Definition
Event history analysis (a.k.a. survival, duration, or failure-time analysis in other disciplines) models the timing of transitions between states — the "hazard" of an event occurring in an interval given survival up to it — as a function of covariates and elapsed time. In its discrete-time form the timeline is split into intervals and each unit-at-risk-per-interval is treated as a separate observation, so the hazard is fitted by logistic (or complementary-log-log) regression. It is the workhorse of demography and event-based sociology, and extends to unobserved heterogeneity (frailty) — individual-specific random effects capturing risk propensity not explained by covariates.
Key Ideas
- Discrete-time hazard as logistic regression (Allison 1984; Holford 1976): with πit the probability that unit i experiences the event in interval t given survival to t, logit(πit)=β0+∑pβpxpit. Each "exposure unit" (e.g. a person-year) enters as an independent Bernoulli observation, so standard GLM machinery fits the model. This is the survival analogue of the Poisson-trick GLM representation of continuous-time hazards.
- Unobserved heterogeneity / frailty: add a unit-specific random effect ui to the linear predictor; ignoring it biases duration-dependence estimates (spurious negative duration dependence from selective attrition of high-risk units). The hierarchical model is typically estimated by MCMC (Lewis-Raftery 1999).
- The "clocks" and age–period–cohort (APC) identifiability: event-history data carry several time axes at once — age, duration in the current state (e.g. since previous birth), cohort, period (calendar time), and state-count "clocks" like parity. Because period = cohort + age, the classic APC collinearity makes the full set non-identified; event-history data compound it with the extra duration/parity clocks.
- Resolving identifiability by modeling + selection: a practical route is to code some clocks parametrically (e.g. via ACE / alternating conditional expectation transformations) and use Bayes factors (BIC-approximated for screening, exact for final models) to decide which clocks the data actually require (Lewis-Raftery 1999).
How It Works
Construct the person-period (episode-split) dataset: one record per unit per interval at risk, with an event indicator and the current values of all time-varying covariates and clocks. Fit the discrete-time hazard by logistic regression; add random effects for frailty and estimate by MCMC. Compare specifications (which clocks, which codings) by Bayes factors — the marginal likelihood of a frailty model can be obtained from MCMC output by the Compound Laplace-Metropolis estimator, extending the Laplace-Metropolis estimator to models with latent variables.
Why It Matters
- Demography and sociology: the standard framework for studying fertility, mortality, marriage, migration, and job transitions, where the timing and its covariate dependence are the objects of interest.
- Separates true from spurious duration dependence: explicit frailty modeling prevents heterogeneity from masquerading as declining hazard — the same heterogeneity-vs-state-dependence issue that arises in panel choice models.
- GLM-native: the discrete-time logistic representation lets survival questions be answered with familiar Bayesian-GLM tools and model comparison.
Open Questions
- Sensitivity of frailty estimates to the assumed random-effect distribution (parametric vs. nonparametric frailty).
- Residual APC non-identifiability after parametric coding — how much is driven by the coding choices themselves.
- Interval-censoring and time-aggregation effects in coarsely-recorded discrete-time data.
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