Survival models and lifetime distributions
CS2 tests whether you can move between the survival function, density, hazard and cumulative hazard, and choose a lifetime distribution that matches a hazard shape rather than a convenient formula.
Core relationships
For a non-negative lifetime T, S(t)=P(T>t), f(t)=−S′(t), h(t)=f(t)/S(t), and H(t)=∫₀ᵗ h(u)du=−ln S(t), so S(t)=exp(−H(t)). Also E[T]=∫₀^∞ S(t)dt when finite. Conditional survival is S(t+u)/S(t). Right-censoring is non-informative only when censoring is independent of T given covariates.
Distributions
Exponential has constant hazard μ and S(t)=e^{−μt}. Weibull has h(t)=λγt^{γ−1}: γ<1 decreasing, γ=1 constant, γ>1 increasing. Gompertz h(t)=Be^{ct}; Makeham adds a constant. Log-logistic and lognormal allow non-monotone hazards. Fractional-age assumptions - UDD, constant force, Balducci - link integer-age life-table values to non-integer ages.
Worked example
Let T be Weibull with λ=0.01 and γ=2 (years). Then H(t)=0.01t² and S(t)=exp(−0.01t²). The median solves 0.01t²=ln2, giving t=8.33 years. At t=5 the hazard is 2×0.01×5=0.1 per year, while the density is f(5)=h(5)S(5)=0.1e^{−0.25}=0.0779. Reporting the hazard as a probability is an error.
Traps
A hazard is a rate, not a probability, and may exceed 1. Check whether a question uses force of mortality μ or hazard λ. Weibull γ>1 is not constant. With censoring, do not divide deaths by the initial number at risk. State independence assumptions before using Kaplan–Meier or Cox.