Survival Models and Life Table Estimation
This domain covers the modelling of time to an event when observation may be incomplete, and the construction of survival and mortality estimates from data.
Core functions
- S(t) = P(T > t); density f(t); hazard mu(t) = f(t)/S(t); cumulative hazard H(t) = integral of mu = -log S(t), so S(t) = exp(-H(t)).
- Right censoring is non-informative when the censoring time is independent of survival given covariates. Left truncation (late entry) arises when lives are only observed from the age at which they buy a policy.
Estimation
- Actuarial estimate: q-hat_x = d_x / E_x where E_x is the initial exposed to risk. Under uniform deaths in the year, E_x = l_x - d_x/2, and mu-hat_x = d_x / E-x where E-x is the central exposed to risk, with q-hat = 1 - exp(-mu-hat).
- Kaplan-Meier: S-hat(t) = product over event times t_i not exceeding t of (1 - d_i/n_i), where d_i is the number of deaths and n_i the risk set. Greenwood's formula gives its variance.
- Nelson-Aalen estimates the cumulative hazard by summing d_i/n_i.
- Cox proportional hazards: mu(t|x) = mu0(t) exp(beta'x), leaving the baseline unspecified and estimating beta by partial likelihood; the proportional hazards assumption must be checked.
Worked example
Five lives are observed: deaths at times 3, 5, 5 and 9, with one withdrawal at time 6. Risk sets are 5 at time 3, 4 at time 5 and 1 at time 9. So S-hat(3) = 1 - 1/5 = 0.8; S-hat(5) = 0.8 x (1 - 2/4) = 0.4; S-hat(9) = 0.4 x (1 - 1/1) = 0. The withdrawal does not itself change the estimate but removes a life from the risk set at time 9.
Traps that cost marks
- Treating censored lives as deaths, or as continuing at risk after withdrawal.
- Mixing initial and central exposed to risk in the same formula.
- Ignoring left truncation in insured-life data.