Utilizing a constrained logistic model to estimate vaccination coverage

Population vaccination coverage is often estimated by simple point estimation or by survival analysis methods; the latter requires knowledge of age at the time of vaccination. For situations when such data cannot be determined, we explore two versions of a logistic model. The first is a logistic growth model; the second model reparameterizes the numerator of the first to naturally constrain parameter estimates. Due to computational challenges in estimation of constrained parameters, we explore three methods of estimation for each model (nonlinear least squares, maximum likelihood estimation, and Bayesian estimation). Simulation results show that all methods of estimation produce comparable estimates and model results, but also that all three are sensitive to data configurations at times yielding unstable estimates. We apply all three methods to the 2003 Kenya Demographic and Health Surveys to estimate the sample vaccination coverage of the combined diphtheria, pertussis, and tetanus vaccine series. While all three methods of estimation have attractive properties as well as limitations, we find Bayesian estimation appealing in its ability to restrict parameter estimates through prior distributions, its relative stability in comparison to maximum likelihood estimation and nonlinear least squares, and associated inference that does not rely on asymptotic distributions of parameter estimates.