Estimability and interpretation of vaccine efficacy using frailty mixing models

Am J Epidemiol. 1996 Jul 1;144(1):83-97. doi: 10.1093/oxfordjournals.aje.a008858.


The authors consider estimability and interpretation of vaccine efficacy based on time to event data, allowing that some of the population might have a very low probability of acquiring disease, and the rest have partial, possibly continuously distributed, susceptibility. The efficacy parameters of interest in the frailty mixing model include the fraction highly unlikely to acquire the infection or disease due to the vaccine, the degree of partial protection in those still susceptible, and the average protection or summary measure of efficacy under heterogeneity. The efficacy estimates can still be usefully interpreted when the heterogeneity results from heterogeneity in contact patterns, contact rates, or infectiousness of the contacts, as long as these are equal in the vaccinated and unvaccinated groups. A likelihood-based method allows estimation of the efficacy parameters of interest from grouped time to event data. Simulated vaccine studies assuming different levels and distributions of efficacy demonstrate that ignoring heterogeneity in susceptibility or exposure to infection generally results in underestimation of vaccine efficacy as well as incorrect interpretation of the estimates. The approach is also applicable to other covariates affecting susceptibility or exposure to infection in infectious diseases. Exploitation of the dependent happening structure of infectious diseases to obtain a shape for the baseline hazard may help identifiability. The authors recommend fitting several models to time to event data in vaccine studies.

Publication types

  • Research Support, Non-U.S. Gov't
  • Research Support, U.S. Gov't, P.H.S.

MeSH terms

  • Effect Modifier, Epidemiologic
  • Humans
  • Infections / epidemiology*
  • Likelihood Functions
  • Proportional Hazards Models*
  • Randomized Controlled Trials as Topic
  • Reproducibility of Results
  • Risk Factors
  • Survival Analysis*
  • Time Factors
  • Vaccines / standards*


  • Vaccines