Think Bayes, 2nd Edition

Chapter 14. Survival Analysis

This chapter introduces “survival analysis”, which is a set of statistical methods used to answer questions about the time until an event. In the context of medicine it is literally about survival, but it can be applied to the time until any kind of event, or instead of time it can be about space or other dimensions.

Survival analysis is challenging because the data we have are often incomplete. But as we’ll see, Bayesian methods are particularly good at working with incomplete data.

As examples, we’ll consider two applications that are a little less serious than life and death: the time until light bulbs fail and the time until dogs in a shelter are adopted. To describe these “survival times”, we’ll use the Weibull distribution.

The Weibull Distribution

The Weibull distribution is often used in survival analysis because it is a good model for the distribution of lifetimes for manufactured products, at least over some parts of the range.

SciPy provides several versions of the Weibull distribution; the one we’ll use is called weibull_min. To make the interface consistent with our notation, I’ll wrap it in a function that takes as parameters $\lambda$, which mostly affects the location or “central tendency” of the distribution, and $k$, which affects the shape.

from scipy.stats import weibull_min

def weibull_dist(lam, k):
    return weibull_min(k, scale=lam)

As an example, here’s a Weibull distribution with parameters $\lambda =3$ and k = 0.8:

lam = 3
k = 0.8