Survival Analysis: Hazard Functions and Censoring for MAS-I

Survival and Hazard Functions

Survival analysis models time-to-event data, where the "event" might be death, claim occurrence, policy lapse, or any transition of interest. The survival function S(t) = P(T > t) gives the probability that the event has not yet occurred by time t. It starts at S(0) = 1 and decreases toward zero. The density function f(t) = -S'(t) describes the rate at which events occur at time t.

The hazard function h(t) = f(t)/S(t) = -S'(t)/S(t) gives the instantaneous rate of the event occurring at time t, given survival to time t. Intuitively, h(t)*dt approximates the probability of the event occurring in the small interval (t, t+dt) given no event before t. The cumulative hazard function H(t) = integral from 0 to t of h(u) du = -ln(S(t)) accumulates the total hazard over time. The key relationship S(t) = exp(-H(t)) connects all these quantities.

Common Parametric Models

The exponential distribution has a constant hazard h(t) = lambda, giving S(t) = exp(-lambda*t). This "memoryless" property means the probability of surviving an additional time period is the same regardless of how long you have already survived. While simple, the constant hazard assumption is often unrealistic for insurance applications.

The Weibull distribution has hazard h(t) = (gamma/theta) * (t/theta)^{gamma-1}. When gamma > 1, the hazard increases with time (aging effect). When gamma < 1, it decreases (early failures or "burn-in"). When gamma = 1, the Weibull reduces to the exponential. The log-logistic distribution allows a non-monotone hazard that first increases and then decreases, which can model the risk pattern of certain diseases or product failures.

Right Censoring

In practice, survival data is often incomplete. Right censoring occurs when observation ends before the event is observed. This happens when a study ends before all subjects have experienced the event, when subjects withdraw from a study, or when policyholders cancel coverage before a claim occurs. A censored observation tells you the event time exceeds the censoring time, but not the actual event time.

Standard methods (like just averaging the observed times) would underestimate the true survival time because they treat censored observations as if the event occurred at the censoring time. Proper survival analysis methods, such as the Kaplan-Meier estimator, correctly handle censored observations by removing them from the risk set at the censoring time rather than treating them as events.

The Kaplan-Meier Estimator

The Kaplan-Meier (KM) product-limit estimator computes the survival function from data that may include censored observations. At each observed event time t_i, the estimated conditional probability of surviving past t_i is (n_i - d_i)/n_i, where n_i is the number at risk (still under observation and event-free) and d_i is the number of events at time t_i. The survival function estimate is S_hat(t) = product over all t_i <= t of (n_i - d_i)/n_i.

For example, consider 10 subjects with event times at 3, 5, 8, and 12 months, and censoring at 4, 6, 9, 10, 11, and 15 months. At t = 3: n = 10, d = 1, so the survival factor is 9/10 = 0.90. At t = 5: the subject censored at 4 has left, so n = 8, d = 1, factor = 7/8. The KM estimate at t = 5 is 0.90 * 7/8 = 0.7875. At t = 8: subjects censored at 6 have left, so n = 6, d = 1, factor = 5/6. The KM estimate at t = 8 is 0.7875 * 5/6 = 0.6563.

The Nelson-Aalen Estimator and Log-Rank Test

The Nelson-Aalen estimator provides an alternative estimate of the cumulative hazard function: H_hat(t) = sum over all t_i <= t of d_i/n_i. The corresponding survival estimate is S_hat(t) = exp(-H_hat(t)). For large samples, this gives similar results to the Kaplan-Meier estimator, but for small samples it tends to give slightly higher survival estimates.

The log-rank test compares survival functions between two or more groups. At each event time, it compares the observed number of events in each group to the expected number under the null hypothesis that the groups have identical survival. The test statistic is (sum of (observed - expected))^2 / (sum of variance terms), which follows approximately a chi-squared distribution with k-1 degrees of freedom for k groups. The log-rank test is most powerful when the hazards are proportional (one group's hazard is a constant multiple of the other's).

Cox Proportional Hazards Model

The Cox proportional hazards model specifies h(t|x) = h_0(t) * exp(beta_1*x_1 + beta_2*x_2 + ... + beta_p*x_p), where h_0(t) is an unspecified baseline hazard and the covariates x act multiplicatively on the hazard. This model is semi-parametric: it does not assume a specific form for h_0(t) but assumes the covariate effects are log-linear and constant over time.

The coefficient beta_j represents the log hazard ratio for a one-unit increase in x_j, holding other covariates constant. If beta_j = 0.3, the hazard ratio is exp(0.3) = 1.35, meaning the event rate is 35% higher for each unit increase in x_j. The Cox model is estimated using partial likelihood, which eliminates the baseline hazard and focuses on the covariate effects.

Common Pitfalls

A frequent mistake in Kaplan-Meier calculations is miscounting the risk set. Censored observations leave the risk set after their censoring time, not before. If a censoring time and an event time coincide, the convention on Exam MAS-I typically specifies the ordering; read the problem carefully. Another error is attempting to interpret the KM estimate beyond the largest observation; if the largest observation is censored, the survival function is undefined beyond that point.

With the Cox model, students sometimes interpret beta directly as the hazard ratio. Remember that exp(beta) is the hazard ratio; beta itself is the log hazard ratio. Also, the proportional hazards assumption (that hazard ratios are constant over time) must be checked. If this assumption is violated, the Cox model may give misleading results.

Study Tips

Exam MAS-I tests your ability to construct Kaplan-Meier estimates from data with mixed censored and uncensored observations, perform log-rank tests to compare survival curves, and interpret Cox regression output. Practice building KM tables step by step, carefully tracking the risk set and the number of events at each time point. Know the relationship between survival, hazard, and cumulative hazard functions, and be able to convert between them. Understand the semi-parametric nature of the Cox model and the interpretation of hazard ratios.