Statistics for ML #98 — Survival Analysis & Hazard Functions

Survival Analysis & Hazard Functions

Survival Analysis models the time until an event occurs (death, disease, recovery) and handles censoring — when the event hasn’t occurred yet at study end.

Key Functions

• Survival function: S(t) = P(T > t) — probability of surviving past time t
• Hazard function: h(t) = instantaneous risk of event at t, given survival to t
• Cumulative hazard: H(t) = ∫₀ᵗ h(u)du

Kaplan-Meier Estimator (non-parametric)

\(\hat{S}(t) = \prod_{t_i \leq t} \left(1 - \frac{d_i}{n_i}\right)\)

Cox Proportional Hazards (semi-parametric)

\(h(t|X) = h_0(t) \cdot \exp(\beta_1 X_1 + ... + \beta_k X_k)\)

from lifelines import KaplanMeierFitter, CoxPHFitter

# Cardiovascular disease survival (from our conference presentation)
kmf = KaplanMeierFitter()
kmf.fit(df['time_to_event'], event_observed=df['event'])
kmf.plot_survival_function()
plt.title('Kaplan-Meier Survival Curve: CVD Dataset')

# Cox model
cph = CoxPHFitter()
cph.fit(df[['time_to_event','event','age','bmi','smoking','hypertension']],
        duration_col='time_to_event', event_col='event')
cph.print_summary()

Series Index | Post #98/100 | Md Salek Miah | saleksta@gmail.com