Urn models and vaccine efficacy estimation

• Published in: Statistics in medicine 2000-03, Vol.19 (6), p.827-835
• Main Authors: Hernández-Suárez, Carlos M., Castillo-Chavez, Carlos
• Format: Article
• Language: English
• Subjects: Adolescent; Adult; Biological and medical sciences; Child; Computerized, statistical medical data processing and models in biomedicine; Disease Outbreaks; Humans; Measles - prevention & control; Measles - transmission; Measles Vaccine - administration & dosage; Medical sciences; Medical statistics; Middle Aged; Models, Statistical; Vaccination - statistics & numerical data
• ISSN: 0277-6715; 1097-0258
• DOI: 10.1002/(SICI)1097-0258(20000330)19:6<827::AID-SIM382>3.0.CO;2-B

We derive the distribution of the number of infections among unvaccinated and vaccinated individuals for model 1 (leaky) and model 2 (all/nothing) vaccines, assuming random mixing of a homogeneous population. For all/nothing vaccines, we show that the distribution of the number of infected vaccinated individuals conditioning on n observed infections follows a hypergeometric distribution, and the vaccine efficacy estimate (VE) can be derived from the usual estimate of the total population size in a capture–recapture sampling program. For leaky vaccines, we show that the number of vaccinated infected follows a distribution that was first derived by Wallenius. We found that the current point estimates of VE for each model perform very well, but the urn model construction presented here provides a strong framework for estimation and hypothesis testing on the parameters, and can be applied when the available data are a sample of the population. Since the method does not require an underlying transmission model, it can be applied to estimate the VE for non‐contagious diseases. Copyright © 2000 John Wiley & Sons, Ltd.