Your first survival curve!

Survival Analysis in Python

Shae Wang

Senior Data Scientist

The survival function

$T$: when the event of interest occurs
$t$: any point in time during an observation

$$\Large{S(t) = Pr(T>t)}$$

$S(t)$: models the probability that the event of interest happens after $t$
$Pr(T>t)$: the survival probability

The survival curve

$$\Large{S(t) = Pr(T>t)}$$

Survival curve example.

The survival curve

$$\Large{S(t) = Pr(T>t)}$$

Survival curve example.

The survival curve

$$\Large{S(t) = Pr(T>t)}$$

Survival curve example.

Survival curve legend.

Interpreting a survival curve

Point$(a,b)$: the probability that an individual survives longer than $a$ is $b$

Survival curve example.

Interpreting a survival curve

Point$(a,b)$: the probability that an individual survives longer than $a$ is $b$
Flatter curve: lower rate of event occurrence
Steeper curve: higher rate of event occurrence

Survival curve example with marked point.

Non-parametric versus parametric models

Non-parametric modeling

Make no assumptions about the shape of the data

Parametric modeling

Make some assumptions about the shape of the data
Described with a limited set of parameters
- i.e. the survival curve may be assumed to follow an exponential distribution

Non-parametric versus parametric models

Non-parametric modeling

Survival curve is usually NOT smooth

Non-parametric survival curve example.

Parametric modeling

Survival curve is usually smooth

Parametric survival curve example.

Relies on the parametric model actually being a good description of the data

Drawing a survival curve

The lifelines package is a complete survival analysis library.

Fit survival functions to data
Plot survival curves based on the fitted survival functions

import lifelines
import matplotlib.pyplot as plt

.fit(durations, event_observed)

.plot_survival_function()

Survival curve example

DataFrame name: mortgage_df

id	duration	paid_off
1	25	0
2	17	1
3	5	0
...	...	...
100	30	1

id: the id of a mortgage loan
duration: the number of years the mortgage is not paid off
paid_off: 1 if the mortgage is fully paid off, 0 if not fully paid off

Survival curve example

import lifelines
from matplotlib import pyplot as plt

kmf = lifelines.KaplanMeierFitter()
kmf.fit(duration=mortgage_df["duration"], 
        event_observed=mortgage_df["paid_off"])

kmf.plot_survival_function()
plt.show()

example kaplan meier survial curve

Let's practice!

Survival Analysis in Python