Lewis-Raftery (1999) Bayesian Analysis of Event History Models With Unobserved Heterogeneity via MCMC

Summary

Lewis and Raftery apply Bayesian discrete-time event-history analysis with unobserved heterogeneity (frailty) — fitted by MCMC and compared via Bayes factors — to explain the marital-fertility decline in Iran (World Fertility Survey data). The methodological interest is (i) handling the age–period–cohort (APC) identifiability problem when it is compounded by two extra "clocks" (duration since previous birth and mother's parity), and (ii) computing Bayes factors for model comparison in a large hierarchical model using the Compound Laplace-Metropolis estimator. Substantively they conclude Iran's fertility decline was a period effect (not cohort), began before the Family Planning Program, and was strongest in Tehran.

Key Claims

• Discrete-time event history via logistic regression. Each married woman-year is a separate "exposure-year" case; the hazard of a birth is $\text{logit}(\pi_{it})=\beta_0+\sum_p\beta_p x_{pit}$ (Allison 1984; Holford 1976), so discrete-time survival is fitted as logistic regression treating each exposure unit as an independent observation.
• Unobserved heterogeneity (frailty). A woman-specific random effect is added to capture heterogeneity in fertility propensity not explained by covariates; the resulting hierarchical model is estimated by MCMC (the random effects would otherwise make the likelihood intractable).
• Five "clocks" and compounded APC identifiability. The model carries up to five time axes — age, duration since previous birth, cohort, period, and parity — so the classic age–period–cohort collinearity is worsened. The fix: model some clocks (age, duration, parity) parametrically, using variable codings suggested by the ACE (alternating conditional expectation; Breiman-Friedman 1985) method when substantively plausible, then let Bayes factors decide which clocks the model actually needs.
• Model comparison: BIC then Compound Laplace-Metropolis. For fast screening, Bayes factors are approximated by BIC $=-\chi^2+P\log N$ (Kass-Raftery 1995). For the final hierarchical (frailty) models, exact Bayes factors are computed by the Compound Laplace-Metropolis estimator — an extension of Lewis-Raftery's (1997) Laplace-Metropolis marginal-likelihood estimator to models with random effects/latent variables.
• Findings. Including period variables alone beats including cohort variables → the decline was a period effect; the period effect is parsimoniously coded by primary-school participation; parity matters at all levels; the decline predates the Family Planning Program, is uniform across women's education but varies with husband's education, and is largest in Tehran.

Concepts Introduced or Extended

• Event History Analysis
• Parametric Survival Model

Entities Mentioned

• Steven M. Lewis
• Adrian E. Raftery

Quotes

"MCMC is used to assess competing explanations of marital fertility decline. Data collected during the World Fertility Study in Iran are analyzed using methods developed to perform discrete time event history analyses in which unobserved heterogeneity is explicitly accounted for."

"We resolved the identifiability problem by modeling some of the clocks (age, duration, parity) parametrically using codings suggested by the ACE method ... We then used Bayes factors to determine which clocks were needed in the model."

My Take

This is a nicely complete applied-Bayes case study: it takes a genuinely hard identification problem (age–period–cohort, already fragile, made worse by two extra clocks) and resolves it not by a clever prior but by substantive parametric coding plus Bayes-factor model choice — a reminder that identification in these models is as much a modeling decision as a statistical one. Methodologically its lasting hook for the wiki is the Compound Laplace-Metropolis extension: Lewis-Raftery (1997) gave a way to get marginal likelihoods from MCMC output for ordinary models, and this paper pushes that to random-effects/frailty models where the latent structure makes the marginal likelihood otherwise very awkward. The discrete-time-logistic representation of survival — each exposure-year an independent Bernoulli — is the same "trick" that connects survival analysis to GLMs (cf. the Aitkin-Clayton Poisson trick in parametric survival), and it is why event-history models slot so naturally into the Bayesian-GLM machinery elsewhere in the wiki.