Survival Analysis

Survival Analysis Overview; Event History

import statsmodels.api as sm
import statsmodels.formula.api as smf

sm.SurvfuncRight#(time, status, entry=None, title=None, freq_weights=None, exog=None, bw_factor=1.0)
sm.PHReg#(endog, exog, status=None, entry=None, strata=None, offset=None, ties='breslow', missing='drop')
smf.phreg#(formula, data, status=None, entry=None, strata=None, offset=None, subset=None, ties='breslow', missing='drop')

 

Probability Distribution

from scipy import stats

# poisson

# exponential
stats.expon.rvs(scale=5, size=300) # sampling: mean = scale, var = scale**2
stats.expon.pdf(0, scale=1) # probability density: pdf = lambda * exp(-lambda * x), scale = 1 / lambda
stats.expon.cdf(1, scale=1) # cumulative distribution: cdf
stats.expon.ppf(.5, scale=1) # percent point function: cdf^(-1): quantile function

# beta

# weibull

 

$${\displaystyle \begin{aligned} \text{Survival Function, Curve} && S(t) &= P(T > t) \\ \text{Hazard Rate} && \lambda (t) &= \lim_{\Delta t \rightarrow 0} \frac{P(t \le T < t + \Delta t \mid T \ge t)}{\Delta t}\\ \text{Cumulative Hazard} && \Lambda (t) &= \int_{0}^{t} \lambda (u) \; du \\ \end{aligned} }$$

# lifelines

 

 

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Non-Parametric: Kaplan-Meier

 

 

 

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Parametric Survival Function Estimation

 

 

 

 

 

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Reference

• https://www.statsmodels.org/dev/duration.html