presidential Dem||1974||3||1|
|712||Iceland||395||395.0||Northern Europe||Europe||Stefan Stefansson||Stefan Stefansson.Iceland.1947.1948.Mixed Dem||Democracy||Mixed Dem||1947||2||1|
6 rows × 12 columns
lifelines library, we’ll need the
KaplanMeierFitter for this exercise:
from lifelines import KaplanMeierFitter kmf = KaplanMeierFitter()
Other ways to estimate the survival function in lifelines are
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
fit to fit the model to
the data. (This is similar to, and inspired by,
KaplanMeierFitter.fit(durations, event_observed=None, timeline=None, entry=None, label='KM_estimate', alpha=None, left_censorship=False, ci_labels=None) Parameters: duration: an array, or pd.Series, of length n -- duration subject was observed for timeline: return the best estimate at the values in timelines (postively increasing) event_observed: an array, or pd.Series, of length n -- True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None entry: an array, or pd.Series, of length n -- relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population were born at time 0. label: a string to name the column of the estimate. alpha: the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only. left_censorship: True if durations and event_observed refer to left censorship events. Default False ci_labels: add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha> Returns: a modified self, with new properties like 'survival_function_'.
Below we fit our data with the
T = data["duration"] E = data["observed"] kmf.fit(T, event_observed=E)
<lifelines.KaplanMeierFitter: fitted with 1808 observations, 340 censored>
After calling the
fit method, the
KaplanMeierFitter has a property
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
property. (The method uses exponential Greenwood confidence interval. The mathematics are found in these notes.)
Alternatively, we can call
plot on the
to plot both the KM estimate and its confidence intervals:
Don’t like the shaded area for confidence intervals? See below for examples on how to change this.
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 #
Interesting that it is only three years. That means, around the world, elected leaders have a 50% chance of cessation in three years!
Let’s segment on democratic regimes vs non-democratic regimes. Calling
plot on either the estimate itself or the fitter object will return
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, ci_force_lines=True) kmf.fit(T[~dem], event_observed=E[~dem], label="Non-democratic Regimes") kmf.plot(ax=ax, ci_force_lines=True) 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 = 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: Democratic Regimes 3 dtype: float64 Median survival time of non-democratic: Non-democratic Regimes 6 dtype: float64
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()
Results df: 1 alpha: 0.99 t 0: -1 test: logrank null distribution: chi squared __ p-value ___|__ test statistic __|____ test results ____|__ significant __ 0.00000 | 208.306 | Reject Null | True
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()
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
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 sublibrary. 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]
datetimes_to_durations is very flexible, and has many
keywords to tinker with.
Fitting to a Weibull model¶
Another very popular model for survival data is the Weibull model. In contrast the the Kaplan Meier estimator, this model is a parametric model, meaning it has a functional form with parameters that we are fitting the data to. (The Kaplan Meier estimator has no parameters to fit too). Mathematically, the survival function looks like:
..math:: S(t) = expleft(-(lambda t)^rhoright), lambda >0, rho > 0,
Apriori, we do not know what \(\lambda\) and \(\rho\) are, but we use the data on hand to estimate these parameters. In lifelines, this is implemented in the
from lifelines import WeibullFitter T = data['duration'] E = data['observed'] wf = WeibullFitter() wf.fit(T, E) print(wf.lambda_, wf.rho_) wf.print_summary()
Estimating hazard rates using Nelson-Aalen¶
The survival curve is a great way to summarize and visualize the lifetime data, however it is not the only way. If we are curious about the hazard function \(\lambda(t)\) of a population, we unfortunately cannot transform the Kaplan Meier estimate – statistics doesn’t work quite that well. Fortunately, there is a proper estimator of the cumulative hazard function:
The estimator for this quantity is called the Nelson Aalen estimator:
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
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, \(\Lambda(t)\). (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:
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. So why did the combination of both regimes have a decreasing hazard? This is the effect of frailty, a topic we will discuss later.
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
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
b = 3. naf.fit(T[dem], event_observed=E[dem], label="Democratic Regimes") ax = naf.plot_hazard(bandwidth=b) naf.fit(T[~dem], event_observed=E[~dem], label="Non-democratic Regimes") naf.plot_hazard(ax=ax, bandwidth=b) plt.title("Hazard function of different global regimes | bandwidth=%.1f"%b); plt.ylim(0, 0.4) plt.xlim(0, 25);
It is more clear here which group has the higher hazard, and like hypothesized above, both hazard rates are close to being constant.
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.)
b = 8. naf.fit(T[dem], event_observed=E[dem], label="Democratic Regimes") ax = naf.plot_hazard(bandwidth=b) naf.fit(T[~dem], event_observed=E[~dem], label="Non-democratic Regimes") naf.plot_hazard(ax=ax, bandwidth=b) plt.title("Hazard function of different global regimes | bandwidth=%.1f"%b);
Other types of censorship¶
Left Censored Data¶
We’ve mainly been focusing on right-censorship, 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.
lifelines has support for left-censored datasets in the
KaplanMeierFitter class, by adding the keyword
False) to the call to
from lifelines.datasets import load_lcd lcd_dataset = load_lcd() ix = lcd_dataset['group'] == 'alluvial_fan' T = lcd_dataset[ix]['T'] E = lcd_dataset[ix]['E'] #boolean array, True if observed. kmf = KaplanMeierFitter() kmf.fit(T, E, left_censorship=True)
Instead of producing a survival function, left-censored data is more interested in the cumulative density function
of time to birth. This is available as the
cumulative_density_ property after fitting the data.
print(kmf.cumulative_density_) kmf.plot() #will plot the CDF
Left Truncated Data¶
Another form of bias that is introduced into a dataset is called left-truncation. (Also a form of censorship).
Left-truncation occurs when individuals may die even before ever entering into the study. Both
NelsonAalenFitter have an optional argument for
entry, which is an array of equal size to the duration array.
It describes the offset from birth to entering the study. This is also useful when subjects enter the study at different
points in their lifetime. For example, if you are measuring time to death of prisoners in
prison, the prisoners will enter the study at different ages.
Nothing changes in the duration array: it still measures time from entry of study to time left study (either by death or censorship)
Other types of censorship, like interval-censorship, are not implemented in lifelines yet.